1. Introduction

Silica-based optical fibers for communication are produced by a two-step process. First, a “preform” with a core-cladding structure is fabricated, followed by the drawing of a light-guiding fiber from this preform. In the case of the widely-used modified chemical vapor deposition (MCVD) process, the preform is formed by the deposition of glassy films on the inside wall of a high-purity silica tube by gas phase reactions and the subsequent collapsing of the tube to a solid rod [1–3]. The collapsing process is normally realized on the MCVD lathe, immediately after the layer deposition, by increasing the temperature and decreasing the velocity of the burner in comparison to the values during deposition. Under the influence of the surface tension of the glass tube and a certain impact pressure of the flames, the rotating tube is collapsed to a solid rod during one or few burner passages with velocities between 1 and 5 cm min⁻¹. Sometimes a slight internal pressure is applied, but mostly no special precautions are necessary in order to ensure a high circularity of the resulting rod preform. In preform/fiber fabrication, the viscosity behavior of the cladding and core glass plays an important role both in the process performance, economy, and fiber properties.

For a long time, it has been well known that quartz glass types prepared by different methods show significantly different viscosities [4–7]. Apart from the many usual techniques available for measuring the viscosity of a liquid or glass, the fiber preparation process offers direct options for deriving viscosity data, either during fiber drawing [8] or in the course of tube collapsing [9,10]. These methods apply to the high temperature region between 1600°C and 2100°C (viscosity region of 10⁷ …10⁴ Pa·s) and have the advantage of not requiring a crucible and, therefore, preventing the contamination of the glass melt. This is a serious problem at the high temperature region of silica glass processing. Moreover, the viscosity is determined in an actual production situation. Effects which influence the viscosity can be elucidated.

Previously, we have shown that during the collapsing of quartz glass tubes with an oxygen-hydrogen burner in the MCVD process, the effective viscosity can deviate from the expected value derived from the measured temperature at the tube surface as a result of a temperature gradient between the outer and inner surface [10]. Based on the previous study, here we have investigated the viscosity of two different silica materials that are important for modern light-guide preparation. Both types of materials are from Heraeus Quarzglas GmbH [11]. The aim of this work is to demonstrate the principles of the method in detail and to determine both the true viscosity-temperature relationships and the thermal behavior of the tube materials under practical collapsing conditions. We investigated both F300 as a highly-pure silica and Fluosil as a fluorine-doped material. F300 is the most important silica material for the current optical fiber technology in general, and Fluosil becomes increasing important in the field of modern speciality fibers. Owing to the high fluorine content of Fluosil, a strong change of the viscous behavior is expected. This makes possible to test our method with two materials with significantly different properties. In the following, the study is extended to a series of different silica materials which are also of importance to this fiber technology [12].

2. The collapsing process of glass tubes

If we limit ourselves to small relative diameter changes, the collapsing process of glass tubes can be well described as radial viscous flow, resulting in the following collapse rate v_ρ [13,14]:

From Eq. (1) it follows that

Taking a certain cross section of the tube into consideration, Eq. (2) can be integrated over time, leading to a change in the wall thickness during collapsing. This can be described as follows

In the MCVD process, the collapse is carried out by a burner traversing the tube with a velocity v_b. The viscosity of the cross section of a tube varies according to the axial temperature profile of the burner (see the example in Fig. 1, which shows a typical tube diameter of about 20 mm). Because we investigate only the stationary state of collapsing, the axial temperature profile with a constant shape is moved across the axial point of the tube with a velocity v_b. In order to integrate Eq. (3) over a full burner pass, we can replace dt by -dz/v_b. Furthermore, if we specify the temperature dependence of η as an Arrhenius expression with an activation energy E and a pre-exponential factor η₀, the time-variable viscosity is reduced to the viscosity at the temperature of the profile maximum η_p and an effective profile width Δz_b [9].

 

Fig. 1 Axial temperature profile measured at the tube surface (z: axial coordinate, Δz_b: typical profile width, see the explanation in the text, v_b: burner velocity).

