1. Introduction

Hollow core multilayer and microstructured optical fibers (MOF) for radiation guiding in the near and mid-infrared (IR) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] have recently received close attention as they promise considerable advantage over their solid core counterparts in applications related to power guidance at almost any IR wavelength for military, industry and medical applications, IR imaging and sensing, and even THz transmission. Due to its complexity, fabrication of such waveguides is an active field of research. Four main methods have been identified for hollow core fiber manufacturing, each offering its own advantages and challenges. First method is a deposition of metallo-dielectric films on the inside of a drawn capillary by liquid-phase coating [5, 6]; technical challenges in enforcing thickness uniformity in the resultant coatings limit fiber length to the distances of ~10m. Second method is a capillary stacking or preform drilling [7, 10, 8, 9, 11, 12, 13] where glass or plastic capillaries are arranged in a predefined manner and then co-drawn, or the predefined array of holes is drilled in a plastic preform; so far such fibers have been mostly demonstrated to guide below 3µm due to the non-transparency of silica and polymer materials used in the fabrication. Third method is a deposition of radially uniform thin films on a drawn substrate fiber by means of physical or chemical vapor deposition methods [14]; main challenge of this technology is presumably uniformity of the resultant coatings, and a throughput due to a relatively slow deposition process. Finally, film rolling process [15] starts with a deposition of a glass (chalcogenide) film on top of a polymer film with consecutive rolling around a mandrel tube, tube etching and drawing; potential challenges include fiber profile optimization which is somewhat nontrivial due to a strictly periodic reflector geometry imposed by the fabrication method, another potential challenge is controlling bio-compatibility of a resultant fiber.

In our research group we study fabrication of all-polymer hollow multilayer Bragg fibers. We find that polymer material combination is very well suited for this purpose. The key issue for Bragg fiber fabrication is a choice of a pair of thermo-mechanically compatible polymer materials with sufficiently different refractive indexes that can be co-drawn into a fiber. Because of the commercial abundance of various moderately priced polymer materials the process of material selection is relatively short time and low cost. Moreover, polymer thermo-mechanical properties can be fine-tuned by simple variation of their molecular mass. Many polymers are available in various physical forms such as rods, tubes, films, granules and powders which enables multiple design decisions for preform fabrication. Many polymers are swelled or solvable in organic solvent which enables incorporation of active materials in their matrix such as dopants, laser dyes, nanoparticles, etc. Polymer materials are generally bio-friendly and some of them are biodegradable which is of potential interest for fiber based bio-medical technologies [18]. Finally, drawing of large core Bragg fibers does not typically require core pressurization which simplifies fabrication infrastructure. Although refractive index contrast between layers in an all-polymer Bragg fiber is relatively small (at most 1.3/1.7), as demonstrated in [19] liquid core all-polymer Bragg fibers can be designed to guide very well both TE and TM like modes, while gas filled all-polymer Bragg fibers can guide effectively a TE polarized mode. Moreover, in liquid filled core fibers high index-contrast omnidirectional designs typically hurt fiber performance making lower index-contrast systems superior in their transmission properties.

Recently, our research group has succeeded in developing two methodologies for fabrication of multilayered all-polymer hollow preforms. One approach uses consecutive deposition of layers of two different polymers by solvent evaporation on the inside of a rotating polymer cladding tube [20]. Orthogonal solvents were found, and solvent evaporation process was developed for both PMMA(Polymethyl methylacrylate)/PS(Polystyrene) and PVDF(Polyvinylidene fluoride)/PC(Polycarbonate) material combinations. In the left of Fig. 1(a), a 30cm long all-polymer preform with 10 consecutive PMMA/PS layers deposited on the inside of a PMMA cladding tube is presented; on the right, preform cross section is shown. In the left of Fig. 1(b), again a 30cm long all-polymer preform with 15 consecutive PVDF/PC layers deposited on the inside of a PC cladding tube is presented; while on the right, preform cross section is shown. Alternative preform fabrication method uses a co-rolling of two dissimilar polymer films similarly to [15], where both commercial and home-made films were used. In the left of Fig. 1(c), an end part of an all-polymer preform with 19 consecutive PVDF/PC layers is presented; in the middle of the figure a cross section of a drawn fiber is shown with a drawdown ration of 1:20; finally in the right of Fig. 1(c) another drawn fiber with 32 layers PMMA/PS layers is demonstrated with a similar drawdown ratio.

