Surface Tension Measurement of SiO₂–Na₂O–NaF System by Maximum Bubble Pressure Method

Abstract

The surface tension of the mold flux is important because it governs the interfacial phenomena between the mold and molten/solid steel. The maximum bubble pressure (MBP) method is often used to determine the surface tension of molten slags. However, for liquid samples with high viscosity, such as molten silicate, the MBP is overestimated. In this study, a simple relaxation function was applied to determine the MBP using the gas flow-rate as a parameter. Consequently, a static MBP was obtained, and the surface tension was reliably measured. The surface tension of the SiO₂–Na₂O–NaF melts were measured over a wide composition range using the developed method. When SiO₂-40 mol% Na₂O was added to NaF, the surface tension of the melts gradually increased with increase in SiO₂–Na₂O concentration. When the NaF in SiO₂-40 mol% NaF was replaced by Na₂O, the surface tension of the melts did not change significantly at the beginning of the addition. It was considered that F⁻ ion was exposed in the surface of melts instead of O²⁻ ion. This result is consistent with the discussion of the relative strength of the bonding force between the constituent particles. The increase in the surface tension upon further replacement of NaF with Na₂O was gradual, indicating that F⁻ was relatively exposed more than O²⁻ on the surface of the investigated melts.

1. Introduction

Continuous casting technology is indispensable for producing high-quality steel.¹⁾ In the continuous casting of steel, a synthetic slag called mold flux is used to produce high-quality steel with high productivity.^2,3,4) The mold flux is composed of silicate and some added fluorides.^2,5) The mold flux functions as a lubricant for smoothly moving cast slabs, as a covering material for preventing atmospheric oxidation, and as an insulation material for cooling molten/solid steel slowly.^6,7) The thermophysical properties of the mold flux, such as the viscosity, thermal conductivity, and specific heat, are important for process control.^3,4) The surface tension of the mold flux that governs the interfacial phenomena between the mold and molten/solid steel is essential.^8,9) By controlling surface tension of mold flux with composition adjustment, the covering and heat-retaining properties are increased, and an improvement in energy efficiency is expected. Therefore, the surface tension and interfacial tension of the molten slag have been extensively measured.^{10,11,12,13,14,15)} In addition to experimental determination, an estimation equation was studied.^{16,17,18,19,20)} The estimation equation for surface tension was originally developed for molten alloys^17,18) and subsequently developed for ionic solutions.^19,20) Mathematical treatment of the estimation equation for the surface tension of ionic liquid is difficult,²⁰⁾ and the estimation model is being developed.

The maximum bubble pressure (MBP) method is used to measure the surface tension of molten silicate.^21,22) The authors used the MBP method to measure the molten silicate.²³⁾ For liquid samples with high viscosity, such as molten silicate, a bubble-forming gas must be supplied at a low velocity,^24,25,26) and the criteria for accurate measurements were unclear. The authors focused on the detachment time of the bubbles and applied a simple relaxation function to obtain accurate data.²³⁾ However, the physical meaning of the detachment time remains unclear.

In the present study, the authors focused on a gas flow-rate whose physical meaning was clear, and applied a simple relaxation function with the gas flow-rate. As a measuring object at high temperatures, SiO₂–Na₂O–NaF system whose viscosities can be varied over wide range was selected. In a previous study, Suzuki et al. extensively measured the surface tension of the Si–Ca–Na–O–F system that is more multi-component system.²⁷⁾ They focused on the exchange of coordination combinations in molten silicate, as shown in the following equation:

  

$$\mathrm{C}\mathrm{a}{\mathrm{F}}_2+\mathrm{N}{\mathrm{a}}_2\mathrm{O}\to \mathrm{C}\mathrm{a}\mathrm{O}+2\mathrm{\hspace{0.17em} }\mathrm{N}\mathrm{a}\mathrm{F}$$(1)

In this study, a simpler system was selected, and the influence of anion exchange was investigated. By measuring surface tension of melts with a wide composition range from the melt containing only O²⁻ to the melt containing only F⁻ as anion, fluoride concentration dependence on surface tension can be quantified with high reliability and the distribution state of F⁻ on melt surface can be estimated. As well, this study provides data for considering suitable amount of fluoride addition, which prevents the excess addition of fluoride and contributes to the reduction of environmental burden.

