Phonon Mean Free Path of Silicate Glasses: A Useful Parameter to Distinguish between Framework and Nonframework Cations

Abstract

Assuming that heat is transported by lattice vibrations (phonons) in silicate glasses, their thermal conductivity is correlated with the product of sound velocity, volumetric heat capacity, and phonon mean free path (MFP). The sound velocity and heat capacity have been studied extensively, but the origin of the composition-induced variation in the MFP remains unclear. The present study investigated MFP in M_2/nO–SiO₂ (Mⁿ⁺: Li⁺, Na⁺, Ca²⁺, Sr²⁺, or Pb²⁺) glasses with a variation of M_2/nO content. The MFP of the silica glass decreased with the addition of M_2/nO. The effect of the type of metallic cation on the MFP was minimal for the selected alkali and alkaline-earth silicate glasses. By contrast, the MFP of lead silicate glasses was higher than those of alkali or alkaline-earth silicate glasses when the metallic cation contents were comparable. Previous studies have demonstrated that alkali and alkaline-earth cations act as nonframework species that break the silicate network structure, whereas lead cations have inconclusive structural roles. Our data indicate that lead cations partly act as framework cations and that phonons tend to be scattered near nonframework cations in silicate glasses. Thus, the phonon MFP is a useful parameter for determining the structural role of metallic cations in silicate glass via phonon propagation.

1. Introduction

Understanding the thermophysical properties of the metal and oxide phases is essential for designing heat transfer in high-temperature industries (e.g., metallurgical extraction^1,2) and glass-making³⁾), directly affecting the efficiency and accuracy of these processes. For example, in the continuous casting process of steel, it is essential to control the rate of heat removal from cast steel to suppress the formation of defects on its surface.^4,5) The mold flux is used as a thermal insulator between the mold and cast steel and plays an important role in controlling the rate of heat removal.^4,6,7) In general, mold flux is composed of at least three phases: silicate glass, crystals, and melt.^8,9,10) Understanding the various thermophysical properties (e.g., thermal conductivity) that dominate the conduction and radiation heat transfer through a multiphase thermal insulator (i.e., mold flux) is important for controlling the rate of heat removal. The dominant factors that control the thermophysical properties of crystalline phases have been studied by numerous researchers;^11,12) however, the mechanism of composition-induced variation in the thermal conductivity of silicate melts and glasses is poorly understood.¹³⁾ Hence, it is difficult to design flux compositions with optimal thermal conductivity. Assuming that the heat carrier in these noncrystalline phases is mainly a lattice vibration wave, i.e., phonon,^14,15) the thermal conductivity (λ [W m^–1 s^–1]) can be expressed as a function of the volumetric heat capacity (C_V [J m^–3 K^–1]), velocity of sound (ν [m s^–1]), and mean free path (MFP) of phonons (l [m]), as expressed in Eq. (1):¹⁶⁾

  

$$\lambda =\frac{1}{3}{C}_{\text{V}}\nu l,$$(1)

An in-depth discussion of these three parameters can provide a complete understanding of the thermal conductivities of noncrystalline materials. The compositional dependence of the C_V and ν of silicate glasses and melts has been studied well in the previous studies,^17,18,19,20) whereas the MFP of phonon is one of the least understood parameters. Previous studies have reported two important findings related to phonon MFP in silicate glasses and melts:

• The MFP of phonon is generally in the range of 0.3–0.7 nm.^21,22,23)

• The phonon MFP decreases with the addition of metallic oxides (e.g., Na₂O).^15,24,25,26)

The former indicates that the phonon MFP should reflect the structure on the length scale between a silicon atom and its second-neighbor cation (Si⁴⁺ or Mⁿ⁺),²⁷⁾ which is schematically illustrated in Fig. 1 for the case of the structure between silicon and the second-neighbor silicon cation (Si–Si_2nd). The latter indicates that the incoherent vibrations near the metallic cations tend to scatter phonons and reduce the MFP. However, it remains unclear whether the phonon MFP depends on the type and content of metallic cations in silicate glasses. In addition, the origin of phonon scattering near metallic cations remains unclear. As our previous study indicated that the compositional dependence of the phonon MFP in silicate glasses is similar to that of these melts,¹⁵⁾ glasses can be used as analog materials to understand the compositional dependence of the phonon MFP in their melts. The present study aimed to investigate the compositional dependence of the phonon MFP for selected binary silicate glasses containing several types of metallic oxides, which are important for mold fluxes, metallurgical slags, and glass materials, as a fundamental study of the metallic cation-induced phonon scattering behavior in silicate glasses and melts.

