1. Introduction

Microstructured optical fibres (MOFs) are distinguished from solid optical fibres by the cross-sectional structure running along their length. The design of this cross-sectional structure gives the fibre certain optical and physical properties which are desirous in a range of applications (see, for instance, [1]). MOFs are fabricated by slowly feeding a preform of suitable geometry (typically 1–3 cm in diameter) into a heated region within a furnace and then stretching the softened glass to the dimensions of a fibre (typical outer diameters of 120–250 $\mu$m and internal channel diameters of the order of the wavelength of light). An operational challenge is to control the cross-sectional hole size in the fibre, and to this end mathematical models are used to predict how the preform geometry deforms as it is drawn.

We previously reported experiments on tubular fibre drawing over a range of operational conditions [2]. Using the ratio of inner to outer diameter, denoted $\rho$, to characterize the geometry of the tube, preforms of different diameter ratios $\rho _0$ were drawn to fibre, and the diameter ratio $\rho _L$ of the final fibre was measured, as draw speed, furnace temperature, and active pressurisation were each systematically varied. Although the dimensions of the resulting fibres were generally in agreement with recent models of the drawing process [3,4], in some cases, for which there was no active pressurisation of the channel, $\rho _L$ was significantly larger than the model’s prediction. This unexpected oversized channel phenomenon was more severe at higher draw speed, at higher pulling tension and for preforms with a relatively large diameter ratio $\rho _0$. An example is ‘Experiment 4’ of [2]; for ease of reference the results for this experiment are reproduced in Fig. 1. In this case $\rho _L$ was close to or greater than $\rho _0$ for all values of the pulling tension, although an oversized channel is simply defined to be a fibre diameter ratio exceeding the corresponding model prediction. Similar examples where, without any applied pressurisation, the relative size of channels in a fibre were larger than those in the corresponding preform, have previously been seen in the drawing of a seven-hole fluoride glass fibre [5] and a low-mode tellurite fibre [6]; see also the six-hole F2 glass preform shown in [2, Fig. 8].

 

Fig. 1. Comparison between experimental measurements (crosses with error bars) and model output (circles) for ‘Experiment 4’ from [2]. Plotted against pulling tension are (a) the fibre outer diameter and (b) $\rho _L$, the diameter ratio of the fibre, with the diameter ratio of the preform, $\rho _0=0.514$, shown by the dashed line. No active channel pressurisation was used and the channel was open to the atmosphere. The preform outer diameter was 10 mm, the feed speed was 1.4 mm/min and the draw speed was 5.9 m/min.

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We posited in [2] that the discrepancies between the model and the experiments might be due to an induced pressure in the channel and used a model for actively pressurised fibre drawing [4] to estimate the magnitude of this extra pressure. Voyce et al. [7] reported a modelling and experimental investigation of the drawing of a tubular preform with sealed ends, which prevents the escape of air and, so, deliberately induces a pressure in the air channel. The effect of sealing was termed ‘self-pressurisation’ and the mathematical model assumed a spatially constant pressure and negligible surface tension. That study showed excellent agreement between model and experiment; the observed diameter of the inner channel was slightly smaller than that predicted by the model, which is to be expected from the model’s neglect of surface tension. Although the experiments in [2] were on unsealed tubes, the drawn fibres with the most severely oversized channels had a diameter ratio close to that of the preform. This suggested self-pressurisation because of the rapid reduction of the size of the channel cross-section through the neck-down during a draw or due to inadvertent sealing of the tube by some unidentified part of the apparatus.

