## Exploration of in-fiber nanostructures from capillary instability |

Optics Express, Vol. 19, Issue 17, pp. 16273-16290 (2011)

http://dx.doi.org/10.1364/OE.19.016273

Acrobat PDF (1165 KB)

### Abstract

A new class of multi-material fiber that incorporates micrometer-thickness concentric-cylindrical sheets of glass into polymer matrix has emerged. The ultimate lower limit of feature size and recent observation of interesting instability phenomenon in fiber system motivate us to examine fluid instabilities during the complicated thermal drawing fabrication processing. In this paper, from the perspective of a single instability mechanism, classical Plateau-Rayleigh instabilities in the form of radial fluctuation, we explore the stability of various microstructures (such as shells and filaments) in our composite fibers. The attained uniform structures are consistent with theoretical analysis. Furthermore, a viscous materials map is established from calculations and agrees well with various identified materials. These results not only shed insights into other forms of fluid instabilities, but also provide guidance to achieve more diverse nanostructures (such as filaments, wires, and particles) in the microstructured fibers.

© 2011 OSA

## 1. Introduction

1. J. Eggers, “Nonlinear dynamics and breakup of free-surface flows,” Rev. Mod. Phys. **69**, 865–929 (1997). [CrossRef]

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7. S. Tomotika, “On the instability of a cylinderical thread of a viscous liquid surrounded by another viscous fluid,” Proc. Roy. Soc. London. **150**, 322–337 (1935). [CrossRef]

8. H. A. Stone and M. P. Brenner, “Note on the capillary thread instability for fluids of equal viscosities,” J. Fluid. Mech. **318**, 373–374 (1996). [CrossRef]

9. X. D. Shi, M. P. Brenner, and S. R. Nagel, “A cascade of structure in a drop falling from a faucet,” Science **265**, 219–222 (1994). [CrossRef] [PubMed]

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_{2}Se

_{3}and As

_{2}S

_{3}), down to sub-micrometer or even nanometer thicknesses, have been successfully fabricated in glass materials, by a drawing process whereby a large-scale “preform” is heated and pulled into a long thread, as depicted in Fig. 1. On the other hand, as the shell thickness is further reduced towards the nanoscale, the thin cylindrical shell (made of the glass material selenium, Se), is observed to break up into an ordered array of filaments; that is, the breakup of the cylindrical shell occurs along the azimuthal direction while the axial continuity remains intact [30

30. D. S. Deng, N. Orf, A. Abouraddy, A. Stolyarov, J. Joannopoulos, H. Stone, and Y. Fink, “In-fiber semiconductor filament arrays,” Nano. Lett. **8**, 4265–4269 (2008). [CrossRef]

31. D. S. Deng, N. Orf, S. Danto, A. Abouraddy, J. Joannopoulos, and Y. Fink, “Processing and properties of centimeter-long, in-fiber, crystalline-selenium filaments,” Appl. Phys. Lett. **96**, 23102 (2010). [CrossRef]

33. J. Eggers and E. Villermaux, “Physics of liquid jets,” Rep. Prog. Phys. **71**, 36601 (2008). [CrossRef]

*μ*m). Another potentially important breakup process, one that is amenable to study even in the simplified case of a cylindrical-shell geometry, is axial instability. Not only does the possibility of axial breakup impose some limits on the practical materials for fiber drawing, but understanding such breakup is arguably a prerequisite to understanding the azimuthal breakup process, for two reasons. First, the azimuthal breakup process produces cylindrical filaments, and it is important to understand why these filaments do not exhibit further breakup into droplets (or under what circumstances this should occur). Second, it is possible that the draw-down process or other effects might couple fluctuations in the axial and azimuthal directions, so understanding the timescales of the axial breakup process is necessary as a first step in evaluating whether it plays any physical role in driving other instabilities.

*r,z*). (Indeed, this model is a satisfactory description of the cylindrical filaments.) We apply capillary instability to the experimental fiber-drawing situation, where we obtain a necessary (but not sufficient) condition for stability that can be used to guide the materials selection and the design of the fabrication process by excluding certain materials combinations from consideration. Our stability criterion is shown to be consistent with the experimental observations. We also find that the stability of the resulting filaments is consistent with the Rayleigh-Tomotika model.

