Billet extrusion is a powerful technique for fabricating soft glass optical fiber preforms. This paper reports progress in the understanding of the relationships between extrusion process parameters and the die geometry. The friction for glass flow within the die is described by a die constant that can be either calculated using die feature dimensions or determined using processing parameters and a glass with known temperature-viscosity behavior. In complex dies in which the glass flows through an array of feed holes the friction can be calculated from the number, length and diameter of the individual channels within the die. The glass flow analysis allows improvement of the extrusion process and guidance of future die design.
©2012 Optical Society of America
Billet extrusion is a powerful technique for the fabrication of structured preforms that are used to produce soft glass and polymer microstructured optical fibers (MOFs) [1–13]. This class of fibers contains an array of air holes that runs along the entire fiber length. The size, shape and arrangement of these holes determine the optical properties that can be achieved with this versatile fiber type . In addition, extrusion has been proven to be a viable technique for the fabrication of preforms for soft glass step-index fibers [14–17] and all-solid microstructured fibers . To fabricate fibers with small core diameter (~1µm) while maintaining practical outer diameter (~100µm), the preforms is first reduced in outer diameter to a few millimeters, then inserted into a jacket tube and finally this assembly is drawn down to fiber, one or more times, depending on the required scale-down ratio. Extrusion has been widely used to fabricate the tubes required for this multi-step drawing process [2–8,10,11].
In the billet extrusion technique, bulk billets are heated up to a temperature where the glass is sufficiently soft to be forced through a die structure by a ram to form an extrudate, which has a transverse profile with millimeter-scale features that is determined by the die exit geometry (Fig. 1 ) . One benefit of the extrusion technique is that it can be employed for a wide range of soft glasses. To date structured preforms have been extruded using different types of lead-silicate [3,4,11], bismuth [5,6], tellurite [2,7–10,12] and fluoride  glasses. In addition, silicate, chalcogenide, fluoride and phosphate glasses have been extruded into simple structures such as rods and tubes [14–17,19]. Another advantage is the virtually unlimited flexibility in size, shape and arrangement of holes that can be achieved in a preform . Of particular importance is the capacity to produce highly porous structures. For example, an extruded preform with four rings of segment-shape, radially aligned holes was demonstrated [20,21]. In addition, extrusion allows the large number of features to be created simultaneously in a single step and in a highly automated and therefore reproducible process. Extrusion also enables long preform lengths, only limited by the billet volume that can be provided. Furthermore, extruded preforms exhibit a fire-polished surface due to cooling of the soft glass in free space (i.e. without being in contact with a liquid or solid).
The key to extrusion of preforms with desired structure and excellent optical quality (i.e. without of scattering centers and with good finish for both internal and external surfaces) is the optimization of the extrusion processing parameters (glass temperature, ram pressure and ram speed) and the die design to prevent detrimental effects such as die breakage and enhanced preform surface roughness at high pressures, glass crystallization and preform thinning at high temperatures. In this paper, we report on analysis of glass flow during extrusion, leading to a more quantitative understanding of the relationships between processing parameters and die geometry. This forms the basis for improved control of the extrusion process and die design, which ultimately paves the way towards a larger variety of preforms with desired structure and enhanced optical quality.
2. Experimental conditions
This section describes the experimental conditions of the billet extrusion trials we performed and for which we analyzed the glass flow. We commence by detailing the setup and processing parameters used for the extrusion trials. Then we describe the die features that are of relevance for the glass flow analysis. Since the temperature is measured during extrusion but the viscosity of a glass is required for the flow analysis, we depict the determination of the temperature dependence of the viscosities of the glasses studied here. Finally, we detail the experimental errors of the extrusion trials.
2.1 Extrusion setup and processing parameters
A sketch of the assembly of the glass billet and extrusion die assembly within the extrusion container is shown in Fig. 1. This assembly is located within a resistance heating furnace, with the die located in the middle of the heated region, the billet above the die and the space ~15cm below the die exit remaining within the hot zone of the furnace. The die temperature is measured via thermocouples placed in contact with the outside of the die near the die exit. As any thermocouple located within the die region would be in contact with the glass and obstructing the glass flow, the actual glass temperature experienced by the glass and the temperature gradient within the die cannot be measured during extrusion without flow interruption or glass contamination. Thus the die temperature is assumed to be equal to the temperature of the glass within the die channel. The same material was used for the lining of the extrusion container and the die to make sure that the glass billet experiences the same interaction with all surrounding materials. The ram force and travel are automatically measured via the load cell and drive system of the extrusion machine.