 

Fig. 1. a) Left - 30cm long all-polymer preform with 10 consecutive PMMA/PS layers deposited on the inside of a PMMA cladding tube. Right - PMMA/PS preform cross section. b) Left - 30cm long all-polymer preform with 15 consecutive PVDF/PC layers deposited on the inside of a PC cladding tube. Right - PVDF/PC preform cross section c) Left - end part of a rolled 19 layer PVDF/PC preform. Middle - cross section of a drawn PVDF/PC fiber with a 1:20 drawdown ratio. Right - cross section of a drawn PMMA/PS fiber with a 1:20 drawdown ratio.

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After preform is fabricated, hollow MOFs are manufactured by preform heating and drawing. Geometry of the final fiber can be significantly influenced by controlling various parameters in the drawing process such as temperature distribution in a furnace, fiber drawing and preform feed velocities, as well as pressurization of the hollow core. When preform is made of polymer materials non-Newtonian nature of polymer viscosity can be of importance. Moreover, if several materials are used in a single preform, drawing process can be influenced greatly by the mismatch in the viscosities of the constitutive materials. For the problem of hollow Bragg fiber drawing, geometries of all the resultant fibers can be simply parametrized by only two parameters - a drawdown ratio, and a hole collapse parameter. We further find that fiber transmission properties depend strongly on a hole collapse parameter which can only be controlled indirectly during drawing process. The purpose of this paper is to characterize hole collapse and multilayer non-uniformity during the draw of a polymer multilayer fiber as a function of standard control variables. We investigate importance of mismatch in the viscosities of two different materials during co-drawing and compare predictions of Newtonian and generalized Newtonian flow models.

Previous studies on fiber drawing have focused mainly on spinning molten threadlines [21, 22] or drawing conventional solid optical fibers [23, 24]. Drawing of hollow fibers was first studied in [25] where the asymptotic “thin-filament” equations were obtained but the effects of surface tension were neglected. A more complete analysis, although confined to Newtonian flow is given in [26, 27, 28, 29, 30, 31]. When preform radius is comparable to neck-down length (which means that the “thin-filament” assumption is no more valid), the surface tension force is relatively small and the tube thickness is comparable to the hole radius, it is demonstrated in [29, 30, 31] than the hole experiences an expansion instead of a collapse. The hole enlargement is also observed in the fabrication of microstructured polymer fibers reported in [17], where a two-stage draw process was used. The analysis presented here is more pertinent to a situation where the geometry is slender and the surface tension effects are important which could be the case of a second draw stage.

2. Effect of a hole collapse on the transmission properties of hollow Bragg fibers

 

Fig. 2. a) Radiation loss of the bandgap guided TE ₀₁ core modes for the high index-contrast (2.0/1.5) air filled fibers with different hole collapse ratios C _r , while the same outside radii $R_o^{\mathit{\text{ft}}}$ . Hole collapse leads to the shift of a bandgap center into the longer wavelength, as well as to a considerable increase in the modal radiation losses. b) Radiation losses of the bandgap guided TE ₀₁ and HE ₁₁ core modes for the low index-contrast (1.6/1.4) water filled fibers with different hole collapse ratios.

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Hole collapse and draw induced non-uniformity of a multilayer reflector have direct impact on the transmission properties of a hollow photonic crystal fiber. We define $R_i^f$ and $R_o^f$ to be the inside and outside radii of a hollow Bragg fiber (insets in Fig. 2), while $R_i^p$ and $R_o^p$ are the corresponding radii of a hollow preform. The first parameter that relates preform and fiber dimensions is a drawdown ratio D which is defined as a ratio of the outside preform diameter to that of a fiber D=$R_o^p$ /$R_o^f$ . Drawdown ratio can be set during drawing process, and is typically well maintained by a feedback loop from a laser micrometer to a tractor assembly. The second parameter characterizes hole collapse in a fiber as compared to a preform, and is defined as C _r =($R_i^f$ /$R_o^f$ )/($R_i^p$ /$R_o^p$ ). This parameter is important when preform core is made of a compressible material such as gas, which is the case during hollow core Bragg fiber fabrication. Thus,C _r =0 signifies that during drawing, the hole collapses completely resulting in a solid core fiber; while C _r =1 signifies that there is no hole collapse and all the fiber dimensions can be calculated from the corresponding preform dimensions by a simple division by a drawdown ratio. Given these two drawing parameters and assuming that all the materials in a melted state are incompressible fluids, then a circular contour of radius r ^p in a preform translates into a circular contour of radius r ^f in a fiber, and they are related by (the ratio between the areas outside and inside the contour is conserved):