The study of tubular fibres detailed in [2] is part of a larger project to investigate the fabrication of MOFs with the techniques of mathematical modelling [2–4,8–11]; this work demonstrates that measurement of pulling tension obviates the need to explicitly specify the viscosity profile along the neck-down region, and enables the modelling of MOF drawing for a sufficiently slender preform with any cross-sectional geometry (not just the capillaries in this study). Previous, less general, modelling of fibre drawing which did not have these features includes studies on drawing tubular preforms to fibre [12], on drawing MOFs which feature channels separated by thin struts of glass [13] and validating the pressurised drawing of a six-hole preform against finite element simulations [14]. Other relevant modelling studies of related processes include modelling of solid fibre drawing with temperature dependence [15] and the drawing of thin glass sheets [16]. None of these previous studies have noted or addressed the phenomenon of oversized channels.

In this paper we provide detailed information on the oversized channel phenomenon for informing future modelling. In particular, we report on our investigations into self-pressurisation as the mechanism behind this phenomenon using tubular preforms with both small and large diameter ratio. In Section 2 we first briefly discuss the effects of surface tension and pressure in fibre drawing and then develop a new model applicable to drawing of sealed tubular fibres with channel pressurisation induced by sealing, which differs from that of [7] in including surface tension. In Section 3 we compare, using experiments and the models, fibre drawing of an unsealed tubular preform with that of an identical preform with a sealed end. In Section 4 we give more detailed information on the evolution of the cross-sectional shape over the neck-down region in the case of an oversized channel in the fibre, obtained from the cooled portion of the preform remaining after completion of ‘Experiment 4’ of [2]. Finally, in Section 5, we present our conclusions and discuss the direction of future modelling.

2. Effects of surface tension and pressure in fibre drawing

In the laboratory reference frame, fibre drawing is essentially a steady-state process and mass conservation requires that the ratio of the cross-sectional area $S_0$ at the start of the neck-down region ($x=0$) to the cross-sectional area $S_L$ at the end of the neck-down region ($x=L$) must equal the ratio of the draw speed $U_{\textrm {draw}}$ to the feed speed $U_{\textrm {feed}}$ (the draw ratio), i.e. $S_0/S_L= U_{\textrm {draw}}/U_{\textrm {feed}}$. In the absence of both surface tension and pressurisation of internal air channels, and where the cross-sectional length scale is much smaller than the neck-down length $L$, mathematical modelling has shown that the cross-sectional geometry of the fibre will differ from that of the preform only in size; however, relative to this case, surface tension will act to reduce the size of holes in the cross-section of the fibre while an internal pressure higher than atmospheric pressure will expand them.

For an axisymmetric tube with initial diameter ratio $\rho _0$ drawn to a fibre with diameter ratio $\rho _L$, this means $\rho _L=\rho _0$ in the absence of both channel pressurisation and surface tension, while $\rho _L<\rho _0$ for non-zero surface tension but no channel pressurisation, and $\rho _L>\rho _0$ for non-zero channel pressurisation but no surface tension. In the case of non-negligible surface tension and pressurisation, the magnitude of $\rho _L$ relative to $\rho _0$ will depend on the relative strengths of the surface tension and the pressure. Let $\gamma$ be the surface tension coefficient of the glass and $p_H$ be the over-pressure above atmospheric pressure in the channel, where both are assumed to be constant. We also define

We now consider what happens when the pressure $p_H$ is due to self-pressurisation because the ends of the tube are sealed so there can be no flux of air into or out of the tube. Following [7], we assume the air to be an ideal gas and the pressure $p_H$ to be spatially uniform throughout the channel. Then, at any two positions $x_1$ and $x_2$ along the length of the preform–fibre, the air temperature $T$ and density $\delta$ must satisfy (using subscripts to denote the position) $\delta _1/\delta _2=T_2/T_1$. Moreover, our steady-state approximation requires that, as with the glass, the air flux be the same at every position $x$. Now, under steady-state conditions, sufficiently far above the neck-down region the air temperature and density are constant values, $T_{\textrm {top}}$, $\delta _{\textrm {top}}$, respectively, and the air, necessarily, moves with the glass because the tube is sealed. Similarly, sufficiently far below the neck-down region, where the tube has the dimensions of the final fibre and has cooled completely, we assume that the air temperature and density attain constant values $T_{\textrm {bot}}$, $\delta _{\textrm {bot}}$ and, again, the air must move with the glass because the tube is sealed. Then, for a preform and fibre with external radii $R_0$ and $R_L$, respectively, $\pi R_0^{2}\rho _0^{2}\delta _{\textrm {top}}U_{\textrm {feed}}=\pi R_L^{2}\rho _L^{2}\delta _{\textrm {bot}}U_{\textrm {draw}}$. Noting that area $S=\pi R^{2}(1-\rho ^{2})$ for a cross-section of external radius $R$ and diameter ratio $\rho$, we use the relationship between the draw ratio and the cross-sectional areas of the preform and fibre to obtain, after some manipulation,