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The outer pressure p_o comprises the atmospheric pressure p_a and a certain impact pressure p_f which is generated by the flowing burner gases depending on the operating conditions of the burner. Replacing p_o by p_a + p_f and inserting Eq. (4) into Eq. (3), we obtain

Because the diameter change and thus the wall thickness change can be measured with high accuracy, the precise determination of the viscosity of the tube material is possible by controlled collapse experiments. For this purpose, we do not need the total collapsing of the tube to a solid rod, but a diameter change of only a small percentage (in general between 1% and 5%), which can be implemented either as diameter decrease (wall thickness increase) or diameter increase (wall thickness decrease) with a suitable pressure setting. On the basis of Eq. (5), both the viscosity at the temperature maximum η_p and the value of the combined influence of surface tension and flame pressure σ* can be determined by two collapsing experiments with changed inner pressure at otherwise constant conditions. The “small collapsing effect” can be adjusted by a suitable choice of the parameters as tube dimensions, burner velocity, pressure etc. in order to measure exact viscosity values between 1600°C and 2000°C, in the viscosity range between 10^4.5 Pa· s and 10⁷ Pa· s.

With Eq. (4), the profile parameter Δz_b can be calculated from the measured axial temperature profile according to

Some additional details concerning the integration of Eq. (1) and the evaluation of collapse experiments are described in [9]. If the viscosity is not constant in the radial direction (e.g., as result of the composition gradient of the tube material or as result of thermal effects as discussed below), the viscosity value which enters basic Eq. (1) is determined by its radius-dependent function η (ρ) according to

In fact, the one-dimensional approach of the collapsing process in Eq. (1) is a simplification. Even if the cross section of the tube always remains circular with high accuracy during the process, there is a change in the viscous flow velocity along the axial coordinate z as a result of the limited width of the heating zone. This effect has been studied numerically via the finite element method (FEM). In [15], comparative FEM calculations have been carried out with the COMSOL(R) Multiphysics 4.2a program package in labor coordinates and physical units, to simulate collapsing as advancing time-dependent process. We can conclude from these calculations that the one-dimensional approach yields a fairly good approximation for nearly all conditions used in our collapsing experiments. Nevertheless, the FEM calculations can be used for a precise correction of our viscosity values in the following manner.

The FEM calculations show that the collapsing velocity is, in general, smaller than expected by Eq. (1). This effect can be easily neglected for large axial heating zones (expressed by large Δz_b), but it has to be considered if the width of the temperature profile Δz_b becomes smaller than the outer diameter of the tube. In order to take this effect into account, we introduced a factor of F (F ≤ 1) into the right hand side of Eq. (1). Comprehensive FEM calculations [15] yield the value of the factor of F depending on the ratio of 2·r_o/Δz_b = α and the ratio r_i/r_o = β, which is the second parameter of importance for the deviation between the one-dimensional and two-dimensional approaches. In Table 1, the range of values of F, important for the experimental conditions, is recorded. This is valid for the “small” collapsing effect of less than 5% of the diameter. The derivations of Eqs. (1)-(5) reveal that the factor of F can be directly used as a correction factor of the viscosity values determined by the collapsing experiments on the basis of Eqs. (5) and (6)

Table 1. Correction factor F depending on α = 2·r_o/Δz_b and β = r_i/r_o for the practically relevant range [15]

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3. Experiments

Experiments were carried out with a series of tubes of different sizes and an outer diameter between 8 mm and 33 mm, wall thicknesses between 0.6 mm and 6 mm, and a uniform length of 500 mm, rotating on an MCVD lathe at 90 rotations per minute. The outer diameter of the tubes before and after the burner pass was automatically recorded with a laser scanner at several axial and angular positions (Laser-Scan-Micrometer LS-3101 (Keyence), 0.1μm resolution, 400 scans/s). In this way, the geometry of the tube and the correct stationary collapse setup could be controlled and the mean diameter used for evaluation. The accuracy of the diameter determination is 0.002 mm. The inner diameter was calculated from the outer diameter, the weight of the tube, and the density of silica glass. The oxy-hydrogen burner consisted of six separate jets, which semi-circularly surrounded the tube. It was operated with gas flows between 10 slm and 50 slm (”standard liter per minute”) of hydrogen and oxygen, respectively, in order to implement different temperatures between 1600°C and 2100°C and different profile widths Δz_b between 10 mm and 35 mm. Depending on the temperature and tube dimensions, the burner velocity v_b was set up to values of between 0.3 cm·min⁻¹ and 4 cm·min⁻¹ in order to carry out a collapsing effect of > 0.1 mm in the diameter change and to prevent non-stationary temperature distributions as a result of a burner velocity that is too high. To ensure the condition of radial flow, the diameter changes used for evaluation were restricted to a maximum of 5% of the outer tube diameter. The temperature of the tube at the outer surface was measured in the wavelength region of 4.8 μm … 5.6 μm with an infrared thermometer, which could be automatically shifted relative to the burner position to record the axial temperature profile. The inner pressure in the tube was set and regulated by a gas system without gas flow through the tube at an accuracy of about 2 Pa.

