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Low-temperature relaxation of various samarium phosphate glasses

  • Mohamed El-Sayed Gaafar EMAIL logo and Samir Yousef Marzouk
Published/Copyright: November 21, 2023
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Journal of the Mechanical Behavior of Materials
From the journal Journal of the Mechanical Behavior of Materials Volume 32 Issue 1

Abstract

Glasses constructed, (1 − x) (0.6595P2O5–0.0958ZnO–0.2447PbO) · xSm2O3 with x = 0.00, 0.0045, 0.0089, 0.0132, and 0.0261 mol%, had been created to investigate the attenuation of longitudinal ultrasonic waves at 2, 4, 6, and 14 MHz frequencies between 120 and 300 K. At a variety of temperatures, clear peaks of a large absorption curve have been seen. These peaks are dependent on the structure of the glass as well as the switching frequency. Maximum peaks have been shown to shift to higher temperatures, and the increase in overall frequency points to the presence of some kind of relaxation process. A thermally induced relaxation process is responsible for producing a calm approach, which has been identified as a result of this mechanism. A quiet approach has been defined as a consequence of a thermally triggered relaxation mechanism. The variance of the mean energy of activation of the mechanism counts on primarily the amount of Sm2O3 mol%. Such dependency has been evaluated based on the loss of normal linear solid form, attaining low dispersion, and a large allocation of Arrhenius kind relaxation through temperature-autonomous relaxation power. The measured acoustical energy of activation values have been quantifiably represented based on the number of loss centers (amount of oxygen atoms that now move at a double-well potential).

Keywords: samarium phosphate glasses; relaxation process; attenuation coefficient; relaxation strength

1 Introduction

In recent years, there has been a lot of interest in phosphate glasses that have been doped with rare-earth (RE) and transition metal (TM) ions. This is due to the fact that there is a chance that these glasses might be used in optical, photonic, and magneto-optical systems [1,2]. Because phosphate glasses often have a lower melting point and dissolve more easily in aqueous solutions such as those that are present in the fluid milieu of the body, they are well suited for use in applications that are concerned with biomedicine. The low physical and chemical stability of the material prevents it from being used for several applications, such as photonic devices and the disposal of nuclear waste. It has been shown that the chemical resistance of these glass structures may be greatly increased by the introduction of other oxides, such as alumina and iron oxide [3–5]. These glasses have been used as the medium for lasers, optical amplifiers, and radiation-resistant optical windows [6]. The dissolution of a significant number of RE and TM elements is enabled by phosphate and aluminophosphate glass matrices, which also permit the dissolution of aluminophosphate glass matrices. The optical properties of these glasses, such as the shape of the luminescence spectra, the intensity of the fluorescence, and the longevity of the fluorescence, are influenced by the environment and the interaction of the RE ions that are present [7]. As a consequence of this, it is essential for comprehending the optical and fluorescence characteristics of these glasses to have a firm grasp of the local environment in which the RE ions are located as well as their interactions with the glass matrix. This is because it is the local environment that determines how the RE ions interact with the glass matrix. Aluminophosphate and iron phosphate glasses were shown to be beneficial in the disposal of nuclear waste [8]. This was due to comparatively low melting temperatures, increased RE and TM solubility, and better chemical durability of glasses. It was discovered that these glasses are also used in the field of optical and photonics technologies. Glasses made of alumina and iron phosphate are also the subject of investigation and are used in the vitrification of radioactive waste [9–12]. It is already well known that the vitrification of radioactive waste may be accomplished using borosilicate glasses.

Low-temperature features of disordered materials form an intriguing thermodynamic combination of specific heat, thermal conductivity, dielectric, and mechanic susceptibilities [13–16]. Despite quantitative disagreement with certain studies, the conventional tunneling model [17,18] appears to account for all of these qualities in a material-independent, universal manner, demonstrating its 50-year-long success. The unexpected crossover of the sound velocities of metallic glasses recorded in the superconducting or normal states of the same sample [19,20] is a particularly striking example.

A sequence of (x/2) Bi2O3–(x/2)Nb2O5–(100–x)Na2B4O7, 5 ≤ x ≤ 30 mol% glasses were mass-synthesized by Mahmoud et al. [21], and their acoustic absorption (α) graphs were examined at various ultrasonic frequency values (f), 2–10 MHz, and at several temperatures (T) extending around 140–300 K. So many aspects of the α–T interaction, such as the average activation energy, depended on the formulation as well as the frequency. The α–T interaction indicated a thermally enabled relaxation of the actions of oxygen atoms. The relaxation is caused by the transition of energy from ultrasonic waves to a double-well potential wherein oxygen atoms pass. This was examined using the concept of the central force of the dual-well structures. Model parameters such as mutual potential energy, loss-centers of the dual-well structure, relaxation intensity, and deformation potential have been influenced by variations in glass formulation.