Fig. 1. Schematic illustrations of the linkage of SiO₄ tetrahedra with its length scale. A typical range of Si–Si_2nd interatomic distance is described according to a reported value.²⁷⁾ (Online version in color.)

2. Principle to Derive the Phonon MFP

The thermal conductivity (λ) of the glass samples is correlated with the thermal diffusivity (α [m² s^–1]) by the following Eq. (2):²⁸⁾

  

$$\lambda =\alpha \rho C_{\text{p}},$$(2)

where ρ [kg m^–3] is the density and C_P [J kg^–1 K^–1] is the specific heat capacity. From Eqs. (1) and (2), the MFP of phonon (l) be expressed by Eq. (3) where C_V=ρC_P:

  

$$l=\frac{3\alpha }{\nu }.$$(3)

The present study has measured the thermal diffusivity (α) and velocity of sound (ν) for the glass samples to determine the phonon MFP (l).

3. Experimental Method

3.1. Sample Synthesis

The nominal and analyzed compositions of the glass samples are listed in Table 1. Glass samples with the composition of x mol% M_2/nO–(100-x) mol% SiO₂ (M: Li, Na, Ca, Sr, or Pb) were labeled as MSx where n represents the valence number of the metallic cation. The glasses were fabricated using melting and quenching methods. Reagent powders of SiO₂ (FUJIFILM Wako Pure Chemical Corporation), Li₂CO₃ (Sigma-Aldrich Co. LLC), Na₂CO₃ (Sigma-Aldrich Co. LLC), CaCO₃ (FUJIFILM Wako Pure Chemical Corporation), and SrCO₃ (Sigma-Aldrich Co. LLC) were mixed using a mullite mortar and pestle to obtain starting powder mixtures for the alkali or alkaline earth silicate glasses. To avoid the precipitation of metallic Pb,²⁹⁾ powder mixtures of PbO (FUJIFILM Wako Pure Chemical Corporation), Pb(NO₃)₂ (FUJIFILM Wako Pure Chemical Corporation), and SiO₂ (FUJIFILM Wako Pure Chemical Corporation) were prepared to synthesize lead silicate glasses. The powder mixtures were placed in a Pt–5mass%Au crucible (silica crucible for lead silicate glass) and calcined at 1073–1273 K for 30 min. The obtained samples were melted at 1273–1873 K in a Pt–5mass%Au crucible, which is less wettable to silicate melts.³⁰⁾ Finally, the molten sample was cast into a graphite mold. The quenched glass was annealed at a temperature close to the glass transition temperature.^31,32) Annealed samples were used for the property measurements. Because it was difficult to melt pure silica (i.e., SiO₂) in the laboratory, a standard quartz glass (5809-a, National Institute of Advanced Industrial Science and Technology)³³⁾ was used for the thermal diffusivity measurements. In addition, the chemical compositions of the synthesized samples were tested using wet chemical analyses; the lithium, sodium, calcium, strontium, and lead contents of the samples were measured using inductively coupled plasma atomic emission spectrometry (ICP-AES), whereas the silicon content was determined by gravimetry.

Table 1. Nominal and analyzed compositions of the samples in mol%. The analyzed compositions are shown in parentheses. f_Si–O–M values were calculated using Eq. (7) with the analyzed compositions.

Samples	SiO2	Li2O	Na2O	CaO	SrO	PbO	fSi–O–M
S	100	–	–	–	–	–	0
LiS28	72.0	28.0	–	–	–	–
	(71.3)	(28.7)	–	–	–	–	(0.34)
LiS33	67.0	33.0	–	–	–	–
	(66.8)	(33.2)	–	–	–	–	(0.40)
LiS36	64.0	36.0	–	–	–	–
	(64.0)	(36.0)	–	–	–	–	(0.44)
NaS18	82.0	–	18.0	–	–	–
	(81.5)	–	(18.5)	–	–	–	(0.20)
NaS23	77.0	–	23.0	–	–	–
	(77.6)	–	(22.4)	–	–	–	(0.25)
NaS28	72.0	–	28.0	–	–	–
	(72.2)	–	(27.8)	–	–	–	(0.32)
NaS33	67.0	–	33.0	–	–	–
	(67.5)	–	(32.5)	–	–	–	(0.39)
NaS36	64.0	–	36.0	–	–	–
	(64.5)	–	(35.5)	–	–	–	(0.43)
CaS42	58.0	–	–	42.0	–	–
	(59.2)	–	–	(40.8)	–	–	(0.51)
CaS45	55.0	–	–	45.0	–	–
	(55.4)	–	–	(44.6)	–	–	(0.57)
SrS40	60.0	–	–	–	40.0	–
	(63.4)	–	–	–	(36.6)	–	(0.45)
PbS33	67.0	–	–	–	–	33.0
	(66.7)	–	–	–	–	(33.3)	(0.40)
PbS50	50.0	–	–	–	–	50.0
	(48.0)	–	–	–	–	(52.0)	(0.70)
PbS60	40.0	–	–	–	–	60.0
	(43.1)	–	–	–	–	(56.9)	(0.80)