Two successive experiments were carried out under identical conditions (tube dimensions, burner gas amounts, temperature, burner velocity, etc.) with the exception of a change in the inner tube pressure. The value of p_a – p_i was set to zero in the first experiment, which led to a diameter decrease via collapsing. In the subsequent experiment, it was set to a negative value (p_i > p_a) in order to create a blow up in the tubes (i.e., an increase in the tube diameter), which results – approximately – in the starting tube dimensions. This procedure is not imperative, but it ensures a low influence of experimental errors on the determined viscosity. The value of p_i – p_a varied between 100 Pa and 500 Pa depending on the respective tube diameters.

In order to implement a defined temperature T_p of the axial temperature profile, two different operating regimes of the burner can be implemented. The first regime (“hydrogen excess”) works with a H₂:O₂ ratio of about 4-5:1 and the second regime (“oxygen excess”) with a ratio of roughly 1. The absolute oxygen flow ranges in the “hydrogen excess” regime between 5 slm and 15 slm and in the “oxygen excess” regime between 30 slm and 50 slm (i.e., much higher).

In order to estimate the magnitude of the flame pressure, a tube with a small orifice was used and the inner pressure was measured during tube rotation. In the pressure-time curves (not shown here), the six burner jets were resolved and an oscillating pressure increase of a few tens of Pa was able to be measured relative to the atmospheric pressure. It is mainly controlled by the oxygen flow, as expected. At most, the mean pressure over one turn can contribute to the collapse. (The effective pressure is most likely lower and the tube is slightly bent rather than collapsed because of the semicircular jet arrangement. This effect, however, is very small and can be neglected for the determination of the tube diameter by the laser scanner.) The surface pressure σ ∙ (r_o⁻¹ + r_i⁻¹) of the tubes ranges from 130 to 30 Pa for tubes with a diameter of 6 mm to 25 mm (with a surface tension of about 0.4 N·m⁻¹ (see below)). Thus, we expect that collapsing with “hydrogen excess flames” is mainly driven by the surface tension (especially for small tubes). To ensure a correct evaluation, in general two experiments were carried out under the same temperature conditions but with both operation regimes of the burner in order to derive the correct information concerning flame pressure and surface tension.

For the experiments, two types of silica glass were used. The synthetic silica tube material F300 is produced via the flame hydrolysis of SiCl₄ as a porous body, which is then dehydrated by gaseous chlorine. The content of metallic impurities (a few tens of ppb) and OH groups (below 1 ppm) is very low, but it contains about 0.16 wt% Cl. Fluosil is a fluorine-doped synthetic silica (3 wt% F, 0.01 wt% Cl), which also has a very low metallic and OH content. Both materials are from Heraeus Quarzglas GmbH [11].

4. Results

In Fig. 2, the determined values of σ* = σ + p_f /(r͞_o⁻¹ + r͞_i⁻¹) are recorded depending on the respective values of (r͞_o⁻¹ + r͞_i⁻¹)⁻¹. As expected, the values for “oxygen excess” flames are, in general, somewhat higher than for “hydrogen excess” flames, which expresses the influence of the flame pressure. With decreasing (r͞_o⁻¹ + r͞_i⁻¹)⁻¹, the values approach those for “hydrogen excess” flames. In fact, the flame pressure depends on the respective values of the oxygen flow, which must be considered in detail in order to derive exact values. The oxygen flow increases when using larger tubes. Thus, the decrease in (r͞_o⁻¹ + r͞_i⁻¹)⁻¹ runs in parallel with a decrease in oxygen flow. The extrapolated value for (r͞_o⁻¹ + r͞_i⁻¹)⁻¹ = 0 can be determined to σ* = 0.39 ± 0.03 N·m⁻¹ both for F300 and Fluosil. Moreover, a systematic influence of the temperature could not be observed (Fig. 2 refers to the same experiments which are shown in Fig. 3 with respect to the viscosity results). At first glance, the extrapolated value is believed to be the surface tension of the silica materials. A comparison with literature, however, shows a certain deviation from the reported values of about 0.3 N·m⁻¹ [16, 17]. A detailed investigation of this difference goes beyond the scope of our study. Note that the numerical values of surface tension and flame pressure are without consequence for the derived viscosity values because the effect of σ* is eliminated by two successive experiments with different p_i as described in paragraph 3.