To understand the effect of a hole collapse on the transmission properties of the resultant fibers, in Fig. 2(a) we first present a set of theoretical curves showing radiation losses of the TE ₀₁ core modes for the high index-contrast air filled fibers drawn with different values of C _r , while featuring the same outside diameter $R_o^f$ . In this example, C _r =1 corresponds to a target hollow core fiber n _c =1 with a strictly periodic 15 layer quarter-wave reflector having material refractive indices n _h =2.0, n _l =1.5 and layer thicknesses $d_h^t=0.25{\lambda }_c^t⁄\sqrt{n_h^2-n_c^2}=144nm$, $d_l^t=0.25{\lambda }_c^t⁄\sqrt{n_l^2-n_c^2}=224nm$, where ${\lambda }_c^t$ =1µm (for more details on design of such fibers see [32]). Target fiber inside and outside radii are chosen to be $R_i^{\mathit{\text{ft}}}$ =5µm, $R_o^{\mathit{\text{ft}}}$ =12.72µm. By design, such a fiber has a large band gap centered at ${\lambda }_c^t$ .

In the presence of a hole collapse C _r <1 (assuming the same value of a drawdown ratio D) two major changes in the the fiber geometry happen. First, while the outside fiber radius is fixed $R_o^{\mathit{\text{ft}}}$ , the fiber core radius is reduced $R_i^f$ =$R_i^{\mathit{\text{ft}}}$ C _r . Second, from Eq. (1) it can be shown that thicknesses of the reflector layers become non-uniform, increasing towards the fiber core, while, on average, layer thicknesses increase as d_h,l⁻~$d_{h,l}^t$/C _r . These geometrical changes can significantly modify fiber transmission spectra.

Thus, as the center wavelength λ _c of a photonic bandgap is proportional to the average reflector layer thickness, then, in the presence of a hole collapse, center of a bandgap is expected to shift to the longer wavelengths λ _c ~${\lambda }_c^t$ /C _r (Fig. 2(a)). We find, however, that in the presence of a hole collapse the ratio of a band gap to a mid-gap (relative bandgap) stays almost uneffected. Another prominent effect of a hole collapse is on the core mode radiation losses. From ([32]), radiation losses of the bandgap guided core modes scale as (λ _c )^p-1/($R_i^f$ ) ^p , where exponent p equals 3 for the TE _0n modes, while for the HE, EH, and TM modes, exponent p is in the range [1,3] and depends strongly upon the fiber core size. Thus, in the presence of a hole collapse, due to reduction of the core radius, and due to shift in the center of a bandgap, we expect core mode radiation loss to increase as Loss~$C_r^{-}$ ^(2p-1), which for TE ₀₁ mode gives Loss ~$C_r^{-5}$. From more detailed simulations we find that for TE ₀₁ mode, actual scaling exponent varies from -5 when C _r ≃1 to almost -7 when C _r ≃0.5 signifying additional degradation of modal confinement due to nonuniformity in the reflector layer thicknesses.

In Fig. 2(b) we present another example of the effect of a core collapse on the radiation losses of the TE ₀₁ and HE ₁₁ core modes of the low index-contrast water filled fibers. In this example, a target water filled core n _c =1.332 fiber features a strictly periodic 16 layer quarter-wave reflector with refractive indices n _h =1.6, n _l =1.4 and layer thicknesses $d_h^t$ =282nm, $d_l^t$ =580nm, where band gap center frequency is ${\lambda }_c^t$ =1µm; inside and outside radii are chosen to be $R_i^{\mathit{\text{ft}}}$ =10µm, $R_o^{\mathit{\text{ft}}}$ =21.90µm. Behavior of modal losses as a function of a core collapse parameter in this case is similar to that of a high index-contrast system presented in Fig. 2(a). However, in this particular case due to accidental degeneracy of the studied modes with higher order ones, for some values of C _r modal losses can exhibit a double dip profile instead of a single dip profile corresponding to a classical band gap.

From the analysis above it follows that hole collapse mainly leads to the linear shift in the bandgap frequency and a super-linear increase in the radiation losses of the core guided modes. In what follows we quantify hole collapse and layer non-uniformity as a function of the standard control parameters during drawing.