35. A. D. Fitt, K. Furusawa, T. M. Monro, and C. P. Please, “Modeling the fabrication of hollow fibers: Capillary drawing,” J. Lightwave. Technol. **19**, 1924–1931 (2001). [CrossRef]

36. 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**, 853–860 (2006). [CrossRef]

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38. I. M. Griffiths and P. D. Howell, “Mathematical modelling of non-axisymmetric capillary tube drawing,” J. Fluid Mech. **605**, 181–206 (2008). [CrossRef]

## 2. Feature Size in Composite Microstructured Fibers

30. D. S. Deng, N. Orf, A. Abouraddy, A. Stolyarov, J. Joannopoulos, H. Stone, and Y. Fink, “In-fiber semiconductor filament arrays,” Nano. Lett. **8**, 4265–4269 (2008). [CrossRef]

31. D. S. Deng, N. Orf, S. Danto, A. Abouraddy, J. Joannopoulos, and Y. Fink, “Processing and properties of centimeter-long, in-fiber, crystalline-selenium filaments,” Appl. Phys. Lett. **96**, 23102 (2010). [CrossRef]

30. D. S. Deng, N. Orf, A. Abouraddy, A. Stolyarov, J. Joannopoulos, H. Stone, and Y. Fink, “In-fiber semiconductor filament arrays,” Nano. Lett. **8**, 4265–4269 (2008). [CrossRef]

## 3. Dimensionless Numbers During Thermal Drawing

*ρ*≈ 10

^{3}kg/m

^{3}is the density of the materials,

*g*≈ 10 m/s

^{2}is gravity,

*U*≈ 5 mm/s is drawing speed,

*η*≈ 10

^{5}Pa · s is viscosity,

*h*≈ 100 nm is the layer thickness, and

*γ*= 0.1 N/m is surface tension between polymer and glass [40, 41]. Therefore, these dimensionless numbers in a typical fiber draw are Small Re number, large Fr number, and large Ca number imply a weak inertia term, negligible gravity, and dominant viscosity effects, respectively. In addition, since the fiber diameter is

*D*≈ 1mm and the length of the neck-down region is

*L*≈ 10 cm, the ratio

*D/L*≈ 1/100 is much less than 1, and thus the complicated profile of neck-down cone is simplified into a cylindrical shape for the purpose of easier analysis.

## 4. Radial/Axial Instability Timescale

8. H. A. Stone and M. P. Brenner, “Note on the capillary thread instability for fluids of equal viscosities,” J. Fluid. Mech. **318**, 373–374 (1996). [CrossRef]

*τ*) for different values of the radius (

*r*) and the viscosity (

*η*) is displayed in Fig. 4. The cross-sectional geometry in the calculation is shown in the inset: interface I is located at radius

*r*, the cylindrical-shell thickness is

*h*, and interface II is at radius

*R*=

*r*+

*h*. The interfacial tension in the calculations was set to

*γ*= 0.1N/m, which was the measured interfacial tension between thermoplastic polymer and chalcogenide glass used in our microstructured fibers [40].

*η*

_{shell}=

*η*

_{clad}), and the other is unequal viscosity (

*η*

_{shell}≠

*η*

_{clad}). In the case of

*η*

_{shell}=

*η*

_{clad}, the instability time scale is calculated exactly from Stone and Brenner’s linear theory [8

8. H. A. Stone and M. P. Brenner, “Note on the capillary thread instability for fluids of equal viscosities,” J. Fluid. Mech. **318**, 373–374 (1996). [CrossRef]

*Ψ(*

_{λ}*λ*,

*R*/

*r*) was found by searching numerically within a wide range of wavelengths

*λ*for a certain value of

*R/r*(Appendix B). Figure 4 plots this time scale versus radius for

*η*= 10

^{5}Pa · s corresponding to As

_{2}Se

_{3}–PES, compared to the dwelling time

*τ*

_{dwelling}≈ 100 sec which is defined by the time of materials in viscous state before exiting hot furnace to be frozen in fiber during thermal drawing [41].

*η*

_{shell}≠

*η*

_{clad}(in the

*η*

_{clad}/

*η*

_{shell}≫ 1 regime, more details in Appendix A), the instability time scale can be roughly estimated from dimensional analysis. Although dimensionless analysis does not give the constant factor, for specificity, we choose the constant coefficient from the Tomotika model [7

7. S. Tomotika, “On the instability of a cylinderical thread of a viscous liquid surrounded by another viscous fluid,” Proc. Roy. Soc. London. **150**, 322–337 (1935). [CrossRef]

