Abstract

We report the first analytical description of the drawing of fibres with holes that does not require knowledge of the viscosity (or temperature) of the glass in the furnace. The model yields expressions for the size of a hole that is isolated from other holes and small compared to the outer diameter of the fibre, but includes the effects of surface tension, pressurisation and arbitrary viscosity profiles. The effect of viscosity is represented by the fibre draw tension which, unlike viscosity, can readily be measured in practice by the fibre fabricator. The model matches experiments without recourse to any adjustable fitting parameters.

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References

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  1. A. L. Yarin, P. Gospodinov, and V. I. Roussinov, “Stability loss and sensitivity in hollow fiber drawing,” Phys. Fluids 6(4), 1454–1463 (1994).
    [Crossref]
  2. A. D. Fitt, K. Furusawa, T. M. Monro, C. P. Please, and D. J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. 43(2/4), 201–227 (2002).
    [Crossref]
  3. S. C. Xue, R. I. Tanner, G. W. Barton, R. Lwin, M. C. J. Large, and L. Poladian, “Fabrication of microstructured optical fibers - Part I: problem formulation and numerical modeling of transient draw process,” J. Lightwave Technol. 23(7), 2245–2254 (2005).
    [Crossref]
  4. S. C. Xue, M. C. J. Large, G. W. Barton, R. I. Tanner, L. Poladian, and R. Lwin, “Role of material properties and drawing conditions in the fabrication of microstructured optical fibers,” J. Lightwave Technol. 24(2), 853–860 (2006).
    [Crossref]
  5. R. M. Wynne, “A fabrication process for microstructured optical fibers,” J. Lightwave Technol. 24(11), 4304–4313 (2006).
    [Crossref]
  6. C. J. Voyce, A. D. Fitt, and T. M. Monro, “Mathematical modeling as an accurate predictive tool in capillary and microstructured fiber manufacture: the effects of preform rotation,” J. Lightwave Technol. 26(7), 791–798 (2008).
    [Crossref]
  7. F. T. Trouton, “On the coefficient of viscous traction and its relation to that of viscosity,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character 77(519), 426–440 (1906).
    [Crossref]
  8. R. T. Knapp, J. W. Daily, and F. G. Hammitt, Cavitation (McGraw-Hill, 1970), p. 108.
  9. S. Wolfram, The Mathematica Book, 5th ed. (Wolfram Media, 2003).
  10. P. J. Roberts, F. Couny, H. Sabert, B. J. Mangan, D. P. Williams, L. Farr, M. W. Mason, A. Tomlinson, T. A. Birks, J. C. Knight, and P. St. J. Russell, “Ultimate low loss of hollow-core photonic crystal fibres,” Opt. Express 13(1), 236–244 (2005).
    [Crossref] [PubMed]
  11. S. G. Leon-Saval, T. A. Birks, W. J. Wadsworth, P. St. J. Russell, and M. W. Mason, “Supercontinuum generation in submicron fibre waveguides,” Opt. Express 12(13), 2864–2869 (2004).
    [Crossref] [PubMed]
  12. W. H. Reeves, J. C. Knight, P. St. J. Russell, and P. J. Roberts, “Demonstration of ultra-flattened dispersion in photonic crystal fibers,” Opt. Express 10(14), 609–613 (2002).
    [Crossref] [PubMed]

2008 (1)

2006 (2)

2005 (2)

2004 (1)

2002 (2)

W. H. Reeves, J. C. Knight, P. St. J. Russell, and P. J. Roberts, “Demonstration of ultra-flattened dispersion in photonic crystal fibers,” Opt. Express 10(14), 609–613 (2002).
[Crossref] [PubMed]

A. D. Fitt, K. Furusawa, T. M. Monro, C. P. Please, and D. J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. 43(2/4), 201–227 (2002).
[Crossref]

1994 (1)

A. L. Yarin, P. Gospodinov, and V. I. Roussinov, “Stability loss and sensitivity in hollow fiber drawing,” Phys. Fluids 6(4), 1454–1463 (1994).
[Crossref]

1906 (1)

F. T. Trouton, “On the coefficient of viscous traction and its relation to that of viscosity,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character 77(519), 426–440 (1906).
[Crossref]

Barton, G. W.

Birks, T. A.

Couny, F.

Farr, L.

Fitt, A. D.