3. Basic equations

Schematic of a hollow multilayer preform profile during drawing is shown in Fig. 3. As the density of polymers varies little in the range of temperatures and pressures considered here we consider an incompressible axisymmetric steady flow. The equations for conservation of mass and momentum in cylindrical coordinates are as follows:

where r and z are the radial and axial coordinates, v _r and v _z are the r and z components of the velocity vector v, ρ is a constant density, p is a pressure, τ _ij is an extra-stress and g is a gravitational acceleration. The components of a total stress tensor σ̿ are

The definition of τ _ij depends upon the polymer model, and is discussed in details later. For these equations, we need to specify the boundary conditions. At the interfaces between different layers the kinematic conditions are

where R _j =R _j (z) denote the interfaces between layers and the index j=1,2…N is used to number them starting from the inner one. The primes denote the derivative with respect to z. Since the first and the N-th interfaces are external interfaces, we will distinguish them by denoting R _i ≡R ₁ and R _o ≡R _N for the inner and outer boundaries respectively.

 

Fig. 3. Schematic of a hollow multilayer preform during drawing. Different colors correspond to different materials in a multilayer.

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Hollow core can be pressurized in order to control its collapse under the action of a surface tension. In this case, at the inner interface the dynamic boundary conditions are

where γ denotes the surface tension coefficient, and

is the curvature, while P _i is the hole overpressure (constant ambient pressure has no effect on the flow). Outward-pointing normal at the inner boundary n _i is defined as:

while

is the unit tangent vector. In a similar way, the dynamic boundary conditions at the outer boundary are

where n _o, t _o and κ _o satisfy the same equations as -n _i, -t _i and κ _i respectively with R _i replaced by R _o .

At the interfaces between the internal layers, we will consider a continuous stress and velocity. In the axial direction the boundary conditions are the known values of the drawing (V _d ) and the feeding (V _f ) velocities. Furthermore, as an initial condition, the values R _j (0) are known.

4. Thin filament equations

One of the basic dimensionless parameters in the problem is the ratio between the preform radius and the length of the neck down region defined as ε. In the case when ε≪1 a so called thin filament approximation can be used. There are two different approaches for simplifying the equations in this case. In the first approach [26, 33], the variables are expanded as power series in ε² and only the dominant terms are retained in the equations. In the second approach [25], which we also follow in this paper, the equations are averaged over the cross-section at each value of z.

The average φ̄(z) of a variable φ(z) is defined as

For the axial velocity, the assumption v _z =v̄ _z is made explicitly. We note first that for a slow varying thin filament; $R_j^{\prime }$ ≪1, and by neglecting terms of the order $R_j^{\prime 2}$ the boundary conditions Eqs. (6) and (10) take the following form:

Multiplying the r-component of the momentum equation (3) by 2πr ², integrating from R _i to R _o , considering $\frac{d}{dz}{\int }_{R_i}^{R_o}2\pi r^2{\tau }_{rz}dr\approx 0$ (see Ref. [34] p.382), neglecting the inertial term because of the small value of the radial velocity and using the boundary values of σ _rr given by Eq. (12), we obtain

Multiplying the z-component of the momentum equation (3) by 2π r, integrating from R _i to R _o , using the boundary values of τ _rz =σ _rz given by Eq. (12) as well as the Eq. (13) and neglecting terms of relative order $R_j^{\prime 2}$ , we obtain

where πQ=π($R_o^2$ -$R_i^2$ )v _z is the constant volumetric flow rate. This is the axial force balance equation.

5. Newtonian flow

The constitutive equation for the Newtonian fluid is

where η is the viscosity which in our case depends on r and z. In this paper we consider a uniform temperature in any given cross-section, thus T is only a function of z. We also assume an axial variation of the temperature dependent viscosity for each constituent material. From the continuity equation one finds:

where A=A(z) is a function to be determined later. The extra-stress tensor takes the form:

where the component ${\tau }_{rz}=\eta \frac{{\partial }_{v_r}}{\partial z}$ has been neglected. From Eq. (17) the following relation holds for the averaged components of an extra-stress tensor

and the axial force balance Eq. (14) can be written in the form

The expression for A is obtained by directly integrating the r-component of the momentum equation (3) from R _i to R _o and using the boundary values of σ _rr given by Eq. (12)

Combining the kinematic boundary conditions Eq. (5) with relation Eq. (16), we obtain

6. Numerical solution for the Newtonian case

The last three equations may be considered as a system of coupled differential equations for v _z (z), A(z) and R _j (z). This system of equations can be solved with an iterative method. Starting from an arbitrary initial distribution of v _z (z), say a linear distribution between the feeding and drawing speed, the initial value equations (21) can be integrated in order to obtain R _j (z) with the value of A given by Eq. (20). This functions are then used to solve the boundary value problem Eq. (19) to obtain a new function v _z (z) passing so at the next iteration. For the examples given later in this paper we have tested this procedure and it converges very fast (less than 200 iterations).