In an extrusion trial, first the assembly is heated up to a temperature, at which the glass has a viscosity of 108-1010 dPa⋅s and therefore is sufficiently soft to be forced through the die. Then the ram is set to move down at a constant speed, thus forcing the glass through the die. The electronic feedback control loop of the extrusion machine is set to automatically adjust the ram force to maintain a set ram speed. A characteristic force profile is shown in Fig. 2 . After the die has filled with glass, and glass starts to emerge from the die exit, the glass flow enters a steady-state regime, where the force typically does not change significantly with time and ram position. As the majority of a preform is extruded in this regime, this part of an extrusion trial is most relevant for the glass flow analysis. Therefore, we used the average force of the steady-state regime for the glass flow analysis (Fig. 2). Note that one can also fix the force (or pressure) and automatically adjust the speed via electronic feedback control loop to maintain the force (or pressure). Distortion of the preform geometry relative to the die orifice geometry (often referred to as ‘die swell’ effect) is determined by the extrusion speed [22,23] and thus constant speed prevents variation of die swell over the preform length.
2.2 Extrusion dies
A simple die used to extrude a rod consists of an inlet funnel and a circular feed hole (or die channel) as depicted in Fig. 3 . In these dies, the length of the die channel is defined as the length of the parallel section of the die channel. For a straight funnel (Fig. 3(a)), this length is well defined, as there is a sharp transition between funnel and channel region. However for a curved funnel (Fig. 3(b)), the transition from funnel to channel region is gradual.
Previously, relatively simple structures such as tubes or wagon-wheel preforms with up to 3 holes were extruded using dies with relatively complex internal structures [3,4,6,8]. It is not readily apparent how these die designs can be adapted and scaled to produce complex preforms. Recently we developed a new, truly scalable die design concept for extrusion of structured preforms that allows the definition of large numbers of transverse features . These dies consist of three parts: a funnel, an array of circular feed holes and a welding chamber with blockages (Fig. 3(c)). The individual glass strands emerging from the feed holes fuse together into a single body in the welding chamber, while the blockages obstruct the flow and thus form the holes in the preform. The feed holes are defined as the die channels in the flow analysis as these are the features within the die that provide the highest constriction for glass flow within the die. Our new die design allows greatly improved flexibility in the selection of the size, shape, number and distribution of the feed holes and blockages, which offers easy design of the die features and independent control of the hole shape and configuration within a preform for the first time. Another advantage of this die design is that, as will be shown in Section 4, the flow through the feed holes can be described by a simple model which allows prediction of the size and number of feed holes to achieve a target preform structure using feasible extrusion conditions.
2.3 Glass types and viscosities
As described in Section 3, the glass temperature affects the glass flow via its impact on the glass viscosity. We studied a range of glasses (Table 1 ) with different temperature-viscosity curves to investigate whether the temperature-viscosity behavior has a significant impact on the glass flow or whether the glass viscosity alone (independent of the temperature and glass composition) is the dominant factor. In addition to the data of extrusion trials we performed using several Schott lead-silicate (LLF1, F2, SF57), bismuth and tellurite glasses, we used data reported in [24–26] for NCS and B270 glass extrusions. The reported data allowed us to study more thoroughly the impact of die channel geometry and interaction at the glass/die boundary on glass flow during extrusion.