*[(1 –*

_{λ}*x*

^{2})Φ(

*x*,

*η*/

_{clad}*η*)] was found numerically by searching a wide range of wavelengths (

_{shell}*x*= 2

*πr*/

*λ*) [Φ(x,

*η*

_{clad}/

*η*

_{shell}) is a complicated implicit function of wavelength and viscosity contrast given in Ref. [7

7. S. Tomotika, “On the instability of a cylinderical thread of a viscous liquid surrounded by another viscous fluid,” Proc. Roy. Soc. London. **150**, 322–337 (1935). [CrossRef]

*η*

_{shell}= 10 Pa · s,

*η*

_{clad}= 10

^{5}Pa · s, corresponding to Se–PSU showing that the observed stability of shells of radius ≈ 250

*μ*m is consistent with the radial stability criterion (

*τ*>

*τ*

_{dwelling}). On the other hand, if

*η*

_{clad}is reduced to 10

^{3}Pa · s with the same shell materials, corresponding to Se–PE, we predict that radial fluctuations alone will render the shell unstable for any radius

*r*≤ 1 mm.

## 5. Applications in Microstructured Fibers

*consistent*with observed stable structures, whether it is sufficient to explain the observed azimuthal breakup, and what materials combinations are

*excluded*. Below, in Sec 5.1 we consider the application of radial stability analysis to the observed stability or instability of cylindrical shells. In Sec 5.2 we look at the impact on materials selections, and in Sec 5.3 we show that the observed stability of the resulting nanoscale filaments is consistent with the Tomotika model. Finally, in Sec 5.4, the observed azimuthal instability is briefly discussed.

### 5.1. Comparison with Observations for Cylindrical Shells

*μ*m). First, the map predicts that feature sizes down to submicrometers and hundreds of nanometers are consistent with radial stability for the equal-viscosity materials combination of

*η*

_{clad}=

*η*

_{shell}= 10

^{5}Pa · s, which corresponds to As

_{2}Se

_{3}–PES or As

_{2}S

_{3}–PEI. For As

_{2}Se

_{3}–PES, Fig. 2(c) shows that a shell thickness of As

_{2}S

_{3}of 1

*μ*m is obtained; in other work, layers of As

_{2}S

_{3}down to 15 nm have been achieved as well [30

**8**, 4265–4269 (2008). [CrossRef]

_{2}S

_{3}–PEI, Fig. 2(d) demonstrates a thickness of As

_{2}S

_{3}down to 32 nm. Second, the map is consistent with thicknesses down to submicrometers for unequal-viscosity materials with

*η*

_{clad}= 10

^{5}Pa · s,

*η*

_{shell}= 10 Pa · s, which corresponds to Se–PSU. Se layers with thickness on the order of 1

*μ*m have been demonstrated in Se–PSU fiber [30

**8**, 4265–4269 (2008). [CrossRef]

*radially*stable down to tens of nanometers. However, we found in experiments [Fig. 2(e), 2(f)] that a Se shell with thickness < 100 nm breaks up into continuous filament arrays [30

**8**, 4265–4269 (2008). [CrossRef]

31. D. S. Deng, N. Orf, S. Danto, A. Abouraddy, J. Joannopoulos, and Y. Fink, “Processing and properties of centimeter-long, in-fiber, crystalline-selenium filaments,” Appl. Phys. Lett. **96**, 23102 (2010). [CrossRef]

_{2}Se

_{3}–PES materials combination, this filamentation of As

_{2}Se

_{3}film was also observed as the thickness is reduced down to 10 nm [30

**8**, 4265–4269 (2008). [CrossRef]

### 5.2. Materials Selection

_{2}Se

_{3}, As

_{2}S

_{3}) are calculated from an empirical Arrhenius formula at the associated temperature during thermal drawing (more details in Appendix C); several thermoplastic polymers (PSU, PES, and PEI) have similar viscosities

*η*

_{polymer}≈ 10

^{5}Pa · s during fiber drawing [41]; and the viscosity of the polymer PE is 10

^{3}Pa · s at temperature T = 250°C [42

42. V. F. Dobrescu and C. Radovici, “Temperature dependence of melt viscosity of polymers,” Polym. Bull. **10**, 134–140 (1983). [CrossRef]

*γ*= 0.1 N/m [40] for all of these materials. Assuming a cylindrical shell of radius ≈ 250

*μ*m and a dwelling time of thermal drawing ≈ 100 sec, we can classify each materials combination by whether it falls in the

*τ*>

*τ*

_{dwelling}yellow region (radially stable) of Fig. 4 or in the

*τ*<

*τ*

_{dwelling}white region (radially unstable).