3.2. Measurements of Thermal Diffusivity and Sound Velocity

The thermal diffusivity (α) of the samples were evaluated using a flash method with a xenon flash lamp, which heats the front surface of the samples by a short energy pulse, and the resulting temperature response of the rear surface is measured with an infrared detector. The present study used a Xenon flash analyzer LFA 467 HyperFlash (NETZSCH, Japan) to determine the α of the samples. The synthesized glass samples were cylindrical (diameter: 10 mm; height: 1 mm). Because the synthesized glass samples were transparent, it was necessary to avoid the penetration of light generated by the xenon lamp. In addition, the flash method requires the surface of the samples to absorb the energy pulse and exhibit high emissivity.³⁴⁾ The top and bottom surfaces of the samples were mirror polished and spray-coated with carbon powder (FC-153) after gold sputtering (~100 nm). The temperature response of the rear sample surface was measured thrice at 298 K for each sample. To determine the thermal diffusivity of the samples, the present study applied the Cape–Lehman model,³⁵⁾ which considers the facial and radial heat losses in the temperature evolution of the rear sample surface over time after a short energy pulse. The model accurately reproduced the temperature evolution observed in this study. The repetitive errors in the three-time measurements were approximately 1% for most of the samples (see detail in Table 2).

Table 2. Measured thermophysical properties of the samples at 298 K.

Samples	α/10–6 m2 s–1	ρ/103 kg m–3	nSi–O–M/nm–3	νL/m s–1	νT/m s–1	ν/m s–1	E#/GPa	l/nm
S	0.824±0.005	2.187	0	–	–	4288*	80	0.58
LiS28	0.574±0.002	2.328	5.22	6160	3710	4090	78	0.42
LiS33	0.555±0.071	2.333	6.21	6283	3658	4079	78	0.41
LiS36	0.531±0.004	2.349	6.90	6372	3701	4130	80	0.39
NaS18	0.563±0.008	2.369	2.91	5404	3228	3569	60	0.47
NaS23	0.515±0.001	2.406	3.58	5386	3157	3513	59	0.44
NaS28	0.481±0.009	2.457	4.53	5352	3070	3439	58	0.42
NaS33	0.439±0.007	2.478	5.33	5410	3050	3433	58	0.38
NaS36	0.419±0.002	2.499	5.86	5381	3007	3393	58	0.37
CaS42	0.500±0.032	2.795	9.07	6298	3481	3938	87	0.38
CaS45	0.485±0.001	2.839	10.2	6325	3494	3954	89	0.37
SrS40	0.386±0.005	3.446	7.59	5366	2965	3355	78	0.35
PbS33	0.393±0.003	4.617	6.07	3720	2204	2444	55	0.48
PbS50	0.270±0.002	6.339	11.1	3109	1732	1956	49	0.41
PbS60	0.256±0.001	6.492	12.0	3030	1682	1901	47	0.40

*  Value reported by Manghnani and Singh³¹⁾

^#  E values calculated using Eq. (4)

The sound velocity (ν [m s^–1]) of the glass samples is defined by the following Eq. (4):³⁶⁾

  

$$\nu =\sqrt{\frac{E}{2\rho }},$$(4)

where E is the Young’s modulus of the samples, and ρ is the sample density [kg m^–3]. Moreover, E can be determined by the longitudinal (ν_L) and transverse wave velocities (ν_T) by the Eq. (5):³⁷⁾