To calculate the viscosity of a glass at the set die temperature, knowledge of the viscosity as a function of temperature is required. Extrusion trials are undertaken within the viscosity range of 108 −1010 dPa⋅s. For such a limited viscosity range, the experimental viscosity data can be well fitted with a two-parameter Arrhenius equation 28,29]: the annealing point ‘T13’ at 1013 dPa⋅s and the softening point ‘T7.6’ at 107.6 dPa⋅s. The knowledge of these two η(T) data points allowed us to calculate the Arrhenius parameter and hence the viscosities at the die temperatures used during extrusion. For the bismuth glass, the known glass transition temperature  was considered to be equal to the annealing point, which is justified considering the similarity of these two temperatures shown for the majority of Schott glasses . To determine the softening point of bismuth glass, we developed the following quick empirical method. The softening point is defined as the temperature at which glass deforms under its own weight within minutes . Therefore, we annealed ~5mm thick bismuth glass plates rested at an angle at gradually higher temperatures (using 10 °C increments) for approximately an hour and identified the temperature at which a plate showed a sharp bend after annealing. As for the Schott glasses, the annealing and softening point were used to calculate the Arrhenius parameters. For the tellurite glass, the Arrhenius parameters published in  were used for the viscosity calculations. The ‘T13’ and ‘T7.6’ temperatures and Arrhenius parameters of the glasses investigated are listed in Table 1. In addition, the upper and lower extrusion temperatures, i.e. the temperatures ‘T8’ and ‘T10’ corresponding to the viscosity range used for extrusion, are included in Table 1. For the NCS glass, the viscosity used for the extrusion trials considered here is given in .
2.4 Experimental errors
The measurement error of the type K thermocouples used to measure the die temperature is ± 1 °C. We measured the die temperature at four points located at the same vertical position and equally spaced around the die. The maximum temperature difference between these four thermocouples was measured to be 3 °C. Taking into account the measurement error for each thermocouple, the total uncertainty in the die temperature values is estimated to be ± 5 °C. For the analysis of the extrusion data in Section 4, this temperature measurement error was used to calculate the error in the viscosity values.
The ram speed and force is measured with high accuracy: <1% of the nominal value. However, the force typically varies by up to 10% around the average value in the steady-state regime. The maximum viscosity variation corresponding to ± 1 °C temperature value uncertainty is 17% for the tellurite glass, which has the steepest temperature-viscosity curve of all glasses studied. Thus the force fluctuations of 10% are attributed to temperature fluctuations and are considered as the relative error in our data analysis in Section 4.
The uncertainty in the billet diameter is significantly <1%. The maximum error in die channel diameter is 6%. The largest error in die channel length (14%) is found for rod dies with a curved funnel, whereas this error is <3% for all other dies.
3. Glass flow analysis
Roeder  indicated that the glass flow through a circular die channel during extrusion follows the Poiseuille law for isothermal laminar flow of an incompressible viscous liquid through a circular pipe with no slip at the fluid/pipe boundary,
In our extrusion setup, the pressure difference between die entrance and die exit plane is approximated to be equal to the extrusion pressure, P, which is given by the ram force, F, and the billet cross-section, A0.Eqs. (3) and (4), the Poiseuille law can be re-written asEquation (6) indicates that the friction increases with increasing die channel length and decreasing die channel diameter.
The above flow analysis assumes no slip at the glass/die boundary (i.e. complete sticking of the glass to the die walls). The majority of oxide glasses are extruded through metal dies made of stainless steel or nickel alloys [11,14,15,23–26]. These metal dies were found to meet the condition of no slip [24–26]. By contrast, graphite dies were observed to cause slip of glass at the die boundary. This correlates with the fact that graphite is a non-wetting material [24–26]. For capillary flow, the impact of slip at the fluid/capillary boundary is taken into account using a constant slip coefficient, α ,Eqs. (8) and (9) reduce to Eqs. (5) and (6).
In our complex dies for extruding preforms with multiple holes, the glass flows first through an array of feed holes, which are considered as an array of die channels. This array of channel provides the highest constriction for glass flow within the die and is therefore considered as the part of the die, which dominates the flow behavior. The good agreement between theoretical and experimental results, as described in Section 4.3, validates this assumption. When entering the die channels, the continuous stream of glass is split up, whereas the pressure at the entrance of each die channel is equal to the extrusion pressure provided by the ram. Hence, for an array of i channels having equal length, L1, and assuming ΔP = P and no slip at the glass/die boundary, Eq. (2) can be re-written to describe the volume flow rate, Qi, within a single die channel with diameter, Di,Eqs. (10) and (4) yieldsEqs. (5) and (6), the die constant of the array of channels within a die is defined as
Equation (14) can be used to calculate the die constant from the die geometry parameters, L1, Nj and Dj under the assumption that the glass completely sticks to the die (i.e. α = 0). The thus calculated Kdie is hereafter referred to as geometrical Kdie. Alternatively, combining Eqs. (4), (12) and (13) allows the calculation of the die constant from extrusion processing parameters,
4. Experimental verification
In this Section, we explore the validity of the simplified glass flow model with experimental results. We commence with investigating the impact of the die channel geometry (diameter and length) for extrusion of glass rods using metal dies with a single circular channel at a fixed temperature. Next we explore the impact of the slip behavior at the glass/die boundary for metal and graphite dies. Finally, the flow of a large number of extrusion trials through stainless steel dies using different glasses, temperatures and die geometries are analyzed.