*τ*=

*τ*

_{dwelling}, divides the map into two areas. The shaded area above the boundary line is the region of potentially suitable materials combinations for fiber drawing (As

_{2}Se

_{3}–PES, As

_{2}S

_{3}–PEI, Se–PSU) [26

26. A. F. Abouraddy, M. Bayindir, G. Benoit, S. D. Hart, K. Kuriki, N. Orf, O. Shapira, F. Sorin, B. Temelkuran, and Y. Fink, “Towards multimaterial multifunctional fibres that see, hear, sense and communicate,” Nat. Mater. **6**, 336–347 (2007). [CrossRef] [PubMed]

**8**, 4265–4269 (2008). [CrossRef]

43. S. Egusa, Z. Wang, N. Chocat, Z. M. Ruff, A. M. Stolyarov, D. Shemuly, F. Sorin, P. T. Rakich, J. D. Joannopoulos, and Y. Fink, “Multimaterial piezoelectric fibres,” Nat. Mater. **9**, 643–648 (2010). [CrossRef] [PubMed]

*τ*), we predict that a wider variety of shell materials with low viscosity may possibly be employed in microstructured fibers, such as the metals Sn and In [26

26. A. F. Abouraddy, M. Bayindir, G. Benoit, S. D. Hart, K. Kuriki, N. Orf, O. Shapira, F. Sorin, B. Temelkuran, and Y. Fink, “Towards multimaterial multifunctional fibres that see, hear, sense and communicate,” Nat. Mater. **6**, 336–347 (2007). [CrossRef] [PubMed]

28. A. Mazhorova, J. F. Gu, A. Dupuis, M. Peccianti, O. Tsuneyuki, R. Morandotti, H. Minamide, M. Tang, Y. Wang, H. Ito, and M. Skorobogatiy, “Composite THz materials using aligned metallic and semiconductor microwires, experiments and interpretation,” Opt. Express **18**, 24632–24647 (2010). [CrossRef] [PubMed]

### 5.3. Stability of Continuous Filaments down to Submicrometer/Nanometer Scale

**8**, 4265–4269 (2008). [CrossRef]

**96**, 23102 (2010). [CrossRef]

**8**, 4265–4269 (2008). [CrossRef]

**96**, 23102 (2010). [CrossRef]

*τ*corresponds to exponential growth of a fluctuation amplitude

*ɛ*according to

*τ*is time varying, then the total amplitude growth is exp[∫

*dt*/

*τ*(

*t*)]. Converting to

*dz*=

*v*(

*z*)

*dt*for a position-dependent axial flow velocity

*v*(

*z*) during thermal drawing, we therefore obtain a total exponential growth factor: where

*z*∈ [0,

*L*] is axial position in the neck-down region with length (L = 6 cm). Γ ≫ 1 corresponds to breakup, while Γ ≪ 1 corresponds to stability.

**150**, 322–337 (1935). [CrossRef]

^{−3}mm/sec is the preform velocity. Again by incompressibility, the filament radius (r) should scale as the fiber radius (

*R*): The temperature distribution during thermal drawing, fit from experiment, is found to be approximately parabolic,

_{2}Se

_{3}–PES fiber drawing are R(0) = 1 cm, s = 20, L = 6 cm, p = 2, T

_{max}= 260 °C, T

_{min}= 210 °C, r(L) = 200 nm,

*η*

_{polymer}= 10

^{6}Pa·s. Figure 6(b)–6(d) presents the corresponding position–dependent variables including radius, velocity, temperature and viscosity. Finally, we obtain This satisfies Γ < 1, but only barely—if this were an accurate estimate of the growth factor, instability might still be observed. However, the assumptions we made above were so conservative that the true growth factor must be much less than this, indicating the instability should not be observable during the dwelling time of fiber drawing. So, the observed filaments are consistent with the Tomotika model, although of course we cannot yet exclude the possibility that there are also additional effects (

*e.g.,*elasticity) that further enhance stability.