In this study, we analyzed the variability of ultrasonic attenuation in glass configuration (1–x) (0.6595P2O5–0.0958ZnO–0.2447PbO) · xSm2O3 with x = 0.00, 0.0045, 0.0089, 0.0132, and 0.0261 mol%, throughout the temperature domain 120–300 K as well as at four various ultrasonic frequencies. Alterations in the formulation in the study of the glass of ultrasonic relaxing will be beneficial to shed some insight on the strength and link these kinds of glass frameworks by replacing Sm2O3 with other constituents.

2 Details of the experiment

2.1 Preparation of glasses

To prepare (1–x) (0.6595P2O5–0.0958ZnO–0.2447PbO) · xSm2O3, x = 0.00, 0.0045, 0.0089, 0.0132, and 0.0261 mol%, a rapid quenching process was used as listed in Table 1. The oxides P2O5, ZnO, PbO, and Sm2O3 with more than 99.9% purity were used in the manufacture of this glass framework. A balance of four digits (HR-200) was used to weigh the powder of different oxides according to each batch. Each batch was grounded in a zirconia grinder for 2 h to obtain a homogenous oxide powder blend.

Table 1

Variation of density (ρ), mean atomic volume (V a), longitudinal ultrasonic wave velocity (Ul), experimental bulk modulus (K e), frequency (F), longitudinal ultrasonic attenuation coefficient (α), peak temperature (T p), attempted frequency (f o), activation energy (E p), theoretically calculated activation energy (E p(th1) according to Eq. (3), deformation potential (D), and relaxation strength (A) for glass system (0.6595–x)P2O5–(0.0958–x)ZnO–(0.2447–x)PbO–xSm2O3

Glass composition (mol%) ρ V m U l K e F α T p f o × 1010 E p E p(th1) D A
[g/cm3] ×10−6 [m3/kg. mol] [m/s] [GPa] MHz [dB/cm] [K] s−1 [eV] [eV] [eV]
Sm0 3,907 39.94 5,052 59.7 2 4.348 186 1.476 0.138 0.137 0.4 0.492
4 4.494 189 0.257
6 5.036 214 0.19
14 6.626 224 0.118
Sm1 3,930 39.92 5,079 60.6 2 4.044 182 1.469 0.135 0.133 0.394 0.46
4 4.08 185 0.235
6 4.673 211 0.177
14 6.033 217 0.108
Sm2 3,953 39.9 5,094 61.1 2 3.698 181 1.233 0.131 0.132 0.386 0.423
4 3.761 183 0.218
6 4.281 211 0.163
14 5.485 213 0.099
Sm3 3,978 39.87 5,121 62 2 3.474 179 1.105 0.127 0.129 0.379 0.4
4 3.716 180 0.216
6 4.046 208 0.155
14 5.114 211 0.093
Sm4 4,048 39.79 5,209 64.8 2 3.122 177 0.889 0.121 0.12 0.367 0.366
4 3.297 175 0.195
6 3.58 202 0.14
14 4.53 210 0.083

The mixtures are heated in an alumina melting pot at 743 K for 2 h to eliminate any moisture in the oxide powder, then melted in an electric furnace at 1,373–1,523 K for 1 h. The molten glass was stirred thoroughly by hand for about 1 min using a holder until it was a liquidly bubble-free transparent solution, then returned to the furnace for an additional 30 min to guarantee that the glass melt reached a high fluidity. After pouring the melt into a preheated stainless steel square mold at 750 K, it was annealed by keeping the melt at the glass transition temperature of +10 K (for each composition) inside the muffle furnace for 3 h to eliminate any thermal strains, then followed by slow cooling to room temperature inside the muffle furnace. For the physical measurements, the glass samples were polished and tagged. One sample of each composition was prepared, and repeated measurements on each sample were taken to keep the reproducibility.

2.2 Density

The Archimedes approach was used to determine the sample density (ρ) using ethyl alcohol (789 kg/m3) as the immersion liquid.

2.3 Ultrasonic attenuation measurements

A cryostat system uses air in liquid form as cooling to position the temperature of the sample among the liquid air temperature and the room temperature. The glass specimen and the transducer that is bonded (nonak – stopcock grease that appeared to be a reasonable coupling) became positioned in the required holder and put within the cooled container. In order to determine the temperature of the sample, a thermocouple was put in close connection with the specimen.