To explore the impact of die channel geometry and material on the glass flow model over a wide range of die channel lengths and diameters, we used data of F2 and SF57 glass extrusion trials we conducted as well as of NCS and B270 glass extrusions reported in [24–26]. The data of these extrusion trials are shown in Figs. 4 -6 and the corresponding extrusion conditions are listed in Table 2 . For the flow analysis of a wide range of die geometries, glasses and temperatures, we used data of extrusion trials we conducted.
4.1 Metal dies with a single circular channel – impact of channel length and diameter
The impact of the die channel length and diameter on the glass flow was investigated using data of F2 and NCS silicate glass rod extrusions through metal dies (made from stainless steel and a nickel-based alloy) at fixed temperature corresponding to a viscosity of approximately 109 dPa⋅s. For the F2 glass rod extrusions, we used dies that had a tapered funnel (Fig. 3(a)) with well-defined channel length compared to dies with a curved funnel. We commence with silicate glasses and metal dies as it was reported that silicate glasses completely stick to metal dies [24–26], resulting in a glass/die slip coefficient of zero. This allows us to study the simplest case with a single circular die channel and α = 0. In this case, according to Eq. (5), the extrusion pressure is proportional to the die channel length and the inverse of the fourth exponent of the channel diameter. However, this dependence is expected to break down at short channel lengths as our glass flow model is based on the Poiseuille law for fluid flow, which assumes that the channel length is longer than its diameter, i.e. L1/D1>1. Note that for the extrusion trials considered here, this condition is not met for all dies.
We first consider the effect of channel length at fixed channel diameter and temperature (Figs. 4(a) and 4(b)). For the F2 rod extrusions, the ram speed is fixed and the pressure is the parameter dependent on the die channel length (Fig. 4(a)). According to Eq. (5) under these conditions, the pressure is proportional to the channel length. To determine agreement of the experimental data with this predicted glass flow behavior, we employed linear regression passing through the axes origin for the data set P = f (L1). The fitted regression line (R2 = 0.7498) is shown as solid line in Fig. 4(a). Even for L1/D1<1, the data follow the predicted proportionality between P and L1. For the NCS rod extrusions, the ram pressure is fixed and the extrudate speed depends on the die channel length (Fig. 4(b)). According to the continuity equation for the flow of an incompressible fluid, the ram speed, v0, and the speed at which the extrudate emerges from the die exit, v1, are related as Q = A0⋅v0 = A1⋅v1, which combined with Eq. (5) yieldsFig. 4(b)) confirms the predicted proportionality of the dependent processing parameter (in the NCS case 1/v1) with the die channel length for L1/D1>1.2.
Another way to explore the validity of our glass flow model is to calculate the dependent processing parameter, P in the F2 case and 1/v1 in the NCS case, using Eqs. (5) and (17), respectively, and the fixed processing parameters. The thus calculated P and 1/v1 values are shown as dashed lines in Figs. 4(a) and 4(b). In the F2 case, the calculated P values are in good agreement with the P values obtained through linear regression of the experimental data, which confirms the validity of our glass flow model. The larger deviation of the experimental pressure from the calculated pressure for the shortest die channel length studied indicates that for the experimental conditions of F2 glass rod extrusion the glass flow model is no longer valid for L1/D1<0.6. In contrast to the F2 extrusions, the calculated values of the dependent processing parameter, 1/v1, of the NCS extrusions are significantly higher than the experimental values. This indicates that the slip coefficient for the NCS rod extrusion trials is non-zero. According to Eq. (8) for constant values of α>0 the dependent processing parameter decreases compared to the case α = 0. The impact of slip coefficient will be further discussed in Section 4.2.