 

Fig. 2 Determined values of σ* depending on the tube dimensions for flames with oxygen excess (•) and hydrogen excess ( + ) for (a) F300 and (b) Fluosil. The error of the single measurement is between ± 0.02 and ± 0,04 N m⁻¹.

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Fig. 3 Determined viscosity values η depending on temperature T for flames with oxygen excess (•) and hydrogen excess ( + ) for (a) F300 and (b) Fluosil. lnη is the natural logarithm, Φ_o the outer diameter and w the wall thickness of the tubes, respectively.

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The determined values of the viscosities η_p are displayed in Fig. 3(a) for F300 and in Fig. 3(b) for Fluosil. All values have been derived on the basis of Eqs. (5) and (6), and corrected according to Eq. (8). For simplification, η_p,corr and T_max are designated as η and T, respectively.

The error of the single measurement is primarily determined by uncertainties and small fluctuations in temperature, profile width, and pressure; other parameters such as Δw and v_b can be specified with high accuracy. The error limits of lnη can be estimated to ± 0.1 (this means a relative error of η of about 10%).

In all cases, the Arrhenius plots lnη against 1/T yield straight lines so that activation energies of the viscous flow can be derived, which show a remarkable difference between F300 and Fluosil. As expected, the experiments with “hydrogen excess” flames and “oxygen excess” flames yield completely agreeable results. This means that the influence of the flame pressure was correctly taken into account. However, we also observe remarkable differences between tubes of different dimensions. Even if the slope is nearly independent of the dimensions, the pre-exponential factor of lnη₀ changes with varying dimensions. At first glance, this seems surprising because the viscosity as material property should not depend on the tube dimensions. In [10], such a phenomenon was already reported and interpreted as a thermal effect (i.e., as a deviation of the mean tube temperature from the measured temperature at the tube surface). In the following, this effect will be discussed in more detail and a way will be presented to derive the true material viscosity of the tubes.

Initially, we will consider the empirical findings in particular. The activation energies E/R are 68000K and 48300K for F300 and Fluosil, respectively. (The influence of the tube dimension on E/R is within the experimental error limits and so small that we have given only the mean value for each material.) With this value, the pre-exponential factors lnη₀ for different tube dimensions have been derived from the curves in Fig. 3. In Fig. 4, all determined values of lnη₀ are plotted as a function of the respective values of the wall thickness w of the tubes. It is obvious that this plot represents a clear correlation in the form of a nearly linear function. In fact, there is a small deviation from the linearity, and the adjustment of a function x + y·w^1.25 provides a somewhat better correlation. The reason for using this specific relationship is discussed in the next paragraph. It is obvious that the difference between the measured temperature at the tube’s surface and the “inner” temperature should vanish if the tube thickness becomes virtually zero. Therefore, we can interpret the extrapolation to w = 0 as yielding the “true” material viscosity. This leads to the following expression for the viscosity of

 

Fig. 4 Pre-exponential factor lnη₀ acc. to Fig. 3 as a function of the wall thickness w for (a) F300 and (b) Fluosil. The colors refer to the experiments and tube dimensions as described in the inset of Fig. 3.

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The error of lnη, calculated from Eqs. (9) and (10) can be estimated to be ± 0.1 (according to the error order of the single measurement) if we ignore for the moment the ambiguity of the extrapolation in Fig. 4. The individual uncertainties of E/R and lgη₀, however, are essentially larger and can be roughly estimated to be ± 1000 K and ± 0.5, respectively. This is the result of the limited temperature range of the investigations. Bear in mind, however, that the errors of E/R and lnη₀ are interdependent in effect on the viscosity. (A change in the activation energy is largely compensated by a change in lnη₀.)