Ultrasonic attenuation tests were conducted using the USIP 20 ultrasonic flaw detection device (Krautkramer – Germany). The approach used during the experiment is the pulse-echo system, in which only the transducer operated concurrently as transmitter and receiver. The tests were performed at temperatures between 120 and 300 K as well as at four ultrasound energy frequencies, comprising 2, 4, 6, and 14 MHz. Elevations of two consecutive echoes have been recorded, and the attenuation coefficient was determined using the following formula:

(1) α = 20 2 d log A 1 A 2 ,

where d is the thickness of the specimen, and the levels of first and second echoes, respectively; A 1 and A 2 represent the intensities of such two echoes.

3 Results and discussion

3.1 Density and molar volume

One of the most essential gadgets for studying the framework of glass systems is the glass density (ρ). Geometrical alterations, coordination number, glass network interstitial changes, and structural hardening or softening all have an impact. As shown in Figure 1, the density of the glass system (1–x) (0.6595P2O5–0.0958ZnO–0.2447PbO) · xSm2O3 enhanced from 3,907 to 4,048 kg/m3, with the amount of Sm2O3 (atomic weight of 348.718 kg/mol) increasing from 0 to 0.0261 mol% at the cost of P2O5 (141.945 kg/mol), ZnO (81.389 kg/mol), and PbO (223.199 kg/mol). However, when the Sm level rose, the molar volume (V m) diminished from 39.94 to 39.79 m3/mol, which was lower than the molar volume of pure P2O5 glass framework (56.33 m3/mol). Sm ions are replacing P, Zn, and Pb ions and creating Sm–O–P links, as evidenced by the decrease in V m and enhancement in density values. In other words, Sm ions filled the gaps in the P2O3 network.

Figure 1 Variation of density and molar volume with Sm2O3 concentrations.
Figure 1

Variation of density and molar volume with Sm2O3 concentrations.

According to Mahmoud et al. [22], vibration bands arise in the 400–600 cm−1 range in the fourier transform infra red (FTIR) spectra of (100−x). (65.95P2O5–24.47PbO–9.58ZnO)−xNd2O3, where x = 0, 0.46, 0.92, 1.38, and 2.71 mol%), as a result of stretching vibration’s contribution to the symmetry of PbO4 and ZnO4 spectral units. Therefore, the structure of the probe glasses changes significantly as a result of the substitution of P2O5, PbO, and ZnO for Nd2O3. Also, the absorption band at 1,100 cm−1 is thought to correspond to Nd–O–P bonds. With increased Nd2O3 and mixed oxide concentration, the relative amount of these bonds increases at the expense of Zn–O–P and Pb–O–P bonds, resulting in an increase in the strength of the 1,100 cm−1 absorption band. As a result, the P–O–Nd bonds can participate in the formation of the pyrophosphate glass by simply swapping oxygen at the Q2 terminal position. Also, Gaafar et al. [23] reported that the amplitude of the band near 1,230 cm−1 in the FTIR spectra of (1–x) (0.6595P2O5–0.0958ZnO–0.2447PbO) · xSm2O3 with x = 0.00, 0.0045, 0.0089, 0.0132, and 0.0261 mol% decreases with increasing Sm2O3 content. This change reveals that the phosphate chains are shortened as the mixed oxides are incorporated into the glass structure, thus leading to a decrease in the relative content of the Q2 species. Such results explained the increase in density and the decrease in molar volume values in the present glass system.

3.2 Thermal expansion coefficient

The values of the glass transition temperature, abbreviated as Tg, together with the mol% consistent with Sm2O3, are shown in Figure 2. The temperature at which the glass transition occurs (Tg) enhanced from 608 to 688 K when the amount of Sm2O3 in the sample increased from 0 to 0.0261 mol%. The FTIR spectra (1–x) (0.6595P2O5–0.0958ZnO–0.2447PbO) · xSm2O3 of the glass system were examined by Gaafar et al. [23], who revealed that the presence of Sm2O3 inside the framework of glass from 0 to 0.0261 mol% resulted in the formation of bridging oxygen molecules, a shift to lower wave numbers. It is possible that this has anything to do with the Tg values getting increased. According to Makishima and Mackenzie [24], it was discovered that the calculated values of the thermal expansion coefficient of the glasses that were the subject of the study diminished from 133.9 to 75.9 1/°C with the rise in Sm2O3 consistent from 0 to 0.0261 mol%, as portrayed in Figure 2. The thermal expansion coefficient of materials is dependent upon the tensile strength of the bonds between the components, according to Srivastava and Srinivasan [25]. As a result, the enhancement in Tg values might be explained by the value of the thermal expansion coefficient, which has likewise dropped over time. In addition, as can be seen in Figure 2, the enhancement in Tg values is supported by the fact that the values of stretching force constant F (obtained by the application of the bond compression model [26]) and averaged bond dissociation energy per unit volume Gi have also improved (estimated using Makishima and Mackenzie [24]), as can be seen in Figure 8.