Both the F2 and NCS rod extrusion trials through metal dies demonstrate that the impact of the die channel length obeys the Poiseuille flow. Next we study the impact of channel diameter for F2 and NCS rod extrusion trials performed using fixed die channel length, temperature and ram speed (Figs. 4(d) and 4(e)). According to Eqs. (5) and (17), P is proportional to 1/D14 for the F2 extrusions performed at fixed v0 or to 1/D12 for the NCS extrusions performed at fixed v1. Linear regressions (shown as solid lines in Figs. 4(d) and 4(e) with R2 = 0.9990 for F2 and R2 = 0.9982 for NCS) of the experimental data agree with the predicted proportionality behavior. We also calculated the pressure (shown as dashed lines in Figs. 4(d) and 4(e)) using Eq. (5) for F2 and Eq. (17) for NCS with α = 0. For the F2 extrusion, the calculated pressure values are just within the measurement errors of the experimental values. However, for NCS extrusions, the calculated values are higher as found for the NCS data with fixed die channel diameter (Fig. 4(b)), which is attributed to α>0.
4.2 Metal and graphite dies – impact of slip coefficient
As demonstrated in the previous Section 4.1, F2 glass extruded through stainless steel dies obeys the Poiseuille flow with complete sticking of the glass to the die, that is the glass/die slip coefficient is zero (α = 0). For NCS glass rods extruded through nickel-based alloy dies, the slip coefficient was found to be >0. The main difference between these two metals is in their iron and nickel content. The stainless steel grade used (303) contains 9% Ni and 71% Fe , whereas the nickel-base alloy contains ~51% Ni and <2% Fe . This compositional difference may affect the bonding of the glass to the metal surface within the die channel and thus the slip behavior between glass and die.
Graphite is known to reduce significantly the friction of glass flow through extrusion dies for some glass materials, as it is a non-wetting material [24–26]. According to Eq. (8) and (9) lower friction is tantamount to α>0 and therefore a lower die constant, Kdie, compared with a die material showing α = 0. Figures 4(b) and 4(e) show the dependent processing parameters, 1/v1 and P, as a function of L1 and 1/D12, respectively, for NCS rod extrusions through metal dies, whereas Figs. 4(c) and 4(f) show the experimental data for the same extrusions through graphite dies. For both data sets, the variable processing parameters of graphite dies are approximately an order of magnitude lower than those of the metal dies. This highlights the impact of the non-wetting behavior of graphite. Note that both data sets 1/v1 = f (L1) and P = f (1/D12) of graphite dies do not obey the proportionality behavior, i.e. the data sets do not follow a line going through the axis origin, as observed for the metal dies (Figs. 4(b) and 4(e)) for L1/D1≥0.6, but the data follow a linear dependency with intercept >0. This behavior disagrees with Eq. (8), which predicts proportionality for constant α value. As will be shown below, one reason for this disagreement is the observation that α of a certain glass/metal pair is not constant for all die geometries.
Figure 5(a) compares F2, SF57 and NCS rod extrusion trials that were undertaken with the pressure as the dependent processing parameter and using the same experimental conditions for each glass except that for one trial a metal die and for the other trial a graphite die was used. As for the NCS data shown in Fig. 4, the extrusion pressure for the graphite dies is significantly lower compared to the metal dies for all 3 glasses considered, which confirms the non-wetting (or low friction) behavior of graphite for different glass compositions.
Using Eq. (8) and the continuity equation, Q = A0⋅v0 = A1⋅v1, we calculated the slip coefficient for the extrusion trials shown in Fig. 5(a). Figure 5(b) illustrates that the slip coefficient for stainless steel and nickel-base alloys are nearly zero, whereas the slip coefficient of graphite is significantly larger. The negative α value for F2 extrusion through stainless steel die is within the measurement error of the extrusion processing parameters.