C. J. Voyce, A. D. Fitt, and T. M. Monro, “Mathematical modeling as an accurate predictive tool in capillary and microstructured fiber manufacture: the effects of preform rotation,” J. Lightwave Technol. 26(7), 791–798 (2008).
[Crossref]

A. D. Fitt, K. Furusawa, T. M. Monro, C. P. Please, and D. J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. 43(2/4), 201–227 (2002).
[Crossref]

Furusawa, K.

A. D. Fitt, K. Furusawa, T. M. Monro, C. P. Please, and D. J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. 43(2/4), 201–227 (2002).
[Crossref]

Gospodinov, P.

A. L. Yarin, P. Gospodinov, and V. I. Roussinov, “Stability loss and sensitivity in hollow fiber drawing,” Phys. Fluids 6(4), 1454–1463 (1994).
[Crossref]

Knight, J. C.

Large, M. C. J.

Leon-Saval, S. G.

Lwin, R.

Mangan, B. J.

Mason, M. W.

Monro, T. M.

C. J. Voyce, A. D. Fitt, and T. M. Monro, “Mathematical modeling as an accurate predictive tool in capillary and microstructured fiber manufacture: the effects of preform rotation,” J. Lightwave Technol. 26(7), 791–798 (2008).
[Crossref]

A. D. Fitt, K. Furusawa, T. M. Monro, C. P. Please, and D. J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. 43(2/4), 201–227 (2002).
[Crossref]

Please, C. P.

A. D. Fitt, K. Furusawa, T. M. Monro, C. P. Please, and D. J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. 43(2/4), 201–227 (2002).
[Crossref]

Poladian, L.

Reeves, W. H.

Richardson, D. J.

A. D. Fitt, K. Furusawa, T. M. Monro, C. P. Please, and D. J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. 43(2/4), 201–227 (2002).
[Crossref]

Roberts, P. J.

Roussinov, V. I.

A. L. Yarin, P. Gospodinov, and V. I. Roussinov, “Stability loss and sensitivity in hollow fiber drawing,” Phys. Fluids 6(4), 1454–1463 (1994).
[Crossref]

Russell, P. St. J.

Sabert, H.

Tanner, R. I.

Tomlinson, A.

Trouton, F. T.

F. T. Trouton, “On the coefficient of viscous traction and its relation to that of viscosity,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character 77(519), 426–440 (1906).
[Crossref]

Voyce, C. J.

Wadsworth, W. J.

Williams, D. P.

Wynne, R. M.

Xue, S. C.

Yarin, A. L.

A. L. Yarin, P. Gospodinov, and V. I. Roussinov, “Stability loss and sensitivity in hollow fiber drawing,” Phys. Fluids 6(4), 1454–1463 (1994).
[Crossref]

J. Eng. Math. (1)

A. D. Fitt, K. Furusawa, T. M. Monro, C. P. Please, and D. J. Richardson, “The mathematical modelling of capillary drawing for holey fibre manufacture,” J. Eng. Math. 43(2/4), 201–227 (2002).
[Crossref]

J. Lightwave Technol. (4)

Opt. Express (3)

Phys. Fluids (1)

A. L. Yarin, P. Gospodinov, and V. I. Roussinov, “Stability loss and sensitivity in hollow fiber drawing,” Phys. Fluids 6(4), 1454–1463 (1994).
[Crossref]

Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character (1)

F. T. Trouton, “On the coefficient of viscous traction and its relation to that of viscosity,” Proc. R. Soc. Lond., A Contain. Pap. Math. Phys. Character 77(519), 426–440 (1906).
[Crossref]

Other (2)

R. T. Knapp, J. W. Daily, and F. G. Hammitt, Cavitation (McGraw-Hill, 1970), p. 108.

S. Wolfram, The Mathematica Book, 5th ed. (Wolfram Media, 2003).

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Figures (6)

Fig. 1
Fig. 1

Undeformed drawdown of a cylindrical glass body from preform radius r1 to fibre radius r2 via intermediate radius r(x) at position x within the furnace. The preform is fed into the furnace at feed rate uf and the fibre is drawn out at draw speed ud.

Fig. 2
Fig. 2

(a) Normalised hole radius a versus local drawdown ratio ρ for normalised pressure P = 0.6 and final drawdown ratio ρ0 = 0.01, with the normalised stresses S marked. Each curve mimics the profile of the normalised hole size along the preform/fibre, given a scaling ρ(x) for the horizontal axis, and their right-hand ends give a0. (b) Final normalised hole size a0 versus S for the same P and ρ0.