Figure 2 Plot of the glass transition temperature (Tg) and thermal expansion coefficient (α) of the investigated glass system with Sm2O3 mol% content.
Figure 2

Plot of the glass transition temperature (Tg) and thermal expansion coefficient (α) of the investigated glass system with Sm2O3 mol% content.

3.3 Ultrasonic studies

V l and K exp analyses with Sm2O3 concentrations are included in Table 1. Such as in Table 1 and Figure 3, the velocity of the longitudinal wave results was enhanced in conjunction with the inclusion of Sm2O3 mol% material at the cost of P2O5, ZnO, and Pb, replacing them by the formation of shorter Sm–O–P bonds instead of longer Zn–O–P and Pb–O–P bonds. Although the variance of the rise in bulk modulus (K exp) with the rise of Sm2O3 mol% concentration is small [23,27,28,29].

Figure 3 Variation of both (longitudinal and shear) ultrasonic wave velocities with Sm2O3 concentrations.
Figure 3

Variation of both (longitudinal and shear) ultrasonic wave velocities with Sm2O3 concentrations.

The dependency of the coefficient of ultrasound attenuation within different working frequencies 2, 4, 6, and 14 MHz is shown in Figures 4 and 5 of the glass formulations 0.6595P2O5–0.0958ZnO–0.2447PbO–0Sm2O3 and 0.6508P2O5–0.0946ZnO–0.2414PbO–0.0132Sm2O3. The other glass configurations displayed the same behavior as all those seen in Figures 4 and 5. Clearly, specified large crests appear at temperatures starting from 120 to 300 K with a shift in Sm2O3 material. This result reveals that, with rising operational frequency, the maximum shifts to higher temperatures and the intensity of this peak increases significantly.

Figure 4 Ultrasonic attenuation coefficient curves of glass composition 0.6595P2O5–0.0958ZnO–0.2447PbO–0Sm2O3.
Figure 4

Ultrasonic attenuation coefficient curves of glass composition 0.6595P2O5–0.0958ZnO–0.2447PbO–0Sm2O3.

Figure 5 Ultrasonic attenuation coefficient curves of glass composition 0.6508P2O5–0.0946ZnO–0.2414PbO–0.0132Sm2O3.
Figure 5

Ultrasonic attenuation coefficient curves of glass composition 0.6508P2O5–0.0946ZnO–0.2414PbO–0.0132Sm2O3.

In contrast, the figures explicitly indicate that the temperature is at its highest point, which moves to a low temperature with a rise in Sm2O3 material. In addition, the tails of the loss graphs coincide with each other, which is primarily in consequence of the thermal extension as the working frequency.

The reciprocal portraits within the operating frequencies and the temperature crests are included in Figure 6, with all the formulations. All these plots display lines that are perfectly straight, showing that the peaks matched the following equation for a given glass composition:

(2) f = f o exp − E p K T p ,

Figure 6 Plot of the logarithm of operating frequency and inverse of temperature peak for the investigated glass system.
Figure 6

Plot of the logarithm of operating frequency and inverse of temperature peak for the investigated glass system.

where f o and E p are the vibrational classical frequency (attempted frequency) and energy of activation, respectively. They became collected from both the line intercept and the slopes of the lines. Figures 4 and 5, along with Figure 6, imply which certain form of relaxing mechanism is operating. The effects of ultrasound attenuation, maximal temperature, attempted classical frequency, and energy of activation of the relaxation mechanism are shown in Table 1. E p quantities diminished within the raise of Sm2O3 consistently, as seen in Figure 7. This drop in the energy of activation suggests that the framework became close and the bond lengths will be lower with the inclusion of Sm2O3 because the average dissociation of bond energy enhances, as seen in Figure 8.

Figure 7 Behavior of the activation energy of the investigated glass system.
Figure 7

Behavior of the activation energy of the investigated glass system.

Figure 8 Behaviors of stretching force constant (F) and dissociation energy (Gi) of the investigated glass system with Sm2O3 mol% content.
Figure 8

Behaviors of stretching force constant (F) and dissociation energy (Gi) of the investigated glass system with Sm2O3 mol% content.

Bridge and Patel [30] confirmed that the maximal temperature dependency of acoustic loss in glasses made of inorganic oxides in the interval of 4 to 300 K was ascribed to the loss processes of the loss of normal linear solid form, attaining low dispersion and Arrhenius-kind relaxing times. Ultrasonic absorption was caused by the effects of atoms passing in potentials with two wells based on two balance arrangements (Figure 9). The measured results of the energy of activation and the attempted frequency indicate wells. In cases of absorption loss maxima, the atoms can vibrate in potentials of two wells attempting to surpass the altitude of the barrier as ultrasound waves (with certain resonance frequencies) are tried to apply and the temperature increases. Implementing high-energy acoustic waves allows a greater amount of atoms to oscillate at potentials of the double well with higher temperatures (as absorption loss levels depend on frequency and temperature). As a result, absorption loss maximal altitudes are enhanced with rising frequency.