In most cases of practical importance, inertial, gravitational and capillary terms in Eq. (19) can be neglected and it takes a simple form

where C is a constant. This equation, which is now uncoupled from the other two, can be easily integrated to give

where L is a furnace length. Once the axial velocity is known, the initial value equations (21) can be easily integrated to obtain the profile of a drawn structure.

One of the key aspects of hollow preform drawing is the partial or even complete collapse of a compressible core as a result of the surface tension forces acting at the free boundaries. As introduced earlier, we characterize the hole collapse by the parameter C _r . Hole collapse typically results in a faster reduction of a smaller core radius compared to the larger outer radius. Thus, starting with identical thicknesses of the same material layers in a preform, in a drawn fiber the inner layers will become thicker than the outer ones. We will characterize the thickness non-uniformity by the ratio $\frac{h_o}{h_i}$ between the thickness of the outer layer and the thickness of the inner one, assuming they were equal in a preform.

 

Fig. 4. Temperature distribution in the furnace.

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In the following we investigate how the hole collapse and layer thickness non-uniformity are influenced by various control parameters. As an example, we consider drawing of a multilayer hollow Bragg fiber preform where cladding tube and one of the two materials of a multilayer is PMMA, while the other material is a different polymer. Materials in consecutive polymer layers are alternated to create a periodic multilayer structure.

6.1. Effects of draw ratio, temperature and viscosity mismatch

In our calculations we assume a uniaxial temperature distribution with a maximum at the furnace center (Fig. 4). In the following when we vary the maximum value of the temperature, we simply rescale the whole profile. We assume that Newtonian viscosity of PMMA obeys an Arrhenius type dependency:

where T(°C) is the temperature,η ₀(Pa) and α(°C) are constant coefficients and T ₀ is a reference temperature. For PMMA their values are given in Ref. [35]; η₀=1.506×10⁵Pa, α=2935°C and T ₀=170°C.

In our first calculation viscosity of the other polymer is assumed to be simply two times higher than that of PMMA. Although we recognize that to model correctly the flow of a particular polymer we need to use its proper temperature dependent viscosity, in our first simulation we rather want to highlight the major effect of adding another material into a preform. Particularly, we want to investigate how the hole collapse is affected by the viscosity of the second material despite of its small volume fraction in the preform.

We consider drawing of a preform with external and internal diameter 31.75 and 25.4mm respectively. PMMA tube is coated on the inside with 25 alternated layers of PMMA and another polymer with a viscosity two times higher, each one of them having a thickness of 50µm. The value of surface tension coefficient is considered constant for exterior interfaces γ=0.032N/m [36] and both densities are considered to be 1195kg/m ³ [35]. We also assume a furnace length L=30cm, a constant preform feeding velocity V _f =2.5µm/s and a zero pressurization P _i =0.

First, we consider the effects of varying the drawing ratio defined as D _r =V _d /V _f (not to be confused with a drawdown ratio defined as D=$R_o^p$ /$R_o^f$ ) and the maximum temperature in a furnace. In Fig. 5(a), solid lines represent the parameter of a hole collapse C _r as a function of a draw ratio D _r =V _d /V _f for different values of the maximal temperature. Dashed lines represent the parameter curves resulting in a constant outside diameter D _o ≡2R _o =125µm and D _o=250µm after the draw. For comparison, in dotted curves we present the hole collapse if no other polymer is present in the preform (drawing of a simple PMMA tube of the same inner and outer radii as a multilayer preform).

 

Fig. 5. (a) Hole collapse parameter Cr as a function of the draw ratio D _r for different temperatures. Solid lines correspond to multilayer preform. Dotted lines correspond to a simple tube with the same thickness as the multilayer preform. Dashed lines represent the curves of a constant outside diameter. (b) Ratio h _o /h _i between the inner and outer layer thicknesses as a function of the draw ratio for different temperatures.