2. Principle

The principle of the MBP method has been reported and reviewed in detail in previous studies.^23,24,25,26) The important points are summarized as follows. A capillary with an inner radius r was immersed in a liquid sample with a density ρ to depth h, and gas was introduced into the capillary. The gas pressure P inside the capillary increased, and a bubble formed on the tip of the capillary. P is expressed by a bubble with a curvature radius R and described by the Laplace equation shown in Eq. (2), considering the hydrostatic pressure on the tip.^24,25,26)

  

$$P=\rho gh+\frac{2\sigma }{R}$$(2)

where g and σ are gravitational acceleration and surface tension, respectively. When R becomes minimum, that is, the bubble radius becomes equal to r, P reaches a maximum value; P_max is referred to as the maximum bubble pressure (MBP).

  

$$P_{max}=\rho gh+\frac{2\sigma }{r}$$(3)

In the experiment, the differential pressure between the gas and atmospheric pressure was measured. P changed in a sawtooth pattern over time, and the maximum in the pattern corresponded to P_max. By plotting P_max against different h, a line between h and P_max was obtained, and the surface tension could be determined from the intercept of the line P₀. However, the bubble was deformed owing to hydrostatic pressure; thus, the Laplace equation must be modified. In this study, the surface tension was determined using Eq. (4), as proposed by Schrödinger.²⁸⁾

  

$$\sigma =\frac{r{P}_0}{2}\left\{1-\frac{2}{3}\left(\frac{r\rho g}{P_0}\right)-\frac{1}{6}{\left(\frac{r\rho g}{P_0}\right)}^2\right\}$$(4)

where P₀ indicates the intercept of the line in an h–P_max plot.

As discussed in a previous report,²³⁾ P_max apparently increases in a short bubble detachment time, namely, at a high gas flow-rate, when a highly viscous liquid is measured. This is because P_max consists of both static and dynamic terms.

  

$$P_{max}=P_s+P_d$$(5)

where P_s is the static pressure independent of the gas flow-rate and P_d is the dynamic pressure dependent on the gas flow-rate. When the viscosity of the liquid increases, the dynamic pressure increases owing to viscous resistance. The viscosity dependence of ultrasonic propagation in a viscous fluid can be explained by a simple relaxation function.²⁹⁾ In this study, the following simple relaxation function was applied to determine the MBP.

  

$$P_{max}=P_s{\left(1+\frac{r{Q}^2}{Q^2+q^2-r{Q}^2}\right)}^{1/2}$$(6)

where P_s is static bubble pressure, Q is gas flow-rate, q is relaxation flow-rate, and r is relaxation intensity. When Q approaches 0, P_max approaches P_s. Experimentally, P_max was measured under various Q, and P_s was determined by curve fitting method.

3. Experimental

3.1. Experimental Apparatus

The experimental apparatus was identical to that used in a previous study.²³⁾ The apparatus comprised a gas introduction system, crucible lifting system, and heating system.

1. Gas introduction system: A capillary with a tip made of nickel (Ni, 98% purity) was used because Ni has high compatibility with molten fluoride.³⁰⁾ The cost of Ni was also considered for the selection of contact material because the capillary was used as a disposable for reliable measurements. The inner and outer diameters of the capillary were approximately 1 and 1.2–1.3 mm, respectively. The inner and outer circles are concentric and form bubbles isotropically. The inner radius was determined using a method identical to that described in a previous study.²³⁾ The inner radius at elevated temperatures were corrected using thermal expansion coefficient of Ni.³¹⁾ Owing to the good wettability between Ni and water, the effective radius was considered to be almost the same as the inner radius of the Ni capillaries.