Figure 9 Double-well potential.
Figure 9

Double-well potential.

Therefore, the cause for the absorption loss maxima found in the present study has been proposed as a consequence of the thermally stimulated particles, and the relaxation mechanisms would also be attributable to particles traveling in asymmetrical double-well atomic dimensional potentials. This motion of a particle would be defined as a vibration around one of the two-well minima’s potential. The penetration of acoustic waves during the substance disrupts the balance and produces a proportional change in energy among the two minima of the double wells by the sum of the atmospheric ΔE = Dε, where D is the potential for deformation, which indicates the energy transfer of the relaxing conditions in the strain field of the unit strength and (ε) the extent of the strain field. The particles relax to the balance configuration by surpassing the gap among both two wells through the thermally enabled operation. Time of relaxation counts on the temperature and the altitude of the barrier (energy of activation). These trends of absorption loss highs have been observed: Bi2O3–Nb2O5 Na2B4O7 [21], MoO3–P2O5 [30] and Li2O–B2O3–WO3 [31].

As per Bridge and Patel [30], El-Falaky et al. [32], and Abd El-Moneim [33], the composition dependency of energy of activation (E p) has been stated with experiential bulk modulus, mean stretching strength constant (F), and mean diameter of atomic rings (l). Trying to apply regression line among E p and F(F/K)0.576, thus, results in an equation:

(3) E p ( th ) = 2.74 × 10 − 4 F ( F / K ) 0.567 − 0.0804 .

A value of correlation is 96.6%.

The amount of loss centers per oxygen atom could be computed using the exponentially decaying process, defined by the following relationship, based on the correlation between the experimental results of specified amounts of loss centers every atom of oxygen and the average bond force constant versus bulk modulus:

(4) N ( th ) = 2.11 × 1 0 − 3 × F ( F / K ) − 2.67697 .

A value of correlation is 96.5%.

The experiential bulk modulus and mean constant of stretching force values were pointed out earlier in the study by Mahmoud et al. [23]. The quantities of the mathematically obtained mean energy of activation and the amount of loss centers for oxygen atoms (E p(th) & N (th)) also are provided in Tables 1 and 2.

The potential of deformation is provided in the form of Sidkey and Gaafar [34]:

(5) D = 1.5 E p 2 / 3 .

The deformational potential (D) values of the glass formulations studied are shown in Table 1. It has been shown that the potential for deformational drops with the enhancement of Sm2O3 mol% material.

Therefore, ultrasound waves have originated in thermal imbalance, and the mechanism of relaxation will regain the balance. In a thermally enabled phase, the particle can pass the barrier among the double wells. Besides, the range of the high temperatures suggests the following relationship: τ = τ o exp ( E p / K T p ) , which is not suited for classification by a single relaxation mechanism. In addition, the energies of activation, E p, collected throughout this analysis (Table 1) distributed values concerning the fixed range of the crystalline (1–x) (0.6595P2O5–0.0958ZnO–0.2447PbO) · xSm2O3 formulations, i.e., that kind of mean among a large allocation of energies of activation. Hence, the loss highs must be defined based on the allocation of relaxational times, among every particle that relaxes passing through a potential with the double well [35]. So, within a series of (n) particles in unit volume passing at equal potentials of double-well barrier altitude (Figure 9), expressing the internal friction using the following form:

Q − 1 = 2 α V ω ,

(6)   = n D 2 ρ V 2 d d Δ 1 1 + e ∆ / KT ω τ 1 + ω 2 τ 2 ,

= D 2 ρ V 2 ∫ 0 ∞ ∫ 0 ∞ d d Δ 1 1 + e Δ / KT × ω τ 1 + ω 2 τ 2 n ( Δ ) n ( E ) d Δ d E ,

where τ = τ o e E / KT ( 1 + e − ∆ / KT ) , at which Boltzmann’s constant (K), the absolute temperature (T), the ultrasound coefficient of attenuation in neper per unit length (α), the angular frequency is (ω), the phase velocity (V), the time of relaxation (τ), the particle’s attempted frequency in either well ( τ 0 − 1 ), the free-energy variance among congruent particle states within the two wells (Δ), i.e., the detachment of the well minima’s, the amount of loss centers (n), and the deformational potential (D). Therefore, with the asymmetrical allocation, where Δ ≥ 2KT and n (Δ) = n o is used, Eq. (6) could be updated via a constant independently among both Δ and E in a simplified way:

(7) Q − 1 = 2 n o D 2 ρ V 2 ∫ 0 ∞ ω τ n ( E ) d E 1 + ω 2 τ 2 ,

where n(E)dE is now the sum of two-well structures (amount of loss centers) within a barrier altitude throughout the scale of E to (E + dE) that is written in terms of vibrating particles.