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We see that to prevent hole collapse higher draw ratios and lower temperatures have to be used. Both cases demand higher draw force which might lead to fiber breakage. We also observe that the collapse of the tube with the same thickness as the multilayer preform is more pronounced because of a lower average viscosity. Effect of the temperature on a hole collapse is easy to understand as viscosity decreases rapidly with reduction of the temperature, thus hindering the hole collapse. Effect of the draw ratio is more subtle. Starting with preforms of the same diameter, draw ratio increase leads to the reduction of a resultant fiber cross-section. As a consequence, the forces of surface tension become more pronounced, thus favoring the hole collapse. This is overcompensated by the fact that increasing the draw ratio leads to the higher axial velocities, thus the time a cross-section spends in a melted zone diminishes which works against the hole collapse.

In Fig. 5(b), thickness non-uniformity parameter is presented as a function of the draw ratio for different values of the maximum temperature. The curves are similar to those for the hole collapse in Fig. 5(a). From mass conservation, the ratio between the cross-section areas of different layers remains constant from which it follows that the thickness non-uniformity parameter is proportional to the hole collapse.

We now describe in more details the effect of mismatch in the polymer viscosities when two different materials are used in the same preform. As seen in Fig. 5(a) the hole collapse is considerably less pronounced for the multilayer structure despite the fact that the higher viscosity material occupies only a very small fraction of the total volume and its viscosity is only two times higher than that of PMMA. In what follows we assume that multilayer preform is made of PMMA and another polymer. For the viscosity of a second material a similar Arrhenius law as for PMMA is assumed.

First, we investigate the effects of changing η ₀ while keeping the other parameters unchanged, which corresponds to the case of using the same polymer, but with a different molecular mass. It should be mentioned here that the polymer’s molecular mass determines whether the fiber is drawable in the first place. Second, we investigate the effects of changing T ₀ which corresponds to the case of varying the polymer material. In Fig. 6 we consider drawing of preforms of various compositions at a fixed maximum furnace temperature of 190 °C and a draw rate D _r =30000. Multilayer preform geometry is the same as described above; mismatch in the polymer viscosities is described in terms of the ratios of the material parameters η ₀/η ₀,_PMMA and (T ₀-T ₀, _PMMA )/T ₀, _PMMA . From Fig. 6 we see that the hole collapse depends significatively on the viscosity of a second material and can be prevented by choosing a polymer with an appropriate viscosity.

 

Fig. 6. Effect of mismatch in the viscosities of materials in a multilayer on hole collapse (solid lines), and layer non-uniformity (dotted lines). Maximum furnace temperature is T=190°C, draw ratio D _r =30000. (a) Effects of η ₀. (b) Effects of (T ₀-T ₀, _PMMA )/T ₀, _PMMA .

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6.2. Effects of the pressurization and preform feeding velocity

Other parameters that influence hole collapse are the hole overpressure and the feeding speed. By increasing the hole pressure we expect to reduce the hole collapse. Also, for a given draw ratio, by increasing the preform feeding speed we expect reduction of the hole collapse as fiber cross-section would spend less time in the melted zone.

 

Fig. 7. The hole collapse and thickness non-uniformity as a function of the hole overpressure and feeding speed. Maximal furnace temperature is T=190°C and draw ratio D _r =5000. (a) Effects of hole overpressure P _i . (b) Effects of feeding speed V _f .

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We consider drawing of the same preform as in a previous section using the draw ratio D _r =5000, while the other draw parameters remain unchanged. In Fig. 7 hole collapse and layer thickness non-uniformity are presented as functions of the hole overpressure P _i and feeding speed V _f . As expected, the hole collapse is very sensitive to pressurization, and in principle, can be reduced by increasing the pressure. Time evolving transient draw simulations, not discussed in this paper, also show that above a certain critical value for an overpressure, which in our case is less than 7Pa, even if the fiber does not blow up immediately, the drawing process never reaches its steady state. A more subtle way of controlling the hole collapse is by changing the preform feeding speed, although, for a given draw ratio, this could be limited by the maximal draw velocity.

7. Generalized Newtonian model

In the previous sections we considered only the temperature dependence of the viscosity, but for polymers such as PMMA and PS the viscosity depends also on the kinematics of the flow. In this case we refer to η as a non-Newtonian viscosity or the generalized viscosity. Polymers in general are also viscoelastic, however we will neglect elasticity effects in our analysis which may be considered as a first step toward a complete modelling of polymeric flow.