Since the top of the preform above the neck-down region must be at least as hot as the room-temperature fibre far below the neck-down region, $T_{\textrm {top}}/T_{\textrm {bot}}=\delta _{\textrm {bot}}/\delta _{\textrm {top}}\ge 1$ so that, from Eq. (3), $\rho _L\le \rho _0$. Therefore, assuming the validity of the assumptions of the self-pressurisation model, in a sealed tube the diameter ratio of the fibre will be no larger than that of the preform. From the discussion above we know that either the (self) pressure $p_H$ will be sufficiently small relative to surface tension that $\rho$ will decrease for all $x\in [0,L]$, or $p_H$ will be sufficiently large that $\rho$ will increase with $x$ for $0\le x<x_C$ for some $x_C<L$, from which point $S(x)$ will have reduced sufficiently that surface tension will dominate pressure and $\rho$ will reduce for $x_C<x\le L$. Either way the outcome will be $\rho _L\le \rho _0$.

Because we expect the temperature at the top of the preform and that of the drawn fibre to be (nearly) equal for much of the fibre draw when the process is essentially steady, we will plot model results for self-pressurisation assuming $T_{\textrm {top}}/T_{\textrm {bot}}= 1$, yielding the prediction $\rho _L=\rho _0$. However, all experimental results satisfying $\rho _L\le \rho _0$ will be deemed consistent with this model.

3. Comparison between an unsealed and sealed tube

In [2] we reported on six experiments of drawing unsealed tubular preforms to fibre. The first two of these experiments (labelled ‘Experiment 1’ and ‘Experiment 2’ in [2]) involved tubes of 10 mm outer diameter and 1.6 mm inner diameter, with fixed feed and draw speeds ($U_{\mathrm {feed}}=1.4$ mm/min, $U_{\mathrm {draw}}=5.8$ m/min) and a furnace temperature ($T_{\mathrm {furnace}}$) which was varied between 940$^{\circ }$C and 880$^{\circ }$C. The two experiments exhibited slightly different behaviour, and the second experiment closely matched the model predictions of [3]. The discrepancy between these experiments motivates a third replication of this experiment in the present study.

For this third repetition an extruded F2-glass tube of 10 mm outer diameter and 1.7 mm inner diameter was drawn to fibre with the operational parameters stated above; we denote the ratio between the inner and outer diameters of the preform $\rho _0=0.17$. The surface tension parameter for F2 glass is $\gamma =0.23\,$Nm$^{-1}$ [17] (which is assumed to be constant over the small temperature range used in these experiments) and the neck down length was $L=0.03$ m. We apply the fibre drawing model for annular tubes from [3] to this new experiment, and then compare the diameter ratio of the fibre yielded by the model to measurements of the drawn fibre. Fig. 2(a) shows the fibre diameter ratio $\rho _L$ of the fibre against pulling tension as given by both experiment and model, where pulling tension increases as furnace temperature is lowered. There is close agreement between the model and the experiments across all four pulling tensions, corroborating the results of ‘Experiment 2’ from [2] which similarly matched the model. ‘Experiment 1’ from [2], which showed inexplicably different results, is henceforth discarded as erroneous.