  

$$E=\rho {\nu }_{\text{T}}^2\frac{3{\nu }_{\text{L}}^2-4{\nu }_{\text{T}}^2}{{\nu }_{\text{L}}^2-{\nu }_{\text{T}}^2}.$$(5)

The present study measured the ν_L and ν_T of the samples using an ultrasonic wave propagation pulse-echo technique with a thickness gauge 38DL PLUS (OLYMPUS corporation, Japan). The glass was cylindrical (diameter: 10 mm, height: 10 mm), and the top and bottom surfaces were mirror-polished. The polished samples were used for measurements. The measurements were performed thrice at a frequency of 5 MHz for each sample at 298 K. The scatter in the measurements was less than 0.5%. The averaged values of ν_L and ν_T were used to derive the sound velocity of the samples. It is clear that Eqs. (4) and (5) enable us to derive sound velocity (ν) without using the sample density (ρ); however, the ρ of the sample is required to interpret the compositional dependence of ν. The present study evaluated the ρ of the samples using a conventional Archimedean method at ~298 K with ethanol (FUJIFILM Wako Pure Chemical Corporation) as the immersion liquid. The density of the samples was measured five times, and the average values were used to determine the sound velocities of the samples. The repetitive error in the measurements was less than 1.2%.

4. Results

4.1. Thermal Diffusivity

Figure 2 and Table 2 show the composition-induced variations in the thermal diffusivities of the glass samples at 298 K. The measured thermal diffusivity of the pure silica glass (0.829×10^–6 m² s^–1) is close to the recommended value (0.837×10^–6 m² s^–1).³³⁾ This validates the reasonability of our methodology for determining the thermal diffusivity. The thermal diffusivity of the silica glass decreased with the addition of Na₂O. This trend agrees well with previous studies that reported the thermal diffusivity of Na₂O–SiO₂ glasses using the laser flash method.^14,15,23) Similarly, the thermal diffusivity of silica glass decreased with the addition of all types of chosen metallic oxides.

Fig. 2. The change in thermal diffusivity of the silicate glasses by adding metallic oxides. The horizontal axis shows the analyzed content of metallic oxides. (Online version in color.)

4.2. Velocity of Sound

Figure 3 shows the effect of adding metallic oxide on the sound velocity in the sample glasses. Manghnani and Singh³¹⁾ measured the sound velocities of silica and sodium silicate glasses with compositions similar to those used in our study. Figure 3 compares their data with our results for the Na₂O–SiO₂ system. The differences between their values and our data for similar compositions were in the range of 3%–5%, which is sufficiently small to compare. Because preparing the sample with a pure silica composition for sound velocity measurement was difficult, a reported value for the silica composition by Manghnani and Singh is plotted in this figure. It was found that the sound velocity of silica glass did not significantly change with the addition of Li₂O or CaO, whereas Na₂O or PbO lowered the sound velocity. As shown in Eq. (4), the velocity of the sound is related to the ratio of Young’s modulus to the density ($\sqrt{E/2\rho }$). Therefore, the glasses with low density and high Young’s modulus (e.g., Li₂O–SiO₂) have a higher sound velocity among the glasses chosen in the present study, whereas the low sound velocity of PbO–SiO₂ systems is likely due to their high density (see Table 2).

Fig. 3. Relationship between the sound velocity and the analyzed content of metallic oxides in the silicate glasses. (Online version in color.)

4.3. Phonon Mean Free Path (MFP)

In the kinetic theory of a gas, the MFP is generally defined as an average distance that a particle will move before a collision with another particle and is correlated with the reciprocal number density of the particle per unit volume (1/n_p).³⁸⁾ Applying this kinetic theory to the phonon propagation, the MFP of phonon can be considered as a distance of phonon propagation without scattering. Assuming that the phonons scattered near metallic cations, the phonon MFP should vary as a function of the number density of the oxygen atoms bonded with silicon and metallic cations (n_Si–O–M). The n_Si–O–M [nm⁻³] can be derived using the following Eq. (6):

  

$$n_{\text{Si}-\text{O}-\text{M}}=n_{\text{total}}\cdot X_{\text{O}}\cdot f_{\text{Si}-\text{O}-\text{M}},$$(6)

where n_total, X_O, and f_Si–O–M are atomic number density of the glass, oxygen atomic fraction, and the fraction of oxygen atoms bonded with silicon and metallic cations, respectively. The present study derived the n_total (=N_Aρ/Y)³⁹⁾ from the measured density (ρ) and the Avogadro’s number (=6.02×10²³) where Y is the average molecular weight of the samples. f_Si–O–M was derived using Eq. (7):^40,41)