In order to investigate whether the axial width of the temperature profile is important for the observed effect, we have tested the correlation of the empirical viscosity values with the ratio of w/Δz_b. The results shown in Fig. 5 demonstrate evidently that there is not a clear correlation in contrast to Fig. 4. Note that a plot of lnη₀ depending on the radii (not shown here) definitely does not reveal a correlation. The same is true for the ratio of radii/profile width. Obviously, the wall thickness w is the crucial parameter for the variation of lnη₀. This empirical finding will be discussed in detail in paragraph 5.

 

Fig. 5 Pre-exponential factors lnη₀ acc. to Fig. 3 as a function of the ratio of wall thickness to profile width w/Δz_b.for (a) F300 and (b) Fluosil. The colors refer to the experiments and tube dimensions as described in the inset of Fig. 3.

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(Figs. 4 and 5 contain some additional results for F300 which are not displayed in Fig. 3 because of the varying tube dimension, which does not allow the setup of a complete Arrhenius plot. In these cases, the activation energy was set to the mean value for the other tube dimensions.)

5. Discussion

Above, it was shown that the viscous behavior of quartz glass tubes can be investigated via collapsing experiments on an MCVD lathe right up to the determination of defined viscosity values. This method, however, is obviously hampered by thermal effects which require in any case a set of experiments with different tubes and an extrapolation of the results to a tube wall thickness of zero. Therefore, this approach will not be very suitable as a simple standard method for glass viscosity measurements. However, it yields process-related data that is valuable for preform and fiber technology. Even the perturbing thermal phenomenon could yield valuable information (e.g., on the temperature in the tube interior, which cannot be directly measured). The main point of discussion here will, therefore, be the principal understanding of the thermal phenomenon discovered during the collapse experiments.

In order to explain the results summarized by Fig. 4 (the increase in the measured viscosity with increasing tube wall thickness), we expect a radial temperature gradient of about 40 K∙mm⁻¹ (as a rough estimation) in the hot zone of the tube (i.e., a temperature drop from the outside surface, where the temperature is measured, to the inner tube surface). It can be characterized by three features:

The temperature gradient implies a continuous heat flow from the outer to the inner tube surface of 8 W·cm⁻² (calculated with a heat conductivity of 2 W·K⁻¹·m⁻¹). Because the temperature of the outer surface is directly measured, only heat transport within the tube wall can account for the compensation of this radial flow. According to the scheme in Fig. 6, both heat conduction in the axial direction (into tube regions of lower temperature) and heat radiation from the hot tube zone (in all directions) will compensate the flow from the surface into the tube bulk. The convectional heat transfer to the gas at the inner surface is very small and can be virtually neglected.

 

Fig. 6 Scheme of the heat transport within the tube wall

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The steady-state energy balance in tubes with cylindrical symmetry in the presence of a distributed internal heat source is described by

First, we will solely consider the influence of heat conduction. In this case, a constant temperature gradient between the outer and inner tube surface (i.e., a constant heat flow from outside to inside) must be maintained by a continuous heat flow in the z direction, from the top of the axial temperature profile to the cooler regions (see Fig. 6). This is immediately understandable and can be calculated without complications by solution of Eq. (11), neglecting the term H(ρ,z), and with the boundary condition ∂T/∂ρ = 0 at ρ = r_i (no radial heat flow at the inner surface). The second boundary condition is simply provided by the measured temperature profile at ρ = r_o (see Fig. 1). The equation was numerically solved with the FEM method using experimentally measured temperature profiles T_o(z) with a variable width. A typical result (for a temperature profile width Δz_b = 1.3 cm) is included in Fig. 8. Within the limits of the applied experimental conditions, there is a decrease in the inner temperature T_i down to about −30 K below T_o for extreme conditions (large w, small Δz_b); however, the gradient is in the majority of cases smaller than 5 K·mm⁻¹ (i.e., much smaller than the observed value of about 40 K·mm⁻¹). This means that the heat conduction in the z direction cannot explain by far the main effect of the temperature drop. This result is in agreement with the experimental findings that the measured viscosity correlates with the wall thickness alone rather than with the temperature profile width (see Figs. 4 and 5). Note that the equation used is independent of the value of the heat conductivity κ; in fact, it does not contain any material parameters at all. The solution depends only on the tube dimensions and the temperature profile T_o(z).