The data reported for NCS and B270 rod extrusions through graphite dies [24–26] allowed us to investigate the impact of die channel diameter and length on the slip coefficient. Figure 6 shows the slip coefficients calculated from the experimental values reported in [24–26] using Eq. (8) as a function of the die channel length for a variety of die channel diameters. The slip coefficient increases significantly with increasing length. Three sets of extrusion trials were performed using the same conditions except for different die channel lengths. For these three data sets, the slip coefficient increases linearly with the die channel length with regression coefficients R2≥0.99 and intercept close to zero. This indicates that glass extruded through short graphite dies exhibits larger degree of wetting to the die surface compared to long graphite dies. The slip coefficient is also somewhat dependent on the die channel diameter (Fig. 6). For example, for the B270 glass, the α values at L1 = 8 mm, 16 mm and 32 mm slightly decrease as D1 increases from 4 mm to 8 mm. Likewise, for the NCS glass, the α value at L1 = 10mm show a slight decrease as D1 increases from 3 mm to 6 mm.
The dependence of the slip coefficient on the die channel length and diameter demonstrates that a simple glass flow model that assumes constant slip coefficient independent on die geometry is not valid for experimental situations of interest in extruding preforms. This result is similar to polymer extrusions, where multi-parameter functions are used to describe the slip behavior of a polymer within a die .
4.3 Stainless steel dies – impact of temperature for different die geometries
In the previous Sections 4.1 and 4.2, we analyzed extrusion trials that were conducted at a fixed temperature and using dies with a single circular channel. In this section, we broaden the data analysis to extrusion trials conducted at different temperatures and using different die geometries. All the trials considered in this section used stainless steel dies, for which we found that the slip coefficient is zero (see Section 4.1). Under this condition, Eq. (16) predicts that the pressure normalized to the total volume flow rate, P/Q, is proportional to the glass viscosity, with the die constant being the proportionality constant. As the viscosity changes exponentially with temperature, we analyzed the logarithms of P/Q and Kdie. In Fig. 7 , the experimentally determined P/Q values are plotted as a function of glass viscosity for a range of rod extrusions using the same die geometry (L1 = 7 mm, D1 = 10 mm, curved funnel) but different temperatures and glasses. In addition, we plotted a line that corresponds to the predicted P/Q values using Eq. (5), which includes the geometrical Kdie. For all five glasses considered, the experimental P/Q values are within the predicted P/Q values demonstrating that the glass flow obeys Poiseuille flow. The agreement also demonstrates that the experimentally inferred Kdie values agree with the geometrical Kdie value within the uncertainty in experimental processing parameters (P, Q, η(T)).
Next we determined the experimental and geometrical Kdie values for a range of die types (Table 3 ) used with different glasses and at different temperatures (i.e. different glass viscosities) (Fig. 8 ). If there were several trials conducted for a certain die and glass type, the average experimental Kdie value is shown. Note that the error bars shown for the experimental Kdie values were determined from the measurement errors of the processing parameters (see Section 2.4). These errors are approximately an order of magnitude larger than the standard deviation of experimental Kdie values obtained for fixed glass and die type. Figure 8 demonstrates that not only for the rod(10) die type considered in Fig. 7 but also for a range of other die types the experimental Kdie values agree well with the geometrical Kdie values. In addition, good agreement of the experimental Kdie values of different glasses is observed.
Another way to determine whether the extrusion flow follows the Poiseuille law is to compare the viscosity data calculated using the Arrhenius parameter given in Table 1 (hereafter referred to as calculated viscosity) with the experimental viscosity values determined from the extrusion P/Q values using Eq. (16) and geometrical Kdie values (hereafter referred to as experimental viscosity). In Fig. 9 , the experimental viscosity values are plotted as a function of the inverse temperature for the three glasses F2, SF57 and Bi using the die geometries listed in Table 3. The solid lines show the result of the linear regression and the dashed lines are the calculated viscosities. Good agreement of the experimental viscosity data determined from extrusion parameters with the calculate viscosity data given for the glasses is observed. This confirms that the glass flow during extrusion follows the Poiseuille law for a range of dies with circular die channels and for different glasses.
It is noteworthy that for complex dies with an array of die channels the good agreement between experimental and calculated viscosity as well as experimental and geometrical Kdie demonstrates that the die part with the feed holes determines the glass flow affecting the relationship between the processing parameters pressure, temperature and speed. The flow within the welding chamber of complex dies may not follow the simple Poiseuille law, but this does not affect significantly the processing parameters though it has an effect on the transverse structure of the preforms.