A Ni crucible (inner diameter: 24 mm) was used and the liquid level of the sample was approximately 20 mm. The gas flow-rate was regulated using a mass flow controller (3200, KOFLOC Inc.), and the differential pressure was measured using digital differential pressure gauges (DP-340BA, COSMO instruments Co., Ltd.). The measurable ranges were 0–0.5 kPa for silicone oils and 0–2 kPa for molten silicates, and the uncertainty was ±1.25 Pa for silicone oils and ±5 Pa for molten silicates. The analog output voltages from the gauges were recorded using a digital recorder (HV GL2000; GRAPHTECH Co.).

2. Crucible-lifting system: The Ni capillary was relatively immersed in the sample liquid by lifting the Ni crucible. The immersion depth of the capillary was measured with a resolution of 0.01 mm using a digital level meter.

3. Heating system: An electrical resistance furnace consisting of MoSi₂ heating elements was used and measurements were performed up to 1873 K. Three stacked heating elements were independently controlled and several tungsten (W) plates were placed above and below the crucible to obtain a homogeneous temperature profile. An excellent temperature homogeneity of ±0.5 K over the crucible height was achieved. The temperature at the top of the crucible was set to 0.5 K higher than that at the bottom of the crucible to suppress the convection of the sample liquid. The atmospheric gas inside the apparatus and bubble-forming gas were argon (Ar, 99.9999% purity) without purification treatment. The oxygen partial pressure in the measurement region was estimated close to the equilibrium oxygen partial pressure (2×10⁻¹¹ atm at 1500 K) governed by an equilibrium W (s) + 3/2 O₂ (g) = WO₃ (s) because W is the most reactive metal in the measurement region.

3.2. Sample Preparation and Experimental Procedure

In room temperature experiments (293 K), silicone oils (Shin-Etsu Chemical Co., Ltd) with a known viscosity and surface tension (see Table 1) were used as sample liquids. The compositions of the silicate samples used for the high-temperature experiments are summarized in Table 2 and their compositions are plotted in a composition triangle shown in Fig. 1. The sample compositions were selected according to the following series: SiO₂-40 mol% Na₂O was added to NaF (Exp. A, B, C, D, E, F); SiO₂ was added to NaF (Experiments A, G, H); and NaF in SiO₂-40 mol% NaF was replaced with Na₂O (Exp. H, I, J, K, L, F).

Table 1. Properties of silicone oil samples and measured values for the verification of a simple relaxation function treatment.

No.	Viscosity, η/mPa·s	Surface tension, σ/mN·m−1	Expected MBP,a) Pr/Pa	MBP without SRFb) treatment at 1.5 mL·min−1 in flow rate,		MBP with SRFb) treatment,
				Pwo/Pa	Difference, D (%)	Ps/Pa	Difference, D (%)
1	80	20.3	176.5	180.0	+2.0	176.9	+0.2
2	800	20.6	178.2	187.7	+5.3	179.6	+0.8
3	8000	20.9	179.8	213.8	+18.9	182.5	+1.5

a)  MBP denotes maximum bubble pressure.

b)  SRF denotes a simple relaxation function.

Table 2. Experimental conditions and results for the surface tension measurement for SiO₂–Na₂O–NaF system.

