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Abstract
Viscosity of optical fiber plays an important role in high temperature applications in harsh environments. A size-dependent viscosity phenomenon of silica optical fiber under high temperature is observed by the stretching method with in-fiber regenerated fiber Bragg gratings (RFBGs). Higher viscosity is derived from optical fiber with larger diameters. The mechanisms of this size-dependent relationship of high temperature viscosity of silica optical fiber are discussed, and the difference in fictive temperature of silica optical fibers is considered as the mainly contribution. An equivalent fictive temperature of optical fiber at a metastable structure state is used to explain the difference in equilibrium viscosity. Then the relationship between equilibrium viscosity and fictive temperature for silica optical fiber at 1000 °C is derived.
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1. Introduction
Viscosity is an important parameter of optical fiber in fiber drawing process and fiber-based high temperature sensing applications [1–3]. Silica optical fiber suffers harsh environments in many application fields such as nuclear reactors, turbines, aerospace, copper and aluminum casting [4–8]. In order to improve and optimize the performance of optical fiber devices, the viscoelasticity of silica optical fiber has received great attention. Some innovative optical devices are effectively promoted by using viscoelasticity of silica fibers, such as ultra-high temperature chirped fiber Bragg gratings [9,10]. The assessment of viscoelasticity property of silica optical fiber can be helpful for correcting the error of strain and stress sensing under high temperature [11]. A new perspective for lifetime prediction of silica fiber Bragg grating directly written by a femtosecond laser is also provided by viscosity [12]. Higher viscosity means better high temperature resistance of fiber Bragg grating, even written by different fabrication technique such as UV or femtosecond laser writing. With the development of the application field of high-temperature optical fiber devices, the using of optical fiber at high temperature will no longer be limited to general single mode fiber but tend to various sizes and special properties fibers. Therefore, it is particularly meaningful to study the viscosity and viscoelasticity behaviors of different sizes silica optical fibers under high temperature.
Viscosity of general telecommunication silica optical fiber is dominated by cladding, which is generally pure silica glass and makes up nearly 99% of entire fiber volume. Nevertheless, it is inappropriate to use the viscosity value of bulk glass as a substitute for that of optical fiber, because the viscosity of silica optical fiber is confirmed to be lower than that of the bulk silica glass with similar chemical composition [13–16]. Simultaneously, the activation energy for viscous flow of silica optical fiber is obtained from a good Arrhenius fit and illustrated to be much lower than that of bulk glass in a similar temperature range, which means a lower viscosity and the high temperature flow occurs more easily in fiber than in bulk glass [15]. The difference in viscosity between silica optical fiber and bulk silica glass is considered to be induced by the different fictive temperatures (Tf), given similar chemical composition. because higher Tf of silica glass generally leads to lower viscosity [14,17]. Tf is related to the cooling rate of glass forming process. Ultrafast cooling rate during optical fiber drawing process leads to a high Tf (generally 1400∼1600 °C) of silica optical fiber, which is several hundred degrees Celsius higher than that of general bulk silica glass (generally 1000∼1200 °C). So that the fiber manufacturing technology and the micron-scale of silica optical fiber make the glass properties of optical fiber distinguish from bulk glass, despite considering a same chemical composition. Meanwhile, silica optical fiber with larger physical size has lower Tf and the Tf of a 125 µm pure silica core optical fiber is about 1600 °C larger than that of 1600 µm fiber [18]. This implies an optimization path of optical fiber devices that larger diameter optical fiber possibly has better thermostability performance because of the higher viscosity. Nevertheless, on the one hand, the equilibrium viscosity of optical fiber will be overestimated than the actual situation if using the Tf dependence relationship of bulk silica glass. On the other hand, it is difficult to establish a relationship between viscosity and geometric size of optical fiber effectively by directly comparing the reported viscosity of silica optical fiber and bulk silica glass. Therefore, it is essential to evaluate the viscosity of silica optical fiber with different diameters under high temperature.
In the present work, to clarify the dependence of geometry size on the viscosity of silica optical fiber, a fiber stretching method was used to demonstrate the viscosity of silica optical fiber with different diameters of 80, 130 and 250 µm, respectively. Various factors that possibly affect viscosity would be discussed seriously, including chemical composition, surface properties and Tf. This work will help to evaluate the high-temperature performance of optical fiber with different sizes and promote the development of various high-temperature optical fiber devices.