### 5.4. Favorability of Azimuthal Versus Axial Instability

*λ*of maximum growth rate for small perturbations, and this

*λ*must be proportional to the characteristic feature size of the system, in this case the film thickness

*d*. As the fiber is drawn, however, the thickness

*d*and hence

*λ*decreases. Now, we consider what happens to an unstable perturbation that begins to grow at some wavelength

*λ*

_{0}when the thickness is

*d*

_{0}. If this is a perturbation along the

*axial*direction, then the fiber-draw process will

*stretch*this perturbation to a

*longer*wavelength, that will no longer correspond to the maximum-growth

*λ*(which is shrinking), and hence the growth will be damped [33

33. J. Eggers and E. Villermaux, “Physics of liquid jets,” Rep. Prog. Phys. **71**, 36601 (2008). [CrossRef]

*any*axial breakup process. In contrast, if

*λ*

_{0}is an

*azimuthal*perturbation, the draw-down process will

*shrink*

*λ*

_{0}along with the fiber cross-section at exactly the same rate that

*d*and

*λ*shrink. Therefore, azimuthal instabilities are

*not*suppressed by the draw process. This simple geometrical argument immediately predicts that the first observed instabilities will be azimuthal (although axial instabilities may still occur if the draw is sufficiently slow).

## 6. Concluding Remarks

*e.g.*, non-cylindrical), temperature-time profiles, or materials (

*e.g.*, Sn–PEI or Se–PE). Finally, by drawing more slowly so that axial breakup occurs, we expect that experiments should be able to obtain more diverse structures (

*e.g.*axial breakup into rings or complete breakup into droplets) that we hope to observe in the future.

## A. Direct numerical simulation of concentric cylindrical shells

**150**, 322–337 (1935). [CrossRef]

**318**, 373–374 (1996). [CrossRef]

*r,z*) plane. Numerical challenges in the simulations arise from the nonlinearity, moving interfaces, interface singularities, and the complex curvature [1

1. J. Eggers, “Nonlinear dynamics and breakup of free-surface flows,” Rev. Mod. Phys. **69**, 865–929 (1997). [CrossRef]

45. R. Scardovelli and S. Zaleski, “Direct numerical simulation of free-surface and interfacial flow,” Annu. Rev. Fluid Mech. **31**, 567–603 (1999). [CrossRef]

*ϕ*(

*x⃗,t*) is coupled with the NS equations to track the interface [46

46. S. Osher and J. A. Sethian, “Front propagating with curvature-dependent speed - alogrithms based on Hamilton-Jacobi formulations,” J. Comput. Phys. **79**, 12–49 (1988). [CrossRef]

49. E. Olsson and G. Kreiss, “A conservative level set method for two phase flow,” J. Comput. Phys. **210**, 225–246 (2005). [CrossRef]

*ϕ*= 0.5 contour and the

*ϕ*evolution is given by

*u⃗*via: The local curvature (

*κ*) at an interface is given in terms of

*ϕ*by:

49. E. Olsson and G. Kreiss, “A conservative level set method for two phase flow,” J. Comput. Phys. **210**, 225–246 (2005). [CrossRef]

*p*=

*γκ*) originating from azimuthal curvature of the cylindrical geometry at interfaces I and II. (ii)–(iv) The interfacial perturbations generate an axial pressure gradient Δ

*p*, and hence a fluid flow occurs that moves from a smaller-radius to a larger-radius region for the inner fluid. Gradually the amplitude of the perturbation is amplified. (v) The

*shrunk*smaller-radius and

*expanded*larger-radius regions of inner fluid further enhance the axial pressure gradient Δ

*p*, resulting in a larger amplitude of the perturbation. As a result, the small perturbation is exponentially amplified by the axial pressure gradient.

*η*

_{clad}) with a fixed shell viscosity (

*η*

_{shell}), in order to help us to identify suitable cladding materials for fiber fabrication. The time-dependent perturbation amplitude curves for various viscosity contrast

*η*

_{clad}/

*η*

_{shell}are obtained by changing the cladding viscosity (

*η*

_{shell}= 10

^{5}Pa · s). (Other parameters in the simulation are

*ρ*= 10

^{3}kg/m

^{3},

*γ*= 0.6 N/m,

*R*= 120

*μ*m.) Instability time scale for the each given viscosity contrast is obtained by exponentially fitting the curves of time-dependent instability amplitude. Instability time scale (

*τ*) as a function of viscosity contrast is presented in Fig. 3. The existing linear theory has only been solved in case of equal viscosity, and predicts that the instability time scale is proportional to the viscosity

*τ*∼

*η*[8

**318**, 373–374 (1996). [CrossRef]

*η*

_{clad}→ 0, the instability time scale should be determined by

*η*

_{shell}, and from dimensional analysis should be proportional to

*rη*

_{shell}/

*γ*, assuming that the inner and outer radius are comparable and so we take

*r*to be the average radius. In the opposite limit of

*η*

_{clad}→ ∞, the time scale should be determined by

*η*

_{clad}and hence should be proportional to

*rη*

_{clad}/

*γ*. In between these two limits, we expect the time scale to smoothly interpolate between the

*rη*

_{shell}/

*γ*and

*rη*

_{clad}/

*γ*scales. In a companion paper [34], we present a generalized analytical linear theory for multi-fluid cylindrical structures, and show that this dimensional analysis is consistent with the exact asymptotic result.