5. Summary and conclusions
In order to gain better understanding of the glass flow during extrusion of optical fiber preforms, we developed a simplified model describing the glass flow through extrusion dies with a single or multiple flow channels. The validity of the model was investigated for a range of oxide glass extrusion trials through metal and graphite dies.
The lead-silicate, bismuth and tellurite glasses investigated were found to adhere to stainless steel dies. In contrast, graphite dies demonstrated non-wetting behavior, resulting in fluid slip at the glass/die boundary, which decreased significantly the friction for glass flow within the die. The slip coefficient describing the slip at the glass/die boundary was found to depend on the die geometry itself. As the channel length increases, the slip coefficient becomes significantly larger, indicating enhanced glass slip at the graphite die surface.
The analysis of lead-silicate, bismuth and tellurite glass extrusion trials through stainless steel dies with different geometries demonstrated that the glass flow during extrusion follows the Poiseuille law for fluid flow, where the pressure to extrude a glass through a die is proportional to the volume flow rate and glass viscosity. The proportionality constant designates the friction of the flow caused by the extrusion die and is thus coined die constant. The proportionality between extrusion pressure and glass viscosity also confirms that the glasses investigated behave as Newtonian fluids over the pressure range used.
For dies with a single or multiple circular channels that constitute the regions of highest friction within the die, the die constant can be calculated from the number, length and diameter of the individual channels. For dies with a complex structure that cannot be approximated as an array of channels, the die constant can be determined from the extrusion processing parameters provided the temperature-viscosity-curve of the glass is known.
The knowledge of the die constant combined with our glass flow model enables the use of the extrusion technique to measure the viscosity of glasses. The flexibility in die channel diameter allows selection of temperatures that are most relevant for the viscosity measurement. For example, a ‘rod’ die with 3mm channel diameter has the same die constant as a complex ‘3-ring’ die used to produce a preform with three rings of holes. Therefore a simple ‘rod’ die can be used to determine the glass viscosity without using complex dies, which are expensive and have a higher risk of breakage at large extrusion pressure.
The main advantage of this new glass flow model lies in the identification of suitable values for ram speed and die temperature (i.e. glass viscosity) via predicting the ram pressure. For example, a high ram pressure can lead to die breakage (e.g. 60 MPa extrusion pressure at 560 °C resulted in breakage of a 4-ring die). This can be avoided by decreasing the ram speed and/or increasing the glass temperature. However, soft glasses may crystallize or form defects such as bubbles at high temperatures. Another option to avoid die breakage or high pressure that can damage the glass itself is to decrease the friction in the die via enlarging the die channels. However, an increase in the size and number of die features results in dies with large die exit cross-sectional area, which can lead to preforms with outer diameters that are too large to be used in a fiber drawing tower. In addition, the glass volume (i.e. billet size) that can be produced is often limited, which would result in preforms being too short to be drawn to fiber. One method to overcome these problems is to choose a die temperature at which the preform emerging from the die exit is sufficiently soft to be pulled under its own weight due to gravity. This tapering effect allows reduction of the preform outer diameter (relative to the die exit diameter) and elongation of the preform, whereby producing preforms with outer dimensions that are suitable for fiber drawing. Another advantage of selecting a die temperature where the glass is relatively soft is reducing the degree of preform bending due to asymmetry in the radial temperature profile. As extrusion is an automated process, a slow ram speed (i.e. long extrusion duration) is practical, which provides flexibility in choosing a suitable regime for ram pressure and die temperature.
This broad range of examples demonstrates that the glass flow model developed in this paper allows optimization of extrusion processing parameters and guidance of the die design to avoid detrimental effects such as die breakage, extreme preform tapering, glass crystallization and formation of defects in the glass. This optimization is crucial to produce glass preforms that have both excellent optical quality and desired dimensions to produce low-loss optical fibers.
We acknowledge the DSTO (Australia) for support for the Centre of Expertise in Photonics, the Australian Research Council for funding this project (DP0665486), and Asahi Glass Co. for the supply of bismuth glass billets. T. Monro acknowledges the support of an ARC Federation Fellowship. We wish to thank the members of the Centre of Expertise in Photonics at the University of Adelaide for their help with the extrusion trials, and Christopher Voyce for helpful discussions.
References and links
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