  

$$f_{\text{Si}–\text{O}–\text{M}}=\frac{2X_{\text{O}}-4X_{\text{Si}}}{X_{\text{O}}},$$(7)

where X_Si denotes the atomic fraction of Si. Previous studies have validated that f_Si–O–M obtained using Eq. (7) agree well with the experimental values obtained using spectroscopic techniques for the silicate glasses.^42,43,44) The phonon MFP of the glasses was derived using Eq. (3), as listed in Table 2. The phonon MFPs of the silicate glasses are plotted against n_Si–O–M in Fig. 4. The phonon MFP for the pure silica glass was close to 0.6 nm, which is similar to the interatomic distance from a silicon atom to its second neighbor silicon atom (Si–O–Si–O–Si),²⁷⁾ as drawn in Fig. 1. The phonon MFP decreased with the addition of metallic oxides. These data indicate that phonon scattering is more likely to occur near metallic cations. Since the phonon MFP is similar for alkali or alkaline-earth cations at a comparable n_Si–O–M (see Fig. 4), herein the phonon MFP evolution was fitted against the n_Si–O–M for the alkali and alkaline-earth silicate systems by the single nonlinear regression curve, which is expressed by the following Eq. (8):

  

$$l=a+\frac{1}{b\left(n_{\text{Si}–\text{O}–\text{M}}+c\right)}.$$(8)

Fig. 4. Evolution of the phonon MFP with n_Si–O–M for the glass samples. The possible error range of the phonon MFP was evaluated based on the scatter in the measured values of the thermal diffusivity and shown as the error bars. For plots without an error bar, the error range is smaller than the size of the plots. The green and black dashed lines represent the fitting curves by Eq. (8) for lead silicate and alkali or alkaline-earth silicate systems, respectively. (Online version in color.)

The physical meaning of the fitting parameters (a and c) in Eq. (8) remains unclear whereas the parameter b (=0.73), which reflects the steepness of the variation in the phonon MFP with increasing the n_Si–O–M, should relate to the probability of phonon scattering near metallic cations. Notably, the phonon MFP values of lead silicate glasses are higher than those of alkali or alkaline-earth silicate glasses with similar n_Si–O–M values. The evolution of the phonon MFP against n_Si–O–M for the lead silicate systems are also well reproduced by the Eq. (8) with the lower b value (0.02), which indicates that the probability of the phonon scattering is lower for the lead silicate system than those for the alkali and alkaline-earth silicate systems. The following discussion considers the origin of the composition-induced variation in the phonon MFP from the viewpoint of the bonding character near the metallic cations.

5. Discussion

In silicate glasses, metallic cations are surrounded by several oxygen atoms and form polyhedral species (MO_N), which link with SiO₄ tetrahedra with different bond strengths and connectivities (e.g., corner- and edge-sharing linkages). The incoherence of the vibration mode near the Si–O–M bond should change depending on its bond character: “bond strengths” and “connectivity.” To evaluate the bond strength, Pauling⁴⁵⁾ proposed the concept of electrostatic valence bond strength, which is described by the cationic valence divided by its coordination number (Z/N).^46,47) The connectivity shows different aspects of the bond character, which, however, should not be completely independent of the Z/N of the metallic cations. Generally, two types of connectivity (linkages) are found in silicate glasses: “corner-” and “edge-sharing” linkages, as depicted in Fig. 5. When a metallic cation (Mⁿ⁺) has a high Z/N, the MO_N polyhedral species prefer to bond with SiO₄ by corner-sharing linkages to retain a long Si–M distance and weaken the cation-cation repulsion.⁴⁵⁾ Meanwhile, a low Z/N ratio of an M cation allows for a shorter Si–M distance and partly for the formation of an edge-sharing linkage between SiO₄ and MO_N. The present study estimated the Z/N ratios for selected metallic cations: Si⁴⁺, Li⁺, Na⁺, Ca²⁺, Sr²⁺, and Pb²⁺. Table 3 shows the estimated Z/N values for these cations, based on the reported coordination numbers (N) for the cations: Si,⁴⁸⁾ Pb,⁴⁹⁾ Ca,⁵⁰⁾ Sr,⁵¹⁾ Li,⁴⁸⁾ and Na.⁴⁸⁾ Si⁴⁺ has a high Z/N of 1 whereas the Z/N of alkali and alkaline-earth cations was lower than that of silicon cation and in the range of 0.17–0.33. The calculated Z/N indicates that the bond strength near the alkali and alkaline-earth cations differs depending on the type of cation; however, the phonon MFPs of the alkali and alkaline-earth silicate glasses do not change with the type of metallic cations when their n_Si–O–M values are comparable, as shown in Fig. 4. This indicates that the effect of the bond strength near the metallic cations on the phonon MFP was small for the chosen alkali and alkaline-earth silicate glasses.