Exp. No.	Concentration of component i, Ci (mol%)a			Basicity, CNa2O/CSiO2	Temperature range measured, T/K	η/mPa·s at 1473 K15)	Surface tension, σ/mN·m−1 = a + bT		σ/mN·m−1 at 1473 K
	SiO2	Na2O	NaF				a	b
A	0	0	100	–	1344–1498	1.2	312.1	−0.1038	159.1
B	15	10	75	0.67	1347–1500	3.3	309.8	−0.0920	174.4
C	30	20	50	0.67	1345–1498	18.8	307.5	−0.0820	186.6
D	45	30	25	0.67	1348–1496	227	339.9	−0.0880	210.3
E	54	36	10	0.67	1347–1495	1445	384.0	−0.0895	252.1
F	60	40	0	0.67	1348–1497	5768	363.4	−0.0544	283.2
G	40	0	60	0.00	1348–1500	N.A.	279.8	−0.0706	175.8
H	60	0	40	0.00	1349–1499	N.A.	387.7	−0.1321	193.2
I	60	10	30	0.17	1350–1500	N.A.	323.8	−0.0922	187.9
J	60	20	20	0.33	1349–1500	N.A.	249.9	−0.0329	201.5
K	60	24	16	0.40	1346–1497	N.A.	403.8	−0.1308	211.1
L	60	30	10	0.50	1348–1498	N.A.	321.4	−0.0568	237.7

a:  Value of initial composition in sample preparation.

Fig. 1. Sample compositions plotted in composition triangle for SiO₂–Na₂O–NaF system.

For Series 1, Na₂CO₃ (>99.5%, Kanto Chemical Co., Inc.) and SiO₂ (>99.5%, Kanto Chemical Co., Inc.) mixtures were placed in a platinum (Pt) crucible and melted at 1673 K in air, and the melt was poured on a copper (Cu) plate for quenching. The solidified samples were crushed into cullets. This melting-quenching-crushing operation was conducted three times. In the melting phase, the termination of CO₂ evolution was confirmed, resulting in the formation of the SiO₂–Na₂O sample. SiO₂–Na₂O sample was added to NaF at 1673 K and melted. Thereafter, SiO₂–Na₂O–NaF mixture was quenched and crushed. For Series 2, SiO₂ was added to NaF at 1673 K and melted. The molten mixture was subsequently quenched and crushed. For series 3, SiO₂-40 mol% Na₂O was added to SiO₂-40 mol% NaF at 1673 K and melted. The molten mixture was subsequently quenched and crushed.

In the high-temperature experiments, a sample cullet in a Ni crucible was placed in a furnace and heated to 1500 K for melting. With a slow flow of Ar gas, the Ni capillary contacted the molten silicate to determine the liquid level. The capillary was immersed to a depth of 3 mm and the MBP was measured at several flow-rates. Furthermore, the capillary was immersed at depths of 6, 9, and 12 mm, and the measurement was repeated. After the measurements were terminated at a specified temperature, the sample temperature was decreased to 50 K and equilibrated for 30 min. The measurements were conducted during a cooling step and successive heating steps. The temperature range was approximately 1350–1500 K (Table 1). Because the mass loss of sample after measurement was less than 1%, it was estimated there is no significant change in composition.

4. Results and Discussion

4.1. Investigation Using Silicone Oils at Room Temperature

The experimental conditions and results are summarized in Table 1. Silicone oils with viscosities of 80, 800, 8000 mPa·s were used. The viscosities differed significantly; however, the surface tensions were approximately the same. Therefore, the expected MBPs were approximately same (176.5–179.8 Pa). MBPs obtained at the flow-rate of 1.5 mL·min⁻¹ were 180.0, 187.7, and 213.8 Pa, which correspond to +2.0%, +5.3%, and +18.9% deviation from the expected values. Thus, high viscosity results in an apparently high MBP.

The flow-rate dependence of the MBP is shown in Fig. 2. The plots show the experimental values, and the line shows the fitting curve using a simple relaxation function. As shown in the figure, the flow-rate dependence of the MBP is well reproduced by the simple relaxation function. The static MBPs determined were 176.9, 179.6, and 182.5 Pa, which correspond to +0.2%, +0.8%, and +1.5% deviation from the expected values. Therefore, a reliable MBP can be obtained using a simple relaxation function.