Obviously, the existence of the temperature drop must be primarily explained on the basis of radiation effects. In contrast to the axial conduction effect, which is closely related to the profile width, the radiation effect on the radial temperature distribution is not reduced to zero for the infinite profile width. In further considerations, we will be neglecting the small contribution of the axial conduction and concentrating on the interaction between radial conduction and radiation (i.e., we will omit the term ∂²T/∂z² in Eq. (11) and consider H as a function of ρ only). The radiation term H(ρ) comprises the emission of radiation from a considered volume element (heat loss) in concurrence with the absorption of radiation emitted from all other volume elements, diminished by absorption on the way, paying attention not only to direct rays but also to reflected rays (heat gain). The main difficulty here is that the radiation term, in turn, depends on the temperature distribution, which has yet to be determined [18]. (Eq. (11) can in principle be transformed into complex nonlinear integro-differential equations with integration over the spatial, spectral, and directional domains [19–22].) Because our aim is an order-of-magnitude estimation, we can greatly simplify the calculation by considering the radiation term H in the first place as a function of the mean temperature in the considered tube region where collapsing proceeds, which is designated in the following as the “effective” temperature. (Afterwards, this “effective” value could be changed depending on the calculated temperature drop in radial direction in order to control its influence on the solution in an iterative manner.)

In our first approximation, we can assume that the absorption coefficient is a constant. Then H(ρ) is represented by

The integration is taken to the body boundaries for rays without total reflection and to infinite for rays which are totally reflected. In the effect, only rays which leave the medium contribute to the radiation term H (the Fresnel reflection is neglected). Because of our assumed constancy of H_e and α, H is only dependent on the radius coordinate. (Our approximation yields the radiation influence for a very large heat zone width.) The value of H_e can be specified with the condition that for an infinite thick body the total energy loss must reach the black body limit.

If we consider for the moment only rays in ρ direction, Eq. (12) can be specified to

 

Fig. 7 T_o – T_i for radiation effects, calculated with Eq. (14) for parallel radiation ––– and for diffuse radiation in a two-plate approximation •

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In reality, we have to consider not only radial rays (simply referred to as “parallel radiation”) but all directions (i.e., the diffuse radiation within the total solid angle). Integration can be carried out similar to the way it was demonstrated earlier for estimation of temperature distributions in glass melts [23, 24]. Because the wall thickness of the tubes is, in general, much smaller than the diameter, we used a two-plate geometry for the calculation. Also in this case, H_e is specified by comparison with the black body radiation. The derived expressions are more complicated than Eq. (13) and (14), but the calculated temperature distribution is nearly identical to the result on the basis of parallel radiation. Due to lack of space, we have not provided details of the calculation but only the result as a comparison between parallel radiation and diffuse radiation in Fig. 7. All further calculations can be implemented for simplification with Eq. (14) as well.

Equation (14) contains the product of K and α and depends on the wavelength of the thermal radiation. K is the usual expression of the Planck formula – the power per area, per solid angle, and per wavelength with a maximum at about 1.5 μm at 2000 K. The temperature difference T_o – T_i (see Fig. 7) can be neglected for λ < 2 μm (K is large, but α is very small) and for λ > 4.5 μm (K is small, and α is very large), it is caused by wavelength regions where the quartz glass is semitransparent. Because the absorption coefficient of quartz glass depends on the temperature, we used α(λ) functions for the integration over the wavelength, determined by [25,26] for water-free silica (“KI” in [25]) at 1600 K, in connection with a value of 2 W·K⁻¹·m⁻¹ for κ, which is believed to represent the pure phonon conductivity of quartz glass [27,28]. The IR spectrum of fluorine-doped silica largely corresponds to that of pure silica (only a small additional band occurs at a wavelength of 10.6 μm [29,30]). Therefore, we used the same α(λ) function for both tube materials for the calculation of the temperature difference. Subsequently, the temperature distribution (equal for both materials) has been used to calculate the viscosity distribution of F300 with Eq. (9) and of Fluosil with Eq. (10) and the respective mean viscosities and their increase with the increasing wall thickness of the tube according to Eq. (7). The results of the calculation are displayed in Fig. 8. It is obvious by comparison with Fig. 4 that our estimation yields an order-of-magnitude agreement with the experimental results. Moreover, the measured difference between F300 and Fluosil is clearly reflected by the calculations. This effect is the result of the remarkable lower activation energy of the viscosity of Fluosil, which leads to a remarkably smaller viscosity increase even at an equal temperature difference.