2. Fiber samples preparation and experimental
The viscosity of silica glass is affected by the thermal history, as well as surface, chemical composition and so on. The impact of thermal history to glass structural state can be characterized by Tf, which is the highest temperature at which the glass is allowed to reach an equilibrium state before a rapid quench to room temperature [19]. It has been known that higher fictive temperature of silica glass is associated with the decrease of average Si-O-Si bond angle [20,21], as shown in Fig. 1. Moreover, a resent molecular dynamics simulation work shows that higher fictive temperature could be correlated with the average length of Si-O bond. Some properties of silica glass such as viscosity η and density ρ also vary with Tf, although follow opposite correlations [22,23]. The fictive temperatures of optical fibers with different diameters can be separated due to different cooling rates during fiber drawing process. Therefore, it is reasonable to use different diameter fibers to clarify the size effect on viscosity.
Three kinds of GeO2-doped silica optical fibers with different diameters as 80, 130, and 250 µm, labeled as F80µm, F130µm and F250µm, respectively (Small Diameter Bend Insensitive SMF (SDBI-SMF), SG1010-A, DG1110-D, YOFC Optical Fiber & Cable Co., Ltd., Wuhan, China), are used for matching our experimental need. The main parameters of optical fibers are shown in Table 1, and the refractive index of fiber core is calculated from NA. For all experimental fibers drawing from high pure silica glass, the outer cladding is pure SiO2 and the inner core is doped with GeO2.
Table 1. Parameters of Silica Optical Fibers
The seed fiber Bragg grating (SFBG) is inscribed into hydrogen-loaded silica optical fiber (Shandong Shenghai Optical Fiber Technology Co., Ltd.). The schematic diagram of experimental set-up is shown in Fig. 2(a). The temperature procedure is performed after the coating-removed SFBG is placed into the uniform temperature zone of a high temperature tube furnace (T1250S/T1225S, Henan Chengyi Laboratory Equipment Co., Ltd., Zhengzhou, China). The length of uniform temperature zone of the tube furnace is ∼5 cm, which is much longer than the length of the grating area (∼1 cm), ensuring that the part of the grating can be uniformly annealed. The temperature is controlled to rise from room temperature to 900 °C with a constant rate bout 13 °C/min. After the sufficient regeneration of FBGs, the temperature naturally cools down to room temperature. An amplified spontaneous emission (ASE)-based broadband source (A-0002, Shenzhen Hoyatek Co., Ltd., Shenzhen, China) and an optical spectrum analyzer (OSA, AQ6317C, Yokogawa Electric Corporation, Tokyo, Japan) are used to monitor the reflection spectrum of the gratings over the entire temperature process. In order to ensure the accuracy of the experiment, two Bragg gratings from the same batch in both three kinds optical fibers were used for regeneration and stretching processes, and denoted as -1 and -2 at the end of fiber label, such as F80µm-1 and F80µm-2. As shown in the Fig. 2(b), the RFBG production process is illustrated by monitoring and displaying the reflection intensity of the seed Bragg grating. During the stage of annealing treatment at 900°C, firstly, the reflection intensity of grating gradually decreased to noise level due to the attenuation of refractive index modulation of SFBG until it was completely erased, named as “erasing process”. Then, the refractive index modulation of grating gradually regenerated and increased to a saturation value, leading to stable reflection intensity. This process is known as “regeneration process”. Ultimately, RFBGs of different fibers were obtained after sufficient regeneration of the seed gratings. The RFBGs have more excellent high temperature resistance than SFBGs. In the F80µm and F130µm, the erasing-regeneration process exhibited good consistency, including similar erasing time and final reflected intensity. Note that the Bragg grating reflection intensity of the F250µm fluctuates when the temperature rises and falls. This phenomenon is due to the interference of coupling modes at different wavelengths when F250µm is spliced with standard single-mode fiber patch cord, which does not affect the monitoring of the center wavelength.