Fig. 2. Maximum bubble pressure measured using silicone oils at 293 K. Plot: measured value. Line: fitting curve based on a single relaxation function treatment.

4.2. Surface Tension Measurement of Molten SiO₂–Na₂O–NaF

The typical flow-rate dependencies of the MBP for molten silicates at high temperatures are shown in Figs. 3 and 4. Viscosities of samples shown in Figs. 3 and 4 were 4600 and 19000 mPa·s, respectively.³²⁾ Similar to those at room temperature, the experimental values at high temperatures were reproduced by a simple relaxation function. High viscosity resulted in a significant drop in the MBP curve.

Fig. 3. Maximum bubble pressure measured in Exp. E (SiO₂- 36 mol% Na₂O-10 mol% NaF) at 1347 K, h: 2.89 mm, η: 4600 mPa·s. Plot: measured value. Line: fitting curve based on a single relaxation function treatment.

Fig. 4. Maximum bubble pressure measured in Exp. F (SiO₂- 40 mol% Na₂O- 0 mol% NaF) at 1348 K, h: 3.01 mm, η: 19000 mPa·s. Plot: measured value. Line: fitting curve based on a single relaxation function treatment.

Uncertainty of surface tension caused by the deviation of physical parameters used for the calculation of the surface tension is summarized in Table 3. The influence of uncertainty of gas pressure (differential pressure) on surface tension was evaluated by following equation.

  

$$u(\sigma )=\frac{r\mathrm{\hspace{0.17em} }u(P_{max,0})}{2}$$(7)

Uncertainty of measured value by differential pressure gauge is directly propagated in that of surface tension.

Table 3. Uncertainty of surface tension caused by the deviation of physical parameters used for the calculation of the surface tension.

No.	Parameter, a	Standard uncertainty, u(a)	Standard uncertainty, u(σ)
1	Gas pressure measured by differential pressure gauge	5.0 Pa	1.25 mN·m−1
2	Surace level determined by contacting tip of capillary	0.10 mm	0.49 mN·m−1
3	Immersion depth measured by digital level meter	0.01 mm	0.05 mN·m−1
4	Inner radius of capillary determined using pure water	0.005 mm	2.50 mN·m−1
The combined standard uncertainty uc is uc(σ) = 2.84 mN·m−1

For the influence of uncertainty of liquid level determination (reference point in depth), propagated uncertainty of P_max was calculated by following equation.

  

$$u(P_{max,0})=\rho g\mathrm{\hspace{0.17em} }u(h)$$(8)

Furthermore, uncertainty of surface tension was calculated using Eq. (7). Namely, uncertainty of liquid level determination is indirectly propagated in that of surface tension.

The influence of uncertainty of depth measurement was calculated using Eqs. (8) and (7). Uncertainty of depth measurement by digital level meter is indirectly propagated in that of surface tension.

The influence of uncertainty of inner radius of capillary was evaluated using following equation.

  

$$u(\sigma )=\frac{u(r)\mathrm{\hspace{0.17em} }{P}_{max,0}}{2}$$(9)

Uncertainty of inner radius of capillary is directly propagated in that of surface tension. Evaluation of deformation of inner contour from true circle is difficult. However, the inner radius was back-calculated with pure water by maximum bubble pressure method. It is considered that the value of inner radius includes the influence of deformation of inner contour.

The measured surface tensions for Series 1, SiO₂-40 mol% Na₂O and NaF, are shown in Fig. 5. The plots show the experimental values, and the lines represent the regression lines determined using the least-squares method. The surface tension of all the samples decreased with an increase in temperature, which is the general behavior of liquids. The surface tension increased with the increase of SiO₂–Na₂O concentration. The temperature dependence (slope of the line) slightly increased with increase of SiO₂–Na₂O concentration.

Fig. 5. Surface tension of samples in SiO₂–Na₂O–NaF system (Exp. A to F). Plot: measured value. Line: fitting curve based on least-square method. Uncertainty bars are omitted for visibility. See Table 2 for corresponding sample compositions.