Fig. 3
Fig. 3

(a) Pressure P versus stress S such that target hole radius a0 = 0.8 is achieved for drawdown ratio ρ0 = 0.01. (b-d) Hole radius variation a(ρ) versus drawdown ratio ρ for a reference hole (broken lines) and holes that are 5% bigger and smaller in the preform (solid lines). In each case S and P are chosen according to plot (a) such that the reference hole is drawn to the same final normalised radius a0 = 0.8. The S and E values are (b) 2.52 and 0.2, (c) 0.36 and 2, and (d) 0.22 and 3 respectively. All quantities are normalised, with R0 being the radius of the reference hole in all cases.

Fig. 4
Fig. 4

Hole radius a(ρ) versus drawdown ratio ρ for a reference hole (broken lines) and holes that are 10% bigger and smaller in the preform (solid lines), where S is the maximum stress described in the text and P is chosen such that the reference hole is drawn to a final normalised hole radius a0 = 1. (a) is for a two-step draw with drawdown ratio ρ0 = 0.1 for each step, and (b) is for a one-step draw with the same overall drawdown ratio ρ0 = 0.01. All quantities are normalised, with R0 being the radius of the reference hole in all cases.

Fig. 5
Fig. 5

Optical micrographs of (a) the preform cane (1 mm scale bar) and (b) a typical fibre drawn from the preform (20 µm scale bar).

Fig. 6
Fig. 6

(a) (points) Measured final diameter 2R versus draw tension F for three holes in a fibre. F was varied by varying T between 1960 - 1890 °C (circles) or the process speeds with uf/ud fixed and uf between 2 - 6.2 mm/min (squares), from a common pivotal case where F = 67 grams-force, uf = 2 mm/min, ud = 12.8 m/min, r2 = 62.5 µm, p0 = 13 kPa and T = 1960 °C. (lines) Corresponding predictions from Eq. (15) taking γ = 0.3 Jm−2. (b) (points) Measured differences between the final diameters of the smallest hole and (upper) the biggest hole and (lower) the intermediate hole versus hole pressure p0, where the other parameters match the pivotal case in (a). (lines) Corresponding predictions from Eq. (17) taking γ = 0.3 Jm−2.

Equations (29)

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u(x)= dx dt = r 1 2 r 2 (x) u f u f ρ 2 (x) ,
ρ(x)= r(x) r 1 1.
ρ 0 =ρ()= r 2 r 1 = u f u d .
F=3π r 2 (x)μ(x) du dx .
dr dx = dr dt / dx dt = Fr(x) 6π u f r 1 2 μ(x) .
dρ dx = Fρ(x) 6π u f r 1 2 μ(x) ,
r(x)= r 1 ρ(x)= A(x) π ,
dR dx = FR(x) 6π u f r 1 2 μ(x) .
p st = γ R .
p v = 2μ R dR dt ,
p o = p H p a .
dR dx = ρ 2 2 u f μ ( R p o γ ).
dR dx = F 6π u f r 1 2 μ(x) { [ 1 3π r 1 2 p o F ρ 2 (x) ]R(x)+ 3π r 1 2 γ F ρ 2 (x) }.
dR dρ +[ 3π r 1 2 p o F ρ 1 ρ ]R(ρ)= 3π r 1 2 γ F ρ.
R(ρ)=ρ e E ρ 2 [ e E R 0 3γ s ρ 0 2 ρ 1 e E z 2 dz ],
s= F π r 2 2 ,
E= 3 p o 2s ρ 0 2 .
ρ 1 e E z 2 dz ={ π 2 E [ erfi( E )erfi( E ρ ) ] E>0 π 2 E [ erf( E )erf( E ρ ) ] E<0 1ρ E=0 .
R 2 R 1 =ρ e E(1 ρ 2 ) [ R 0,2 R 0,1 ].
lnρ(x)= F 6π u f r 1 2 x dz μ(z) .
ln ρ 0 = F 6π u f r 1 2 dz μ(z)
lnρ(x)=ln ρ 0 × x dz μ(z) dz μ(z) ,
F= 6π u f r 1 2 μln ρ 0 L ,
ρ(x)= ρ 0 x/L = e (ln ρ 0 )x/L ,
a= R ρ R 0 .
S= R 0 s ρ 0 2 3γ = R 0 ρ 0 2 F 3πγ r 2 2 = R 0 F 3πγ r 1 2
P=ES= p 0 R 0 2γ
a(ρ)= e E(1 ρ 2 ) e E ρ 2 S ρ 1 e E z 2 dz ,
S= e E ρ 0 2 ρ 0 2 1 e E z 2 dz e E(1 ρ 0 2 ) a 0 .

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