Viscosity of Se–Te glass-forming system

Introduction

Amorphous chalcogenides are interesting materials with unique properties which found many applications, e.g., in electronics and optoelectronics [1]. They can be used as threshold and memory switches or infrared optics elements (detectors, lenses or fibers). Some of them are also well known phase change materials (PCM) which are used for rewritable CD or DVD discs [2]. The laser induced changes between crystalline and amorphous phases are used in that case. Also electric induced phase changes of chalcogenides are investigated due to the potential application in memory media [2].

Viscosity is one of the basic physical properties and knowledge of it for given material is very important. Fundamental temperature points necessary for successful preparation of bulk glasses are calculated from temperature dependence of viscosity. Viscosity is also important for description of structural relaxation and cold crystallization of glasses and undercooled melts, respectively. Structural relaxation is very slow rearrangement of thermodynamically unstable glass toward equilibrium. This process is in fact a very slow flow of material. The connection between viscous behavior and relaxation can be derived from the similarity between apparent activation energies of both processes which was described for example for selenium glass [3]. Also structural relaxation time τ is proportional to viscosity η through well known Maxwell equation

(1) τ   =   η   /   G ∞  (1)

where G_∞ stands for the shear modulus at infinite frequency. Connection between viscosity and cold crystallization which takes place in undercooled melts is also very important. Knowledge of viscosity is necessary for calculation of reduced crystal growth rate [4]. This quantity can be used for determination of growth mechanism [5]. Direct connection between viscosity and crystal growth rate for several organic and inorganic materials was described by Ediger et al. [6]. They also mentioned deviation of Stokes-Einstein equation [7] for strongly undercooled systems.

This work is mainly focused on the study of viscosities of selenium-tellurium system. This system is very interesting. It is formed by two chalcogenide elements which are chemically relatively similar and both form hexagonal crystals [8]. However, the glass forming ability of the elements is quite different. Pure selenium is good glass-former and has been subject of plenty experimental studies and works. Tellurium can be hardly prepared as bulk glass (quenching rate around 10¹⁰ K/min is necessary [8]). Also glass-forming ability of selenium and tellurium based materials is very different [8]. Se–Te glasses form miscible solid solutions [9] and adding of tellurium to pure selenium increases temperature of glass transition and microhardness [10], conductivity [11] etc. On the other side band gap energy decreases with Te content [11]. Enthaplic relaxation of this system was studied by Svoboda et al. [12]. Same group of authors studied also crystallization kinetics of glasses in this system [13–15]. Viscous behavior of Se–Te melts was studied by Perron et al. [16]. We also published review of almost twenty papers dealing with viscous behavior of pure selenium in broad temperature interval [17]. But according to our best knowledge viscous behavior of Se–Te undercooled melts and glasses has not been part of any published work.

Experimental

Bulk selenium-tellurium samples with 5, 10, 20 and 30 at.% of tellurium were prepared by classical melt quenching method. The pure elements (99.999 % purity; metals basis) were weighted into the fused silica ampoule (6 mm inner diameter) which was subsequently degassed to the final pressure of approximately 10^–3 Pa and then sealed. The rocking furnace was used for melting and synthesizing at 920 K for 20 h. Subsequently temperature was decreased to 770 K for 4 h and ampoule was quenched into water afterwards. Iced water was also used in the case of Se₇₀Te₃₀ as coolant. The amorphous character of prepared samples was confirmed by X-ray diffraction. Thin plates (2.5 mm in height) with parallel planes were cut from the bulk ingot (6 mm in diameter). These samples were ground by corundum abrasive powder and some of them were also polished to optical quality.