A RFBG stretching method has been proved for observing viscosity of silica optical fiber [15]. When the temperature of the tube furnace reached 1000 °C and stabilized, the optical fiber was loaded for stretching while the change in the central wavelength of the RFBG was recorded by OSA using a resolution of 0.05 nm. Since tensile stress affected the relaxation process of silica fibers [16,24], we needed to use different loads to match fibers with different diameters to ensure that the tensile stress was almost equal. The loads of 6.98, 18.43 and 67.76 g were applied to optical fibers with diameters of F80µm, F130µm and F250µm, respectively. The results of two repeated experiments are shown in Fig. 3. The fiber will undergo several stages, including heating up, applying load and unloading, and cooling down. The vertical dashed line is used to distinguish the stretching part during the load is applied. The central wavelength of the RFBG is continuously red-shifting under constant tensile stress. In order to observe the long-range variation of silica fiber at high temperature, the stretching time of every RFBG is designed to be as long as possible. The longest stretching time exceeds 465 min under the premise that the reflection spectrum maintains a good Gaussian shape. The nonlinear change of wavelength shifts can be observed in the first few dozen minutes. Then after long time stretching, the wavelength shifts of all fiber samples increase almost linearly with time. Meanwhile, in order to demonstrate the permanent deformation caused by stretching, the load would release for nearly half an hour at the same annealing temperature, then the programmed cooling procedure was carried out. As shown in the inset of Fig. 3, wavelength shift of the relaxing part of F250µm-1 is magnified for the eyes. After a slightly blue-shifting, the central wavelength of RFBG tends to slowly relax to a stable value, but much larger than the initial value. The unrecoverable part of wavelength shifts reveals the viscoelasticity permanent elongation of optical fiber.
3. Size-dependent viscosity of optical fiber
The fibers are considered to be uniformly stretched in the constant temperature region, and the tensile strain can be calculated as ε = ΔλB/[(1-pe)λB], where λB is the initial wavelength, and pe = 0.22 is the effective elasto-optical coefficient. As an important parameter to describe the viscoelastic properties of materials at high temperature, viscosity can be expressed as the ratio of stress to strain rate, η = σ0/(dε/dt), where tensile stress σ0 = mg/(πr2), g = 9.8 m/s2, r is the radius of fiber, and m is the load mass. Tensile stress is considered constant because the change in fiber cross-sectional area caused by stretching is too small so that it can be completely ignored. Strain-time curves of stretching processes were divided into different segments, and the slope of each individual segment was obtained by using a linear fit. Viscosity of different fibers at 1000 °C are shown in Fig. 4, and the equilibrium viscosities are shown in Fig. 5.
The viscosity of silica fibers gradually tends to approach equilibrium state because of structural rearrangements induced by the high temperature structural relaxation [16,17], and the glass structure of silica optical fiber reaches a metastable equilibrium. Every fiber sample in this work was fully annealed and considered to be sufficiently relaxed after stretching for 90 min. Meanwhile, each fiber was stretched for as long as possible if the central wavelength of the RFBG reflection spectrum allows to be well distinguished. For different diameter fiber samples, the equilibrium viscosities are shown in Fig. 5. The error bars correspond to the fluctuation range of the standard deviation of the equilibrium viscosity. The log of equilibrium viscosities of F80µm, F130µm and F250µm at 1000 °C is averaged as 13.343, 13.436 and 13.561, respectively. A clear size dependence can be observed, which represents the correlation of viscosity on silica fiber geometry diameters: a larger diameter corresponding to the larger viscosity.
4. Origin of size-dependent viscosity
Potential factors possibly affecting viscosity must be removed for confirming objectivity of the size-dependent viscosity and tracing back the origin. The viscosity of silica glass could be affected by many factors such as manufacturing technology, temperature (including experimental temperature and thermal history) and chemical composition [25]. It is difficult to ensure that the thermal history of all experimental samples are completely consistent and that is why viscosities obtained by different researchers tend to slightly different. For example, Viscosities of YOFC silica optical fibers in this work are slightly lower than that of reported Corning's SMF-28 fiber using same RFBG stretching method, i.e., about 13.64 at 1000 °C [15,16]. Another reported viscosity of silica optical fiber is obtained by a fiber bending method and that is lower than the viscosities measuring under same temperature by RFBG stretching method [13]. At all events, viscosity of silica optical fiber is invariably much lower than that of bulk silica glass, the log of the viscosity of the latter is about 15.82 at 1000 °C [14]. In this work, the effect of production processes on the experimental samples can be excluded firstly, given that all optical fibers were drawn from the same manufacturer. The temperature dependence of viscosity needs to be evaluated. As demonstrated in the experimental procedure, all fiber samples undergo the same temperature program before stretching so the thermal history can be considered identical. Furthermore, the effect of the thermo-optic coefficient on the central wavelength shift of the RFBGs can be removed because the stretching processes of all fiber samples are dwelled at 1000 °C. During the long time annealing process at a constant high temperature, the axial stress in cross section of optical fiber and the frozen-in strain along the fiber are both significantly decreased when sufficient relaxation occurs, which allowed the impact on viscosity minimizing to a negligible level [26–28]. On the basis of excluding these above factors, there are three possible impact factors on viscosity need to be further discussed, i.e., chemical composition, surface properties and Tf.