Since phosphate glass is considered to be a three-dimensional matrix of cation–oxygen–cation (A = cation, O = anion) links, there is now an allocation of the thermally mean cation–anion–cation distance of the equilibrium level as well as the related distributions of (A–O–A) angles. In addition, there would still a range of A–A locations (bond spacings), including values either greater or lesser than the ideal (crystalline) amount. The oxygen density is thus related to the average number of well systems per unit volume [30].

Considering the spread of dual-well structures of vibrating particles arising from the expansion of anion–cation space within a spread of barrier altitudes that are relative to the bond’s energy. All the longitudinal and transverse vibrations of oxygen atoms are the sprightly vibratory ions (in the same order that phosphate, zinc, and lead atoms have much greater clusters). So, with each of the dual-well A–O–A configurations, more oxygen atoms positioned to the opposite side of cation A would be positioned at slightly different locations at varying distances from the A. This also implies that the heterogeneity of P–O–P, P–O–Zn, P–O–Pb, P–O–Sm, Zn–O–Pb, Pb–O–Pb, and Sm–O–Sm bond angles implies that there is a variation of the size of the atomic ring in the glass structure under scrutiny. That spread of asymmetric forms Δ would occur as the second-order consequence. At the maximal temperature of T p, the oxygen atoms composed of a dual-well structure can change the configuration by thermal activation energy, when jumping through the barrier.

Thus, the overall number of dual-well potential structures (loss centers, i.e., the number of oxygen atoms receiving ultrasonic waves) per unit volume (n) is calculated by:

(8) n = ∫ 0 ∞ n ( E ) d E ,

(9) = ρ V 2 2 n o D 2 ∫ 0 ∞ c ( E ) d E ,

where the integrated part ∫ 0 ∞ c ( E ) d E would be the area under the curve pertaining to α and T in Figures 4 and 5 of them.

Gilroy and Phillips [36] had believed that the asymmetrical dual-well potential n(E) is taking shape:

(10) n ( E ) = ( 1 / E p ) exp ( − E / E p ) .

Presuming that if n o = 1 / E p , the activation energy (E p), Eq. (9) will come in the shape:

(11) = ρ V 2 E p 2 D 2 ∫ 0 ∞ c ( E ) d E .

As stated by Bridge and Patel [30], it is difficult to explain this assertion in the context of n(E), and it will have the benefit that n is based on the concepts of the experiential factors E p and ∫ 0 ∞ c ( E ) d E within the ultrasonic loss mechanism individually. The goal of selecting a particular shape with n(E) would be to create a way to address the structural dependency of such dual-well structures and no longer.

For this reason, the overall amount of two-well structures per unit of volume would be determined for such absorption loss highs found in all the glass compositions analyzed at the adjusted 2 MHz ultrasonic frequency, which were chosen like a model (to address the assembly dependence of the quantity of dual-well systems per unit volume) and portrayed in Table 2 and Figure 10.

Table 2

Representation of the number of loss centers per unit volume (n), oxygen density [O], number of loss centers per oxygen atom (N), theoretically estimated number of loss centers (Nth) according to Eq. (4), average bond length (R), longitudinal modulus (q), percentage elongations (e), percentage contractions (c), theoretically determined deformation potential (Dth), and mutual potential energy (U o)

Glass composition mol% n × 1027 [m−3] [O] × 1028 [m3] N% N th R [nm] q [GPa] e% c% D th [eV] U o [eV]
Sm0 3.909 5.487 0.712 0.669 0.1847 99.7 64.4 −4.865 0.4 5.197
Sm1 3.245 5.485 0.592 0.579 0.185 101.4 62.4 −4.81 0.393 5.2
Sm2 2.758 5.483 0.503 0.559 0.1853 102.6 62.1 −4.741 0.386 5.203
Sm3 2.524 5.484 0.46 0.484 0.1855 104.3 60.4 −4.685 0.379 5.205
Sm7 1.675 5.482 0.306 0.281 0.1863 109.8 57.2 −4.58 0.368 5.214
Figure 10 Plot of the number of loss centers per unit volume (n).
Figure 10

Plot of the number of loss centers per unit volume (n).