## B. Linear theory of concentric cylindrical shells with equal viscosities

**318**, 373–374 (1996). [CrossRef]

*σ*) for a wave vector

*k*= 2

*π*/

*λ*is a solution of the following quadratic equations

*r*and

*R*are the radii of the unperturbed interfaces I and II,

*γ*

_{1}and

*γ*

_{2}are the interfacial tensions, and

*η*is viscosity. Λ(

*a,b*), where

*a*≤

*b*, is associated with the modified Bessel function, For the case of

*γ*

_{1}=

*γ*

_{2}=

*γ*, the growth rate has the following formula, where the growth factor of Ψ(

*λ*,

*R*/

*r*) in Eq. (17) is a complicated function of instability wavelength [8

**318**, 373–374 (1996). [CrossRef]

*τ*∼

*σ*

^{−1}∼

*ηr*/

*γ*is scaled with radius. For the case of

*R*= 2

*r*, this growth factor is calculated in Fig. 8. A positive growth factor indicates a positive growth rate (

*σ*> 0), for which any perturbation is exponentially amplified with time. Instability occurs at long wavelengths above a certain critical wavelength. Two critical wavelengths exist for the co-axial cylinder shell. One is a short critical wavelength

*λ*= 2

_{f}*πr*for a faster-growth mode (red line). The other is a long critical wavelength

*λ*= 2

_{s}*πR*for slower-growth mode (blue line).

## C. Viscosity of Materials During Thermal Drawing

_{2}Se

_{3}, and As

_{2}S

_{3}) and thermoplastic polymers (PES, PEI, and PSU). The viscosity of chalcogenide glass-forming melts depends on temperature and is calculated from an empirical Arrhenius formula [50

50. P. G. Debenedetti and F. H. Stillinger, “Supercooled liquids and the glass transition,” Nature **410**, 259–267 (2001). [CrossRef] [PubMed]

*R*is the ideal gas constant,

*T*is the temperature in Kelvin, and

*η*is viscosity in Pa · s. The parameters of log

*η*

_{0},

*C*, and

*D*for our materials are listed below: −2.0, 6651, 770.82 for Se, 3.09, 18877.8, 875.56 for As

_{2}Se

_{3}, and −3.62, 33744, 650.8 for As

_{2}S

_{3}[51

51. A. S. Tverjanovich, “Temperature dependence of the viscosity of chalcogenide glass-forming melts,” Glass Phys. Chem. **29**, 532–536 (2003). [CrossRef]

_{2}Se

_{3}, or As

_{2}S

_{3}films is around 220, 260, or 300

^{o}C, respectively, with the corresponding viscosities of 10, 10

^{5}, or 10

^{5}Pa · s, respectively.

## Acknowledgments

## References and links

1. | J. Eggers, “Nonlinear dynamics and breakup of free-surface flows,” Rev. Mod. Phys. |

2. | P. de Gennes, F. Brochard-Wyart, and D. Quere, |

3. | J. B. Fournier and A. M. Cazabat, “Tears of wine,” Europhys. Lett. |

4. | J. Plateau, |

5. | L. Rayleigh, “On the capillary phenomena of jets,” Proc. Roy. Soc. London |

6. | L. Rayleigh, “On the instability of a cylinder of viscous liquid under capillary force,” Philos. Mag. |

7. | S. Tomotika, “On the instability of a cylinderical thread of a viscous liquid surrounded by another viscous fluid,” Proc. Roy. Soc. London. |

8. | H. A. Stone and M. P. Brenner, “Note on the capillary thread instability for fluids of equal viscosities,” J. Fluid. Mech. |

9. | X. D. Shi, M. P. Brenner, and S. R. Nagel, “A cascade of structure in a drop falling from a faucet,” Science |

10. | A. M. Ganan-Calvo, R. Gonzalez-Prieto, P. Riesco-Chueca, M. A. Herrada, and M. Flores-Mosquera, “Focusing capillary jets close to the continuum limit,” Nat. Phys. |

11. | M. Moseler and U. Landman, “Formation, stability, and breakup of nanojets,” Science |