4.1 Chemical composition
The chemical composition directly leads to different physical properties of glass materials, and dopants will significantly affect the viscosity of silica glass. Whereas, a single material hypothesis is generally considered for the researches on viscosity of silica optical fibers [15]. The core with GeO2 is unlikely to affect the results to any appreciable level because of the small volume proportion (about 1% or less) [11]. A concern regarding the chemical composition of optical fibers is that their high-temperature resistance can be impacted by dopant diffusion, particularly when annealed at temperatures close to the glass transition temperature region. It is known that, under the same thermal conditions, optical fibers with higher viscosity have better thermal stability because the reduce of dopant diffusion effect. At elevated temperatures, the dopant diffusion occurring in radial direction will theoretically reduce the refractive index difference between optical fiber core and cladding, while the axial diffusion will reduce the refractive index modulation of the grating. As a result, this will lead to a decrease in the reflection intensity and the central wavelength shift of SFBG/RFBGs. One advantage of using RFBG to monitor viscosity is that it is embedded within the optical fiber, offering excellent high-temperature stability. When annealing at a high temperature, the central wavelength shift of RFBG without load is particularly smaller than that of loaded RFBG (the wavelength shift of RFBG without load is about 0.15 nm when it is kept at 1000 °C for 270 min), which induce negligible effect on measurement of viscosity. In addition, the thermal dopant diffusion has a clear effect on GeO2-doped silica fiber which usually occurs at not less than 1200 °C [29]. To sum up, when studying viscosity using RFBG, the influence of dopant diffusion can be disregarded. Considering that the difference in the log of viscosities of three fiber samples is ∼0.1, the effect of the chemical composition of the three types of GeO2-doped silica optical fiber was scrupulously evaluated. Doping of GeO2 will induce a decrease in the viscosity of silica glass and an increase in the refractive index, and the relationship between the refractive index and GeO2-dopant content is linear [30]. A linear relationship was approximatively established using the refractive index of standard SMF-28 (cladding index is 1.4615 and core index is 1.4682 [31]) versus the GeO2-dopant content (0 mol% in cladding and about 3mol% in fiber core). we define the equivalent GeO2-dopant content as Ce = Ccore(Vcore/Vfiber), where Ccore = 0.0067 × (ncore-1.4615)/3 (mol%), ncore is the core index of fiber sample, Vcore is volume of fiber core and Vfiber is of whole fiber. The relationship between equivalent viscosity and GeO2-dopant content at 1000 °C can be fitted by extracting and extrapolating the data from Ref. [15,28,32] as
the correlation coefficient R2 = 0.9374. Taking in the parameter of different fiber samples to Eq. (1) and the equivalent viscosity can be evaluated, as shown in Table 2.Table 2. Equivalent Dopant Content and Viscosity Under 1000°C
We note that the equivalent viscosities are slightly higher than the experimental results and without tendency of size dependence. Meanwhile, the difference of the order of magnitude in the log of viscosity between the different experimental fiber samples is less than 0.01. Based on these estimates, the different GeO2-dopant contents in our samples are unlikely to be sufficient to explain the size dependence of viscosity. This means that the single material hypothesis can be considered reasonable and the effect of GeO2-dopant content on the three samples is not significant.