The density of oxygen atoms [O] has also been determined for the glassy structures undergoing examination in the form of an equation [36]:

(12) [ O ] = C N A 16 G ,

where C is now the sum amount of oxygen in 100 g of glass, G becomes a volume of 100 g of glass, and N A is now the quantity of Avogadro. Table 2 displays the results of the number of loss centers per unit volume, the number of loss centers per oxygen atom N, and the oxygen density. Relaxation strength A signifies the strength of the internal friction maximum, which, in addition, is a measurement of the quantity of relaxing units present throughout the glass framework and the magnitude of unrelaxed strain added along with every unit. The relaxing strength A with the varying glass set inspected was determined based on the information provided by Carini et al. [35]:

(13) A = 2 α V π f ,

where α is the coefficient of ultrasound attenuation in dB/cm (maximal relaxing loss at T p), V l is the longitudinal ultrasound wave velocity, and f is the operational frequency. The strength of relaxation results is displayed in Table 1, which indicates the concentration of the defects. It indicates that perhaps the relaxation strength diminishes with the increase in the inputs of the Sm2O3 modification at the same operational frequency. In comparison, the relaxation strength diminishes with the decrease in operational frequency with the same content adjustment.

Figure 11 indicates the alteration in the number of loss centers per oxygen atom N with Sm2O3. It is apparent from such a figure that perhaps the increase in Sm2O3-modifier input has contributed to a diminishment for all: the quantity of two-well structures per oxygen atom (quantity of oscillating atoms), the strength of relaxation, as well as the deformational potential attributable to the substitution of P2O5 (with a coordination number near 2), ZnO (with a coordination number around 4), and PbO (with a coordination numbers around 4) by Sm2O3 (with a higher coordination number). As a result, the activational energy of the relaxation mechanism is diminished by increasing the density of cross-linking, which could be described by taking two variables into account. First, the rise throughout the allocation of samarium ions here between P–O sequences as a system modification contributes to an improvement in the density of cross-linking of the glass topology. Second, there is now the enhancement in the proportional strength of the bonds [37,38], which implies a drop in the number of two-well structures for each atom of oxygen (the number of oscillating atoms).

Figure 11 Plot of the loss centers per oxygen atom (N).
Figure 11

Plot of the loss centers per oxygen atom (N).

3.4 Systematic analysis of longitudinal and shear oscillations

The model of the central force was suggested as per Bridge and Patel [32]; the potential for deformation was conceptually meant to be determined using the following formula:

(14) D th = q 2 M n b N A ρ δ y δ x ,

where the symbol ρ denotes the density, the symbol n b denotes the amount of the number of anion–oxygen–anion components per formula unit (around 2 for P2O5, 4 for ZnO, 4 for PbO, and 8 for Sm2O3), δx is the bond space multiplied by two, and δy is the spacing of minima’s throughout the two-well potential, which can be seen in Figure 9. K and G are the bulk and shear elastic moduli, and δx was supplied in accordance with the study by William et al. [39]. On the basis of the theoretically treated longitudinal dual-well structures shown in Figure 12, these quantities have been calculated. The longitudinal modulus values, q, are shown in Table 2, which may be found here.

Figure 12 Plots of the potential wells for the longitudinal motion of the two-well systems at different elongations from 0 up to 100%, as examples in order to guide the viewer.
Figure 12

Plots of the potential wells for the longitudinal motion of the two-well systems at different elongations from 0 up to 100%, as examples in order to guide the viewer.

The mutual energy potential of the structural form of the three atoms, which consists of such an anion throughout the center of two cations (or vice versa) and which is detached by the space Ro by the position r of the oxygen atom O in the glass framework, 0.6595P2O5–0.0958ZnO–0.2447PbO–0Sm2O3, 0.6536P2O5–0.0950ZnO–0.2425PbO–0.0089Sm2O3 and 0.6423P2O5–0.0933ZnO–0.2383PbO–0.0261Sm2O3, is portrayed in Figure 12. They became evaluated at distinct elongation quantities, as per the following formula [30]:

(15) U = a 1 r + 1 ( 2 e r o − r ) + b 1 r m + 1 ( 2 e r o − r ) m .

There is 6 <m> 12. The categories of bonds present in the glass structure undergoing examination are: (P–O–P), (P–O–Zn), (P–O–Pb), (P–O–Sm), (Zn–O–Pb), (Pb–O–Pb) and (Sm–O–Sm). The constant a for a given structural model is defined as:

(16) U o = a r o 1 − 1 m .

That constant b becomes provided by b = ( a r o m − 1 ) / m ; consequently, r o is now the bond distance, U o is now an anion–cation bond energy, and e is its elongation, which is equivalent to e = (R/2r o), which is the same thing as saying that A–A. The region is split off by the symmetry of the distance 2r o. The longitudinal and transverse minimum space quantities, as well as the constants a and b, and U o throughout the dual-well potential, and theoretically deformational potentials, are included in Table 2 by each glass formula. This is based on the assumption that m is equal to 9.