 

Fig. 8 Viscosity increase (──) and temperature drop (- - - -) with increased wall thickness w, calculated on the basis of radiation effects for F300 and Fluosil with an effective temperature of 1800°C. For comparison: ……. T_o-T_i, calculated for axial heat conduction without radiation contribution (Δz_b = 1.3 cm).

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Note that the temperature drop T_o – T_i depends on the effective temperature as expected, a change of ± 200 K (i.e., 2000°C and 1600°C) leads to a variation in T_o – T_i of about ± 25% for a wall thickness of 6 mm. The influence on the calculated viscosity increase is, however, smaller; it amounts to about ± 5%, so that the viscosity curves in Fig. 8 are well represented for an effective temperature region between 1600 K and 2000 K. This covers the full range of experiments, except the Fluosil tubes with a wall thickness of 1 mm, where the viscosity increase is still insignificant.

In the conclusion, our estimations support that the measured viscosity increase with increasing wall thickness is a result of an increasing temperature drop between the outer and inner surface. The main reason for this drop is the thermal radiation in regions where the quartz glass is semitransparent. The calculated increase in the viscosity with the wall thickness follows nearly a w^1.25 dependence which was used for the extrapolation to a wall thickness of zero in paragraph 4, leading to a slightly improved extrapolation value. Even if the estimation is rather simplified, it yields a principal understanding of the observed effect and information on the parameters of importance.

The estimations further support our assumption that the real viscosity can be determined by the extrapolation of measured values for different wall thicknesses to the thickness of zero. When we compare our determined viscosity relationship in Eq. (9) with present literature, we find good agreement in general. A comprehensive survey of different viscosity values is provided by Nascimento [16]. Our viscosity-temperature relationship in Eq. (9) fits well with the literature results for silica types I. With 565 kJ∙ mol⁻¹, the activation energy E, which is derived from Eq. (9), is in close agreement with the highest reported data of 570 ± 5 kJ∙ mol⁻¹. This can be expected because of the very high purity of F300 and the crucible-free method which prevents contaminations during the viscosity measurement. According to the classification of Brückner [31], F300 is rather a Type III b glass. The activation energy of the viscosity of type III glasses, cited in [7] is with 450 kJ∙ mol⁻¹ remarkably lower than the value determined for F300. This is an obvious consequence of the high hydroxyl content of the glasses reported in [7] (i.e., the reported data is type III a). A literature value for F300 or for a comparable silica is currently not available to our knowledge. For Fluosil, our investigation of Eq. (10) yielded an activation energy of 402 kJ∙mol⁻¹ (i.e., a much lower value than for F300, as a result of the fluorine content). The decrease in the activation energy with increasing fluorine content is, in principal, in agreement with the literature data [30]. A detailed discussion of the viscosity dependence on the silica types and composition, however, is outside the scope of this paper and will be done together with additional investigations of a series of different fluorine-doped high-silica glasses in a following paper [12].

6. Conclusions

In conclusion, we have demonstrated that collapsing experiments can be successfully applied for the viscosity measurement of glass tubes in the high temperature region (i.e., for viscosity values between 10⁷ Pa∙ s and 10^4.5 Pa∙ s). The method which, in principle, has been known for a long period of time, was theoretically established here in detail and practically demonstrated using the example of two synthetic silica materials which are important for modern fiber technology. This approach has the advantage of not requiring a crucible and, therefore, preventing the contamination of the glass melt, which is a serious problem at the high temperatures of silica glass processing. For the exact determination of real viscosity values, however, a set of experiments with tubes of different wall thicknesses is necessary. This requirement is due to the fact that a thermal effect leads to a deviation between the measured temperature at the tube surface and the temperature within the tube bulk, which is responsible for the measured viscosity value. It could be shown that the reason for this thermal effect is the radiation loss in regions where the quartz glass is semitransparent (i.e., in a wavelength range between 2 μm and 4.5 μm). It could be further shown that the extrapolation to a wall thickness of zero is a well-suited procedure for derivation of correct viscosity-temperature relationships. Even if the demonstrated method seems to be more difficult than classical methods such as rotating crucible and falling ball, it yields process-related data and, beyond that, valuable information to the temperature in the tube interior, which cannot be directly measured but is very important for preform and fiber technology.