4.2 Surface properties
Surface properties are important implications for both microscopic and macroscopic properties of glass, but sometimes it is difficult to quantify how much these above surface contribute to viscosity due to the complexity. At a nanoscale, surface tend to have a significant impact on the properties of the glass [33,34], as well as the phenomenon of size effect [35,36]. The sharply increased specific surface area and high surface energy of nano-silica will bring the melting point down to hundreds degrees Celsius, but it is not clear what extent the surface energy of micron sized silica glass fibers affects viscous flow. Avramov I [37] reported that surface structure of glasses is inferred contributing to make the experimental viscosity (so called apparent viscosity) decrease from the bulk viscosity, because the viscosity on the glass surface is different from that in the bulk. In this situation, it is reasonable that a fiber with smaller diameters has lower viscosity, since silica fibers with smaller radii have relatively larger specific surface areas. Besides, the micro-crack and surface damage can also possibly produce a huge impact on the mechanical strength of glass and always induce experimental errors. These conditions are difficult to be perfectly ruled out in laboratory operations however, tend to be minimized for well-manufactured fibers.
Furthermore, similar size effects on surface structural relaxation have been demonstrated in [38,39]. The faster surface structural relaxation kinetics of micrometer sized fibers and thin films than that of bulk glass of the same composition are derived from its experimental and simulation results, but the experimental conditions require silica glass samples to be heated in high water vapor pressure. When the RFBGs in fibers are heated at 1000 °C but exposed to environment dry atmosphere, so it seems that the surface structural relaxation occurs reasonably but too slightly to assess the effect on viscosity. In addition to these above factors, another concern point is that the surface tension of silica fiber might cause the calculated viscosity to be lower than the actual value, because the surface tension resists the stretching force during the stretching process at high temperature. This effect can be simply removed by evaluating the magnitude of the surface tension. The surface tension of silica fibers, which is independent of fiber diameter (no less than micrometer-scale), can be estimated as 0.29 N/m at 1000 °C by reasonable extrapolation from the literatures’ experimental results [40,41]. Then the stress induced by surface tension can be calculated, which is in the opposite direction of stretching stress. A same surface tension can be assumed for fibers with different diameters but the same pure silica cladding. The effect of surface tension on tensile stress can be easily calculated as 0.11%, 0.07%, 0.03%, corresponding to fibers with different diameters as 80, 130, and 250 µm, respectively. Surface tension has almost no effect on viscosity of optical fibers. So that the effect of surface tension of fiber samples can be excluded.
4.3 Fictive temperatures
It is recognized that an elevated Tf can lead to a reduction in viscosity [17], and the differences in fabrication between optical fiber and other bulk silica glass lead to significant difference in both viscosity and Tf [13]. The Tf of single mode optical fiber is about 1600 °C since optical fiber is rapidly quenched at a very fast cooling rate during the drawing process, by contrast, which is several hundred degrees Celsius higher than that of general bulk silica glass. At a given experimental temperature of 1000 °C, the difference in the log of equilibrium viscosity between an as-drawn silica glass fiber (Tf is 1600 °C) and a bulk silica glass (Tf is 1000 °C) can be estimated at 2.18, through the extraction of data in [14,15]. This approximation relationship provides a preliminary judgment that the Tf could be an important factor leading to the diameter size-dependent viscosity of silica optical fibers. It should be noted that the relationship between Tf and viscosity of silica glass is non-linear [14], which will be further elaborated later on.
A diameter dependence of Tf for pure silica core fibers (diameter range from 125 µm to 1600 µm) is demonstrated as this fitting curve [18]:
D is diameter of optical fiber, Tf is in Kelvin and refers to the fictive temperature of the as-drawn fiber (without undergoing high temperature annealing such as the regeneration procedure). It means that the present optical fiber samples with different diameters have different initial fictive temperatures. By the way, the Tf is generally uniform on the cross-section of a pure silica core fiber, but is seemly distributed along the diameter of optical fiber with a core-cladding structure [42,43]. The Tf of a GeO2-doped core is always lower than that of the pure silica cladding but that can be ignored, given that the cladding plays a decisive role in the viscosity of fiber as well as the Tf gradient between fiber core and edge will eliminate during high temperature annealing [44]. Therefore, it is reasonably surmised that if the fibers in present work follow a similar curve as Eq. (2), and the Tf can be exported as1607, 1593 and 1565 °C of F80µm, F130µm and F250µm, respectively. As a means of verification, a natural air-cooling process of optical fiber during fiber drawn was simulated by finite element method, then the Tf can be estimated. The initial temperature of optical fiber is set to 1850 °C and the parameters listed in Table 1 are utilized. The cooling process during 200 ms is shown in Fig. 6(a), and the qc should be selected when fiber cooling through the glass transition temperature (∼1210 °C), as shown in Fig. 6(b). The cooling rates of F80µm, F130µm and F250µm are about 34350, 18700, 8000 °C/s, respectively, and the corresponding Tf is 1614, 1585, 1545 °C. This is quite similar to the results estimated by Eq. (2), and further proved that the initial Tf of three kinds of fiber are different.