12. | M. E. Toimil-Molares, A. G. Balogh, T. W. Cornelius, R. Neumann, and C. Trautmann, “Fragmentation of nanowires driven by Rayleigh instability,” Appl. Phys. Lett. |

13. | S. Karim, M. E. Toimil-Molares, A. G. Balogh, W. Ensinger, T. W. Cornelius, E. U. Khan, and R. Neumann, “Morphological evolution of Au nanowires controlled by Rayleigh instability,” Nanotechnology |

14. | J. T. Chen, M. F. Zhang, and T. P. Russell, “Instabilities in nanoporous media,” Nano. Lett. |

15. | Y. Qin, S.M. Lee, A. Pan, U. Gosele, and M. Knez, “Rayleigh-instability-induced metal nanoparticle chains encapsulated in nanotubes produced by atomic layer deposition,” Nano. Lett. |

16. | H. A. Stone, A.D. Stroock, and A. Ajdari, “Engineering flows in small devices: Microfluidics toward a lab-on-a-chip,” Annu. Rev. Fluid. Mech. |

17. | T. M. Squires and S. R. Quake, “Microfluidics: fluid physics at the nanoliter scale,” Rev. Mod. Phys. |

18. | R. Huang and Z. Suo, “Wrinkling of a compressed elastic film on a viscous layer,” J. Appl. Phys. |

19. | E. Cerda, K. Ravi-Chandar, and L. Mahadevan, “Thin films—wrinkling of an elastic sheet under tension,” Nature |

20. | E. Cerda and L. Mahadevan, “Geometry and physics of wrinkling,” Phys. Rev. Lett. |

21. | D. B. Keck, R. D. Maurer, and P. C. Schultz, “On the ultimate lower limit of attenuation in glass optical waveguides,” Appl. Phys. Lett. |

22. | G. P. Agrawal, |

23. | S. D. Hart, G. R. Maskaly, B. Temelkuran, P. H. Prideaux, J. D. Joannopoulos, and Y. Fink, “External reflection from omnidirectional dielectric mirror fibers,” Science |

24. | B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO |

25. | M. Bayindir, F. Sorin, A. F. Abouraddy, J. Viens, S. D. Hart, J. D. Joannopoulos, and Y. Fink, “Metal-insulator-semiconductor optoelectronic fibres,” Nature , |

26. | A. F. Abouraddy, M. Bayindir, G. Benoit, S. D. Hart, K. Kuriki, N. Orf, O. Shapira, F. Sorin, B. Temelkuran, and Y. Fink, “Towards multimaterial multifunctional fibres that see, hear, sense and communicate,” Nat. Mater. |

27. | M. Yaman, T. Khudiyev, E. Ozgur, M. Kanik, O. Aktas, E. O. Ozgur, H. Deniz, E. Korkut, and M. Bayindir, “Arrays of indefinitely long uniform nanowires and nanotubes,” Nat. Mater. |

28. | A. Mazhorova, J. F. Gu, A. Dupuis, M. Peccianti, O. Tsuneyuki, R. Morandotti, H. Minamide, M. Tang, Y. Wang, H. Ito, and M. Skorobogatiy, “Composite THz materials using aligned metallic and semiconductor microwires, experiments and interpretation,” Opt. Express |

29. | J. N. Winn, Y. Fink, S. H. Fan, and J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. |

30. | D. S. Deng, N. Orf, A. Abouraddy, A. Stolyarov, J. Joannopoulos, H. Stone, and Y. Fink, “In-fiber semiconductor filament arrays,” Nano. Lett. |

31. | D. S. Deng, N. Orf, S. Danto, A. Abouraddy, J. Joannopoulos, and Y. Fink, “Processing and properties of centimeter-long, in-fiber, crystalline-selenium filaments,” Appl. Phys. Lett. |

32. | S. Chandrasekhar, |

33. | J. Eggers and E. Villermaux, “Physics of liquid jets,” Rep. Prog. Phys. |

34. | X. Liang, D. S. Deng, J.-C. Nave, and S. G. Johnson, “Linear stability analysis of capillary instabilities for concentric cylindrical shells,” J. Fluid. Mech. (in press). |

35. | A. D. Fitt, K. Furusawa, T. M. Monro, and C. P. Please, “Modeling the fabrication of hollow fibers: Capillary drawing,” J. Lightwave. Technol. |

36. | 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. |

37. | 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 |

38. | I. M. Griffiths and P. D. Howell, “Mathematical modelling of non-axisymmetric capillary tube drawing,” J. Fluid Mech. |