Figure 6 shows plots of the surface tension at 1473 K extracted from Fig. 5. The surface tension of SiO₂-40 mol% Na₂O determined by Appen and Kayalova³³⁾ and that of NaF determined by Hara and Ogino³⁴⁾ are also shown. A value estimated using Butler’s equation,^{16,17,18,19,20)} where the partial molar excess energy term was set to zero (i.e., an ideal solution was assumed), is shown by the dotted line. The surface tensions of the pure components were estimated from values at higher temperatures²²⁾ by assuming a supercooled liquid. Surface tension of SiO₂-40 mol% Na₂O was 283 mN·m⁻¹, which was 96.7% of value determined by Appen and Kayalova. In particular, the present value was 3.3% lower than the literature value, indicating that the dynamic term in the MBP was removed in the high-viscosity composition region. The surface tension was significantly lower than that estimated for all the compositions. Surface tension did not significantly increase as expected in the beginning of addition of SiO₂-40 mol% Na₂O into NaF because F⁻ ion was exposed on a liquid surface as discussed later.

Fig. 6. Composition dependence of surface tension of samples in SiO₂–Na₂O–NaF system (Exp. A to F) at 1473 K. C_Na2O/C_SiO2 = 0.67. Literature values (Appen,²³⁾ Hara²⁴⁾) are also shown. Dotted line is drawn using the Butler’s equation based on an ideal solution assumption.

The surface tension of Series 2 when SiO₂ was added to NaF is shown in Fig. 7. The surface tension of all samples decreased with an increase in temperature, indicating a general behavior. Surface tension increased with increasing SiO₂ concentration. The temperature dependence did not exhibit a systematic change. This may be because a fully homogeneous liquid was not obtained because of the slow dissolution rate of SiO₂, although these mixtures form a homogeneous liquid in the equilibrium phase diagram.³⁵⁾

Fig. 7. Surface tension of samples in SiO₂–NaF system (Exp. A, G, H). Plot: measured value. Line: fitting curve based on least-square method. Uncertainty bars are omitted for visibility. See Table 2 for corresponding sample compositions.

Figure 8 shows plots of the surface tension at 1473 K extracted from Fig. 7. The estimated value is indicated by the dotted line. The surface tension was significantly lower than the estimated values. Surface tension did not significantly increase as expected in the beginning of addition of SiO₂ into NaF.

Fig. 8. Composition dependence of surface tension of samples in SiO₂–NaF system (Exp. A, G, H) at 1473 K. Dotted line is drawn using the Butler’s equation based on an ideal solution assumption.

For the surface tensions for Series 3, NaF in SiO₂-40 mol% NaF was replaced by Na₂O, as shown in Fig. 9. The surface tension of all samples decreased with an increase in temperature. However, the temperature dependence did not exhibit a systematic change. This may be owing to the inhomogeneity of the sample liquid; further investigation is required.

Fig. 9. Surface tension of samples in SiO₂–Na₂O–NaF system (Exp. F, H, I, J, K, L). Plot: measured value. Line: fitting curve based on least-square method. Uncertainty bars are omitted for visibility. See Table 2 for corresponding sample compositions.

Figure 10 shows plots of the surface tension at 1473 K extracted from Fig. 9. The estimated value is indicated by the dotted line. The surface tension was significantly lower than the estimated value.

Fig. 10. Composition dependence of surface tension of samples in SiO₂–Na₂O–NaF system (Exp. F, H, I, J, K, L) at 1473 K. Dotted line is drawn using the Butler’s equation based on an ideal solution assumption.