It has obviously been shown that the elementary minimal potential for elongation will be less than (e < 1) among all glass structures. The two-well potential begins to grow for elongation quantities past 30%. With elongations ranging from 64.4 to 57.2%, the theoretical barrier diminishes from 0.362 to 0.296 eV, where the Sm2O3 value is about 0 and 0.0261 mol%. In contrast, with an increase in Sm2O3 concentration in the range of 0.0261 mol%, the mutual energy potential (Table 2) would be enhanced as a major cause by enhancing the dissociation energy of bonds.

The mathematically estimated deformational potential quantities congruous to the elongations seen in Table 2 became noticeable as being in close alignment with those experientially established. They concluded that the introduction of Sm2O3 within the structure between 0 and 0.0261 mol% resulted in a regular diminishment in the elongation of P–O bonds from 64.4 to 57.2%, as portrayed in Figure 13.

Figure 13 Elongation effect of Sm2O3 addition to the glass system (1–x) (0.6595P2O5–0.0958ZnO–0.2447PbO) · xSm2O3.
Figure 13

Elongation effect of Sm2O3 addition to the glass system (1–x) (0.6595P2O5–0.0958ZnO–0.2447PbO) · xSm2O3.

Also, the well had a potential energy U and a contraction c based on the shear oscillations of O atoms: 0.6595P2O5–0.0958ZnO–0.2447PbO–0Sm2O3, 0.6536P2O5–0.0950ZnO–0.2425PbO–0.0089Sm2O3, and 0.6423P2O5–0.0933ZnO–0.2383PbO–0.0261Sm2O3, which are portrayed in Figure 14. They became evaluated at distinct contraction quantities, as per the following formula [30]:

(16) U T = − 2 a ( c 2 r 0 2 + d 2 ) 1 / 2 + 2 b ( c 2 r 0 2 + d 2 ) m / 2 ,

where d is the shear displacement.

Figure 14 Plots of the potential wells for the shear motion of the two-well systems at different contractions from 0 up to −18%, as examples in order to guide the viewer.
Figure 14

Plots of the potential wells for the shear motion of the two-well systems at different contractions from 0 up to −18%, as examples in order to guide the viewer.

It has obviously been shown that the elementary minimal potential for contraction, c, will be less than (e < 1) among all glass structures. With contractions ranging from −4.865 to −4.580%, the theoretical barrier drops from 0.138 to 0.121 eV, where the Sm2O3 value was about 0 and 0.0261 mol%. In contrast, with an increase in Sm2O3 concentration in the range of 0.0261 mol%, the mutual energy potential (Table 2 and Figure 15) would enhance.

Figure 15 Contraction effect of Sm2O3 addition to the glass system (1–x) (0.6595P2O5–0.0958ZnO–0.2447PbO) · xSm2O3.
Figure 15

Contraction effect of Sm2O3 addition to the glass system (1–x) (0.6595P2O5–0.0958ZnO–0.2447PbO) · xSm2O3.

Figure 16 shows a decrease in percentage longitudinal elongations with a decrease in percentage transverse contractions, which means that the glass network structure decreased horizontally and decreased vertically in dimension. This indicates that Sm2O5 filled the interstices, confirming the diminishment in molar volume with the increase in Sm2O5 content.

Figure 16 Elongation and contraction effects on structure of Sm2O3 addition to the glass system (1–x) (0.6595P2O5–0.0958ZnO–0.2447PbO) · xSm2O3.
Figure 16

Elongation and contraction effects on structure of Sm2O3 addition to the glass system (1–x) (0.6595P2O5–0.0958ZnO–0.2447PbO) · xSm2O3.

4 Conclusions

It could be argued that low-temperature longitudinal ultrasound absorption led to the identification of well-defined highs, whose altitudes increased with an increase in the frequency introduced. These highs contributed to the thermally enabled relaxation of the interaction with the Arrhenius equation. Furthermore, the maximum temperature profiles decreased with an increase in the Sm2O3 concentration. The activation energy and attempted frequency quantities have diminished with the increase in the Sm2O3 modification.

In comparison, the quantities of the number of loss centers diminished by the diminishment in the energy of activation despite a diminishment throughout the zone underneath the longitudinal ultrasound absorption against the temperature relation as the Sm2O3 percent rose.

Acknowledgements

The authors wish to dedicate this search work to the soul of Prof. Dr. S. A. Gaafar Faculty of Science-Cairo University.

  1. Funding information: The authors state no funding involved.

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors declare no competing interests.

  4. Compliance with ethical standards: The authors say very clearly that neither people nor animals were tested during this research.

  5. Data availability statement: The datasets generated during and/or analyzed during this study are available from the corresponding author on a reasonable request.

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Received: 2022-10-13
Revised: 2023-05-11
Accepted: 2023-05-26
Published Online: 2023-11-21

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/jmbm-2022-0292
Keywords for this article
samarium phosphate glasses; relaxation process; attenuation coefficient; relaxation strength
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BY 4.0

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