39. | G. K. Batchelor, |

40. | S. D. Hart and Y. Fink, “Interfacial energy and materials selection criteria in composite microstructured optical fiber fabrication,” Mat. Res. Soc. Symp. Proc. |

41. | S. D. Hart, Multilayer composite photonic bandgap fibers, PhD thesis, MIT (2004). |

42. | V. F. Dobrescu and C. Radovici, “Temperature dependence of melt viscosity of polymers,” Polym. Bull. |

43. | S. Egusa, Z. Wang, N. Chocat, Z. M. Ruff, A. M. Stolyarov, D. Shemuly, F. Sorin, P. T. Rakich, J. D. Joannopoulos, and Y. Fink, “Multimaterial piezoelectric fibres,” Nat. Mater. |

44. | M. F. Culpin, “The viscosity of liquid indium and liquid tin,” Proc. Phys. Soc. |

45. | R. Scardovelli and S. Zaleski, “Direct numerical simulation of free-surface and interfacial flow,” Annu. Rev. Fluid Mech. |

46. | S. Osher and J. A. Sethian, “Front propagating with curvature-dependent speed - alogrithms based on Hamilton-Jacobi formulations,” J. Comput. Phys. |

47. | S. Osher and R. P. Fedkiw, “Level set methods: An overview and some recent results,” J. Comput. Phys. |

48. | J. A. Sethian and P. Smereka, “Level set methods for fluid interfaces,” Annu. Rev. Fluid Mech. |

49. | E. Olsson and G. Kreiss, “A conservative level set method for two phase flow,” J. Comput. Phys. |

50. | P. G. Debenedetti and F. H. Stillinger, “Supercooled liquids and the glass transition,” Nature |

51. | A. S. Tverjanovich, “Temperature dependence of the viscosity of chalcogenide glass-forming melts,” Glass Phys. Chem. |

**OCIS Codes**

(060.2280) Fiber optics and optical communications : Fiber design and fabrication

(060.2290) Fiber optics and optical communications : Fiber materials

(230.1480) Optical devices : Bragg reflectors

(230.4000) Optical devices : Microstructure fabrication

(060.4005) Fiber optics and optical communications : Microstructured fibers

(220.4241) Optical design and fabrication : Nanostructure fabrication

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: May 31, 2011

Revised Manuscript: July 25, 2011

Manuscript Accepted: August 1, 2011

Published: August 10, 2011

**Citation**

D. S. Deng, J.-C. Nave, X. Liang, S. G. Johnson, and Y. Fink, "Exploration of in-fiber nanostructures from capillary instability," Opt. Express **19**, 16273-16290 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-17-16273

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### References

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- M. Yaman, T. Khudiyev, E. Ozgur, M. Kanik, O. Aktas, E. O. Ozgur, H. Deniz, E. Korkut, and M. Bayindir, “Arrays of indefinitely long uniform nanowires and nanotubes,” Nat. Mater. 10, 494–501 (2011). [CrossRef] [PubMed]
- A. Mazhorova, J. F. Gu, A. Dupuis, M. Peccianti, O. Tsuneyuki, R. Morandotti, H. Minamide, M. Tang, Y. Wang, H. Ito, and M. Skorobogatiy, “Composite THz materials using aligned metallic and semiconductor microwires, experiments and interpretation,” Opt. Express 18, 24632–24647 (2010). [CrossRef] [PubMed]
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- A. D. Fitt, K. Furusawa, T. M. Monro, and C. P. Please, “Modeling the fabrication of hollow fibers: Capillary drawing,” J. Lightwave. Technol. 19, 1924–1931 (2001). [CrossRef]
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- S. Egusa, Z. Wang, N. Chocat, Z. M. Ruff, A. M. Stolyarov, D. Shemuly, F. Sorin, P. T. Rakich, J. D. Joannopoulos, and Y. Fink, “Multimaterial piezoelectric fibres,” Nat. Mater. 9, 643–648 (2010). [CrossRef] [PubMed]
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- E. Olsson and G. Kreiss, “A conservative level set method for two phase flow,” J. Comput. Phys. 210, 225–246 (2005). [CrossRef]
- P. G. Debenedetti and F. H. Stillinger, “Supercooled liquids and the glass transition,” Nature 410, 259–267 (2001). [CrossRef] [PubMed]
- A. S. Tverjanovich, “Temperature dependence of the viscosity of chalcogenide glass-forming melts,” Glass Phys. Chem. 29, 532–536 (2003). [CrossRef]

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