Fig. 5. Schematic illustrations of (a) corner-sharing and (b) edge-sharing SiO₄–MO_N linkages. (Online version in color.)

Table 3. Formal cationic valence (Z) and reported coordination number (N)^48,49,50,51) for the chosen cationic species. The fraction of the edge-sharing linkage was categorized into three categories depending on our assumption based on the results of structural characterization: Non, Low, and High.

	Si4+	Pb2+	Ca2+	Sr2+	Li+	Na+
Z	4	2	2	2	1	1
N	4	3.5*	6	7.5*	4	6
Z/N	1	0.57	0.33	0.26	0.25	0.17
Fraction of edge-sharing	Non	Low	High	High	High	High

*  Intermediate values in the range of the reported coordination numbers^49,51)

The remaining aspect of the bond character is the connectivity of the polyhedral linkages. Since it has been reported the repulsive force between two cations in the edge-sharing linkage affects the thermal vibration mode in oxide crystalline materials,⁵²⁾ the fraction of the minor edge-sharing linkage in silicate glasses should affect the phonon MFP, whereas the corner-sharing linkage is the main type of the connection in oxide glasses.⁵³⁾ Based on the calculated Z/N, the polyhedral unit with Pb²⁺ (e.g., PbO₄) tended to have a lower fraction of edge-sharing linkages than the polyhedral species of alkali and alkaline-earth cations. Some authors have clarified the atomic arrangement of the 22.7Na₂O–77.3SiO₂ (mol%),⁵⁴⁾ 33PbO–67SiO₂ (mol%),⁵⁵⁾ and 50PbO–50SiO₂ (mol%)⁵⁵⁾ glasses using X-ray and neutron scattering with the aid of computer simulation based on reverse Monte Carlo (RMC) modeling. The present study recharacterized the RMC-modeled atomic arrangement and derived the fraction of edge-sharing connections for the SiO₄–MO_N linkage from the partial pair distribution functions for Pb–Si and Na–Si. The obtained fraction of edge-sharing linkages was 9% for the SiO₄–NaO_N in the 22.7Na₂O–77.3SiO₂ (mol%) glass, whereas those of the SiO₄–PbO_N were 4% and 2% for the 33PbO–67SiO₂ (mol%) and 50PbO–50SiO₂ (mol%) glasses, respectively. Although we did not characterize the fraction of edge-sharing linkages for the SiO₄–MO_N for all the glass systems, Toyoda et al.⁵⁶⁾ have recently reported that the fraction of edge-sharing linkages for the SiO₄–LiO_N in a lithium silicate-based multicomponent glass was close to 10%, which is relatively similar to the fraction in the sodium silicate glass. Assuming that the fractions of edge-sharing linkage in the lithium, calcium, and strontium silicate systems are similar to that of the sodium silicate glass, the polyhedral unit with Pb²⁺ tended to have a lower fraction of edge-sharing linkages than the polyhedral species of alkali and alkaline-earth cations, as estimated from Z/N. The current MFP data allow us to hypothesize that the repulsive force between Si and metallic cations in the edge-sharing linkage generates an incoherent vibration mode, which tends to scatter phonons in the silicate glass structure.