Fig. 6. (a) The natural air-cooling process of different diameter fibers during drawn process within 200 ms; (b) cooling rate and fictive temperature.
It is known that the correlation between equilibrium viscosity and Tf is demonstrated as [45,46]:
Fig. 7. The difference in log of viscosity comparing to the result of F80µm. The dotted lines are for the eyes.
Furthermore, it is possible to utilize a curve similar to Eq. (4) for describing the dependence of viscosity on the Tf of silica optical fibers under a temperature of 1000°C. This is demonstrated in Fig. 8 by fitting the data derived from both our work and reported literatures [13–17]. In the new curve fitting, the correlation coefficient R2 = 0.9997, and the adjustable parameters f, k and log(η∞) are 5354, 7.41 × 10−3 and 11.56, respectively. The fitting curve of optical fiber indicates that there is a limiting minimum viscosity, i.e., about 1011.56 Pa·s, when the Tf is equal to or exceeds 1600 °C. Note that in our work, the measured viscosity has arrived at an equilibrium state clearly which means a metastable state of glass structure. The Tf’ can be regarded as the characterization of this metastable structure and is different from the equilibrium structure of glass liquid.
Fig. 8. Fitting of the fictive temperature dependence relationship of log(η). Viscosities are all at 1000 °C. The blue line is fitting curve of general bulk silica glass and black dotted line is relation to silica optical fiber.
In summary, the possible factors impacting the size dependence of viscosity of optical fibers with different diameters are summarized in Table 3, and the magnitude of changes in viscosity are also indicated. The influence of various factors on viscosity may be a combined effect, but the present authors believe that this size dependence is mainly induced by the differences in the fictive temperatures.
Table 3. Possible Factors Impacting the Size-Dependent Viscosity
There is no doubt that the smaller the size of the glass, the more obvious the impact of the size effect on the physical or chemical properties would be. Although the size boundaries of size effect on silica glass are still inconclusive, the scale of size dependence phenomenon has been extended from the nanoscale to the microscale. It is supported not only by the present work but also by researches in other fields, for example, the 20 µm orders of magnitude fused silica micro-pillars are demonstrated to exhibit nearly 10 times stronger compressive strength of bulk-sized counterparts [49]. In the future, the mechanism of the size effect on silica glass or silica optical fiber could be further elucidated through experimental methods such as infrared spectroscopy [50] and Raman spectroscopy [44], or molecular dynamics simulation [51–53].
5. Conclusion
The size-dependent viscosity of silica optical fiber is illustrated in present work. The viscosity of silica fibers with different diameters at high temperature was observed by the stretching method using in-fiber regenerated fiber Bragg gratings, and a clear size dependence was observed as that a higher viscosity could be obtained from an optical fiber with a larger diameter. The log of equilibrium viscosities of fibers with 80, 130 and 250 µm diameters at 1000 °C are 13.343, 13.436 and 13.561, respectively. After a thorough discussion and exclusion of others possible impact factors, the tuning of fictive temperature of silica optical fiber is considered mainly contribute to the size-dependent viscosity. An equivalent fictive temperature of optical fiber at metastable structure state is used to explain the difference in viscosity. Then the relationship between equilibrium viscosity and fictive temperature for silica optical fiber at 1000 °C is derived. The clarifying of size dependence of optical fiber will help to the selection of suitable optical fibers under high temperature harsh environment, and make a contribution to the development of high-temperature optical fiber devices in harsh environment.
Funding
National Natural Science Foundation of China (61775045, 62005063, 62105078); Science Fund for Distinguished Young Scholars of Heilongjiang Province (JQ2022F001); National Key Research and Development Program of China (2021YFC2802202); Fundamental Research Funds for the Central Universities (3072022JC2702).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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