 

Fig. 2. Comparison between experimental measurements (crosses with error bars) and model output (circles) for the preform with diameter ratio $\rho _0= 0.17$. Plotted against the pulling tension is the fibre diameter ratio $\rho _L$ for the (a) unsealed preform and (b) sealed preform, with $\rho _0$ shown by the dashed line.

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To establish how the self-pressurisation effect described in Section 2 manifests under operational conditions similar to those used in the experiments described above, a sealed tube was also drawn to a fibre. The preform was nearly identical to that used in the unsealed experiment and, in fact, both these preforms were short sections (around 17 cm long) of one longer extrusion. In this case, the top end was sealed by filling the first few millimetres of the channel with epoxy resin, while the fibre was drawn from the lower end (it being assumed that the diameter of the channel through the drawn fibre was so small that the preform was effectively sealed at the bottom). The operational parameters were identical to those used in the unsealed experiment. The diameter ratio for the sealed-tube draw is shown in Fig. 2(b). Both model and experiment agree in showing this to be essentially independent of pulling tension with the experiment showing $\rho _L$ to be a little smaller than $\rho _0$ which is consistent with the model.

Noting that the most severe cases of oversized channels in [2] correspond to the experiments on preforms with large diameter ratio (see Fig. 1 above where $\rho _L>\rho _0$), an experiment similar to those just described was also performed for both unsealed and sealed preforms with large diameter ratio. Specifically, each preform had nominal outer and inner diameters of 10 mm and 6 mm, respectively, such that $\rho _0= 0.61$. Fig. 3 compares the experimental measurements with the relevant model for the unsealed and sealed preforms, respectively. In both cases we see a large difference between the model and experiment with $\rho _L$ exceeding $\rho _0$ by a significant amount which is not consistent with either model. Moreover, there is little difference between the sealed and unsealed tubes. A comparison of the results in Fig. 3 for the unsealed preform with Fig. 1 also shows the difference between model and experiment to be larger for larger diameter ratio.

 

Fig. 3. Comparison between experimental measurements (crosses with error bars) and model output (circles) for the preform with diameter ratio $\rho _0= 0.61$. Plotted against pulling tension is the fibre diameter ratio $\rho _L$ for the (a) unsealed preform and (b) sealed preform, with $\rho _0$ shown by the dashed line.

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In summary, the experimental results for the preforms with small diameter ratio $\rho _0=0.17$ agree with the models, showing a marked difference between the sealed tube, where $\rho _L$ is just a little less than $\rho _0$ and is independent of pulling tension, and the unsealed tube, where $\rho _L$ is significantly less than $\rho _0$ and changes with pulling tension. For the preforms with large diameter ratio $\rho _0=0.61$ there is poor agreement with the models and a similar outcome for both sealed and unsealed preforms, namely oversized holes with $\rho _L$ substantially larger than $\rho _0$ and little dependence on pulling tension. This suggests that the phenomenon of oversized holes is not simply explained by self-pressurisation of the channel caused by the rapid reduction in the size of the channel or inadvertent sealing by some unidentified part of the draw apparatus.

4. Investigation of geometry through a tubular neck-down

Although the preform neck-down remaining at the conclusion of a fibre drawing experiment may deform slightly as it cools, it is likely that the geometry of the cooled neck-down is representative of the geometry in the neck-down during fibre drawing. Assuming this we examined the exact shape along the length of a cooled neck-down to assist with identification of the mechanism leading to the phenomenon of oversized channels. Since ‘Experiment 4’ of [2], which used an unsealed preform with diameter ratio $\rho _0=0.514$, showed a pronounced oversizing of the diameter ratio relative to the model prediction (see Fig. 1), we selected the remainder of this preform for detailed examination.