Metallic cations in silicate glasses and melts are categorized into framework and non-framework cations. The former have a high Z/N ratio and constructs a silicate network structure by forming a corner-sharing linkage between SiO₄ and MO_N, as proposed by Zachariasen.⁵³⁾ The latter, which generally have a low Z/N, tends to break the silicate network structure. The structural roles of metallic cations in silicate glass have been extensively studied through structural characterization^57,58) and physical property measurements.^59,60) Silicon cations were determined to act as framework cations, whereas alkali and alkaline-earth cations act as nonframework cations.^61,62) In contrast, the structural role of lead cations in silicate glasses remains inconclusive. Kohara et al.⁵⁵⁾ reported that Pb²⁺ cations partly act as framework cations (network formers) based on the extraordinarily large amount of free volume (voids) in lead silicate glass, which cannot be found in conventional binary silicate glasses containing nonframework cations (e.g., alkali silicate glasses). Kacem et al.⁶³⁾ reported that lead cations act as nonframework cations (network modifiers) based on the glass-transition temperature (T_g) data for binary silicate glasses, and the larger decrease in the T_g of silica glass with the addition of lead oxide than that with the addition of a comparable calcium oxide content. Our finding contradicts the compositional dependence of T_g, where lead silicate glasses have higher phonon MFPs than alkali and alkaline-earth silicate glasses at comparable n_Si–O–M values. In particular, these results indicate that phonons propagate through Si–O–Pb bonds with a higher probability than through Si–O–R bonds, where R represents an alkaline or alkaline-earth atom. This tendency indicates that lead cations partly act as framework cations from the viewpoint of phonon propagation. The structural role estimated by the phonon MFP is consistent with the conclusion from the free-volume analysis but contradicts the interpretation from the T_g data. These different conclusions could be ascribed to the differences in the dominant parameters of T_g and phonon MFP. Assuming that T_g is the temperature at which the structure of glass starts to relax via viscous flow, T_g should be influenced by the mobility of the oxygen atoms near the metallic cations.^64,65) As the oxygen atoms between silicon and lead atoms (Si–O–Pb) have a higher mobility than those between two silicon atoms (Si–O–Si), Si–O–Pb bonds should behave in the breakage of the silicate network structure from the viewpoint of viscous flow (T_g). In contrast, phonon MFP was dominated by the connectivity of the polyhedral species. The corner-sharing Si–O–Pb linkage acts as a connected part of the network structure for phonon propagation. Although the compositional dependence of T_g and phonon MFP obtained opposite conclusions on the structural role of lead cations, the structural role of lead cations is categorized differently depending on the types of properties. Zachariasen⁵³⁾ proposed empirical rules to distinguish oxide components based on the glass-forming abilities. According to these rules, oxygen polyhedra of metallic cations must share corners rather than edges or faces to form glass. Interestingly, the classification of the structural role of metallic cations by phonon MFP conforms to Zachariasen’s rules⁵³⁾ for the polyhedral connectivity in the chosen compositions. Further studies on the phonon MFP of oxide glasses containing other types of framework and nonframework cations with detailed structural information are required to fully understand the origin of the composition-induced variation in phonon MFP.

6. Conclusion

This study investigated the variation in the phonon MFP of binary silicate glasses at 298 K by adding M_2/nO (Mⁿ⁺: Li⁺, Na⁺, Ca²⁺, Sr²⁺, or Pb²⁺). The MFP decreased with an increase in SiO₄–MO_N linkages in the glasses. The MFPs of alkali and alkaline earth silicate glasses are similar when the number densities of oxygen atoms between the silicon and metallic cations (n_Si–O–M) are comparable. This indicates that the MFP does not strongly depend on the type of alkali or alkaline earth cation. However, the MFP of lead silicate glass was higher than those of the alkali and alkaline-earth silicate systems when n_Si–O–M is comparable. This tendency can be explained by the differences in the fraction of edge-sharing linkages where phonons tend to be scattered. Our phonon MFP data indicate that lead cations partially act as framework cations from the viewpoint of phonon propagation.

Conflict of Interest

The authors declare that there is no conflict of interest.

Acknowledgments

This work was done as an activity of the ISIJ Research group I “Improvement of measurement accuracy in thermal conductivity of molten oxides (2022–2024),” which was financially supported by the Iron and Steel Institute of Japan (ISIJ). We appreciate the members of the research group for their fruitful discussion. This work was partially supported by JSPS KAKENHI (JP24K01153 and JP21H01854), the cooperative research program of “Network Joint Research Center for Materials and Devices (MEXT), and the research grant from the Nippon Sheet Glass Foundation for Materials Science and Engineering. We thank Mr. Shigeru Suzuki (Japan Steel Works, Ltd.) for his contribution to the chemical analysis of the samples. The authors are grateful to Mr. Yoshihiro Nakamura (Tohoku University) for carefully polishing the glass sample surfaces. We appreciate the constructive comments of the anonymous reviewer and editor for improving our manuscript.

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