The surface tension of the melt did not change significantly when NaF was replaced with Na₂O at the beginning of the replacement, indicating that the particles exposed on the liquid surface did not change. It is known that O²⁻ ion is exposed on a liquid surface in oxide melts.^36,37) It is reasonable to consider that F⁻ ion is exposed on a liquid surface in fluoride melts. The surface of the melt is in a high-energy state and unstable compared to the bulk because the particles exposed on the surface lose a neighboring particle to be combined. The excess energy corresponds to surface tension.³⁷⁾

The Coulomb force F acting between the charged particles is given by the following equations:³⁸⁾

  

$$F=\frac{(Z^{+}{Z}^{-})e^2}{4\pi {\text{ϵ}}_0{a}^2}-\frac{nB}{a^{n+1}}$$(10)

  

$$I=\frac{Z^{+}{Z}^{-}}{a^2}$$(11)

  

$$a=r^{+}+r^{-}$$(12)

where Z⁺ is valence of cation, Z⁻ is valence of anion, e is charge of electron (C), ε₀ is permittivity of vacuum (F·m⁻¹), a is interionic distance (m), r⁺ is cationic radius, r⁻ is anionic radius, B is constant (N·mⁿ), n is constant (–), and I is Coulomb force index (m²). The literature values of the ionic radii in solid crystals (6 coordination)³⁹⁾ were used to calculate the Coulomb force index. The Coulomb force indices for some combinations of cations and anions used in the present study are listed in Table 4.

Table 4. Coulomb force index for some combinations of cation and anion related to the present study.

Combination No.	Cation	Cationic radiusa, r+/pm	Anion	Anionic radiusa, r−/pm	Interionic distance, a/pm	Coulomb force index, I/m−2
1	Si4+	54	O2−	126	180	2.47×1020
2	Si4+	54	F−	119	173	1.34×1020
3	Na+	116	O2−	126	242	3.42×1019
4	Na+	116	F−	119	235	1.81×1019

a:  Value under six coordination

Coulomb force index for Si⁴⁺ and O²⁻ is approximately twice of that for Si⁴⁺ and F⁻. This is because the ionic radii of O²⁻ and F⁻ are approximately the same but the valence of O²⁻ is twice of that of F⁻. Namely, the bonding force between Si⁴⁺ and O²⁻ is significantly larger than that between Si⁴⁺ and F⁻. Coulomb force index for Na⁺ and O²⁻ is also approximately twice of that for Na⁺ and F⁻. Consequently, the bonding force loss when O²⁻ was exposed on the surface was larger than that when F⁻ was exposed on the surface. Therefore, the excess energy is relatively small when F⁻ is exposed on a surface. The experimental result demonstrated that the surface tension did not change by replacing NaF with Na₂O at the beginning of the addition and was consistent with the above discussion. The increase in the surface tension upon further replacement of NaF with Na₂O was gradual, indicating that F⁻ was relatively exposed more than O²⁻ on the surface of the melt. This tendency is considered to be common in oxyfluoride melts.

5. Conclusions

A simple relaxation function was applied to determine the MBP when the gas flow-rate dependence of the MBP was analyzed; static MBP was obtained, and surface tension was reliably measured. The surface tensions of SiO₂–Na₂O–NaF melts were measured over a wide composition range using the developed method. When SiO₂-40 mol% Na₂O was added to NaF, the surface tension of the melts gradually increased with increase in SiO₂–Na₂O concentration. When the NaF in SiO₂-40 mol% NaF was replaced by Na₂O, the surface tension of the melts did not change significantly at the beginning of the addition. This was because F⁻ ion was exposed in the surface of melts instead of O²⁻ ion. This result is consistent with the discussion of the relative strength of the bonding force between the constituent particles. The increase in the surface tension upon further replacement of NaF with Na₂O was gradual, indicating that F⁻ was relatively more exposed than O²⁻ on the surface of the investigated melts.

Statement for Conflict of Interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.

Acknowledgment

This study was financially supported by a grant for fundamental research from the Advanced Research and Education Center for Steel (ARECS) at Tohoku University.

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