This neck-down was sectioned into 1 mm cross-sectional slices. First the neck-down was encased in epoxy resin and then sliced from the end with largest diameter using a high precision glass saw. The newly cut end was polished after each slice was cut off. Measurements were made of the inner and outer diameters of the polished side of each slice, and their ratio $\rho$ was computed. These measurements are shown in Figs. 4(a) and (b), where $x=0$ corresponds to the first slice. The sectioned length was around 15 mm of the neck-down, which had a total length of around 30 mm. Part of the extremely fragile fibre end had snapped off prior to sectioning. The gap in the data at 10 mm is where a slice cracked during this delicate sectioning process.

 

Fig. 4. Geometry through the upper 15 mm of the cooled neck-down region remaining after ‘Experiment 4’ of [2], where $x=0$ mm corresponds to the first slice at the end with largest diameter and $x=15$ mm corresponds to the last slice. (a) Physical external and internal diameters, and (b) the inner to outer diameter ratio $\rho$. The value of $\rho$ at $x=0$ compares with the diameter ratio of the initial preform, $\rho _0=0.514$.

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10.1063/1.3002555The results for the cross-sectional measurements of $\rho$, shown in Fig. 4(b), display some unexpected behaviour. For the initial $5$ mm of the neck-down $\rho$ is within measurement error of the preform. Beyond this $\rho$ decreases to $0.34$ at $x=13$ mm and then increases sharply to $\rho =0.46$ at the last measurement position, $x=15$ mm. The fibre that corresponds most closely to this neck-down was that obtained at the end of the experiment using the final pulling tension value of 30 g (see results in Fig. 1(b) above and [2] for details on the experimental procedure) having $\rho _L=0.546$, which is larger than the diameter ratio of 0.46 for the last measured point of the neck-down. This implies that $\rho$ continued to increase beyond the last measurement position. This behaviour is surprising and not explained by any model of fibre drawing developed to date. As discussed in Section 2, the model of [4] shows that in the absence of internal pressure, $\rho$ must decrease along the neck-down due to surface tension. On the other hand, with pressurisation of the internal channel, either surface tension dominates pressure for all time, and $\rho$ must decrease with distance along the neck-down, or pressure initially dominates surface tension, and $\rho$ will initially increase and then decrease when the cross-sectional area reduces (due to stretching) sufficiently that surface tension becomes dominant. Neither of these behaviours correspond to that shown in Fig. 4(b), where $\rho$ decreases before a sharp increase as the neck-down radius approaches fibre dimensions. Thus, this analysis adds to the evidence that self-pressurisation is not the mechanism for oversized air channels in fibres.

5. Conclusions

We have developed a new steady-state model for drawing of a sealed preform, where both an induced pressure in the channel due to sealing and surface tension are important, and have performed several targeted experiments to explore self-pressurisation as the mechanism behind the phenomenon of oversized channels seen in the drawing of some preforms to fibre. Relevant models together with experiments using tubular preforms of small diameter ratio, agree and show the behaviour of sealed and unsealed tubes to be markedly different. For the former self-pressurisation results in a fibre with diameter ratio comparable to, and not larger than, that of the preform; for the latter the fibre diameter ratio is significantly smaller than that of the preform. New experiments on unsealed and sealed preforms with larger diameter ratio, yielded similar fibres with diameter ratio significantly larger than the preform in both cases. In neither case did the outcome correspond with the relevant model. This shows that oversized channels are not explained by self-pressurisation. The sectioning of a cooled neck-down from an experiment that yielded a fibre with a significantly oversized channel also revealed a profile through the neck-down that is inconsistent with inflation due to a uniform pressure in the channel arising from self-pressurisation.

Although eliminating self-pressurisation as the mechanism for oversized channels in fibres drawn from unsealed preforms, our experiments also suggest an alternative cause that warrants future investigation. Noting that oversized channels depend on both preform geometry and furnace temperature which controls pulling tension, and that the neck-down sectioning revealed expansion immediately after the region of rapid change in the diameter, hence viscosity, we will investigate 3D effects due to a sharp neck-down as the cause of apparently oversized geometry and the influence on this of temperature/viscosity gradient through the neck-down.