For the non-Newtonian viscosity we will use the Carreau-Yasuda model [34] which was also considered in [37] and the values of different coefficients for some important polymers are given in that paper. According to this model the viscosity is given in the form

where η ₀, K ₁, a, and n are constant coefficients, $\stackrel{̿}{D}=\frac{1}{2}(\nabla \mathbf{v}+\nabla {\mathbf{v}}^T)$ is the rate of the deformation tensor, II _D =tr(D̿·D̿) is its second invariant and the term f gives the temperature dependence of the viscosity in the Arrhenius form $f=e^{\alpha \left(\frac{1}{T}-\frac{1}{T_0}\right)}$. Considering the Eqs. (17) and (15) and the fact that A is negligible compared to $v_z^{\prime }$ , one obtains $I{I}_D=\frac{3}{2}{v}_z^{\prime 2}$.

The numerical procedure is the same as in the Newtonian case. The system of coupled partial differential equations (19–21) can be solved again by the same iterative procedure. At each step of the iteration we must solve the boundary value problem Eq. (19) which, in this case, becomes more complex because the viscosity depends on the rate of deformation. Thus, this equation must be solved itself by an iterative procedure (so in overall there are two nested iterations). However, since the inertial and gravitational terms are usually negligible, the axial velocity v _z (z) can be evaluated separately by using repeatedly Eq. (23) with a viscosity update according to Eq. (25), until the convergence is attained. Once v _z (z) is known, Eq. (21) can be easily integrated.

As an example, we consider drawing of the same preform as for the Newtonian case, which is a PMMA tube with OD and ID diameters 31.75 and 25.4mm respectively, coated on the inside with 25 alternated layers of PMMA. Viscosity of PS is assumed to be two times higher than that of a PMMA. Layer thickness is assumed to be 50µm for all the layers. We consider γ=0.032N/m, ρ=1195kg/m³, L=30cm, V _f =2.5µm/s and P _i =0. Parameters for the non- Newtonian viscosity Eq. (25) are K ₁=0.0861 n=0.1401 and a=0.7347 for PMMA, and K ₁=0.3891 n=0.2194 and a=0.6097 for PS [37]. Finally we assume α=2935, T ₀=170°C, and the value of the maximum temperature in the furnace being T=180 °C.

Numerical results are presented in Fig.8(a) where hole collapse parameter is plotted as a function of the draw rate and maximum furnace temperature for Newtonian and non-Newtonian models. We observe that hole collapse is more pronounced in the non-Newtonian case because, as seen from Eq. (25), the non-Newtonian viscosity is lower than the Newtonian one, making it easier for the surface tension to reduce the hole. As expected, non-Newtonian nature of polymer viscosity becomes more apparent with increase in D _r . This can be clearly seen in Fig. 8(b) where the viscosity distribution is plotted as a function of z for different draw ratios. For comparison, we have also included the Newtonian viscosity at the same temperature.

 

Fig. 8. a) Comparison between Newtonian and generalized Newtonian model. Solid lines correspond to generalized Newtonian model and dotted lines to Newtonian model. b) Viscosity distribution in the furnace for different draw ratios at T=180°C. Solid lines correspond to generalized Newtonian model and dotted line to Newtonian model.

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8. Conclusions

Drawing of multilayer hollow polymer fibers is studied using the thin filament approximation. Both, the Newtonian and generalized Newtonian models of polymer flow are considered. Hole collapse is identified as a key parameter effecting transmission properties of the resultant hollow Bragg fiber. The hole collapse caused by surface tension is characterized as well as the closely related layer thickness non-uniformity. It is demonstrated that by varying various control parameters such as furnace temperature, feeding speed and pressurization it is possible to reduce hole collapse. While hole pressurization provides a very effective way of compensating for the hole collapse, it is found that the final fiber dimensions are very sensitive to the value of an overpressure. Moreover, the draw process could not reach a steady state if the overpressure was larger than a critical value. Finally, under the same draw conditions, the hole collapse is more pronounced when non-Newtonian viscosity is taken into account.

Comparison between the experimental results and theoretical modelling is currently underway. In general, qualitative agreement between theory and experiment of the dependence of a core collapse on the draw parameters is readily observable. As the results of our numerical analysis correspond to the equilibrium drawing conditions, to make a quantitative comparison with experiments we have to ensure that our experimental drawing has reached equilibrium and is not dominated by transients.