The viscous behavior of each mentioned material was studied by thermomechanical analyzer TMA CX 03 (RMI, Czech Republic). The TMA CX 03 is able to measure the height of the sample with resolution 0.01 μm in dependence on temperature and time. Other details about the instrument can be found elsewhere [18]. This instrument can be used for determination of viscosity by two methods, penetration [19] and parallel-plate one [20]. The details about experimental and temperature arrangements of measurement can be found elsewhere [21]. Here can be only mentioned that penetration method was used in setup with hemispherical and cylindrical indenters and is suitable to cover viscosity interval from 10^7.5 to 10¹³ Pa.s. We used stainless steel cylindrical indenter (1 mm in diameter) and corundum hemispherical indenter (3.98 mm in diameter). Parallel-plate method in our experimental setup (corundum plates) is suitable for viscosity values approximately from 10⁵ Pa.s to 10⁹ Pa.s (it depends on quality of sample/plate interface, so called “slip and stick” conditions). Combination of both mentioned methods allow to cover viscosity interval of whole measurable region of undercooled melt and a part of glass region. The lowest viscosity limit is bound to crystallization of sample. If the temperature is high enough the nuclei are formed and crystals start to grow. The growing crystals strongly influence flow of undercooled melt and make impossible to obtain the correct viscosity value. All materials studied in this work starts to crystallize at higher viscosity value than 10⁵ Pa.s (the lowest viscosity point which can be determined by parallel-plate method). The higher viscosity point, on the other side, is limited by the measuring time which is necessary to obtain equilibrium viscosity value. Here it should be mentioned that this time does not correspond to the relaxation time but in fact it is much higher. For the highest viscosity value determined in this work (log η/Pa.s = 12.6) the time of equilibrating was 90 h.

Results and discussion

Viscosity values of Se₉₅Te₅, Se₉₀Te₁₀, Se₈₀Te₂₀, Se₇₀Te₃₀ compositions measured by penetration and parallel-plate methods are summarized in Table 1. The measuring method for each point is also given in the table. In the experimental part we mentioned that some samples were polished to optical quality and some of them not. The reason for this is mentioned in our previous work [17] which deals with viscosity of pure selenium. We observed the deviation of viscosity values between samples with polished and non-polished surfaces. The samples with polished surfaces embodied the lower viscosity values than non-polished samples and deviated from data measured by other authors (there are plenty of data for pure selenium in literature, see [17]). The highest deviation was almost one order of magnitude. It was mainly observed for parallel-plate method and higher temperatures. We concluded that this behavior is caused by “slip and stick” conditions. Especially parallel-plate method is very sensitive for these effects. These conditions are also responsible for measuring range of this method (mentioned in experimental part). The problem with sample/plate interface causes that we always test if there is any deviation in viscosity values between polished and non-polished samples which was not observed for Se–Te materials. Here should be also mentioned that composition containing 30 at.% of tellurium exhibits the strong tendency to surface crystallization. Hence determination of viscosity values for this composition, especially for non-polished samples, is complicated and can be affected by higher experimental error. It is in accordance with observation published in the work dealing with surface and volume crystallization of Se–Te system [22].

Figure 1 depicts dependencies of logarithm of viscosity on reciprocal temperature for all measured compositions from selenium-tellurium system. The experimental data are fitted by use of two most typical viscosity equations. First one is simple Arrhenius type equation (Arr)

Fig. 1

Viscosity dependencies of Se–Te system on reciprocal temperature. Lines represent VFT fits through experimental data for compositions with 5 and 10 at.% of tellurium. Data points for Se₈₀Te₂₀ and Se₇₀Te₃₀ are fitted by use of Arrhenius type equation.

(2) η   =   η 0 exp ( E A   /   R T )  (2)

where η₀ and E_A are empirical parameters and R stands for universal gas constant. E_A is often denoted as apparent activation energy of viscous flow. This equation is represented by straight line in the plot of logarithm of viscosity against reciprocal temperature. The most chalcogenide glass-formers can be well described by this Arrhenius type equation in region of undercooled melt. Hence coordinates according to Fig. 1 are the most typical form of viscosity data depiction. The minority of chalcogenide materials exhibits behavior similar to behavior of Se₉₅Te₅ and Se₉₀Te₁₀. One of them is pure selenium. Hence it is not surprising that materials with high selenium content resemble such behavior. Temperature dependencies of viscosity of these materials cannot be described by simple Arrhenius equation even at region of undercooled melt. In these cases it is necessary to use equation with more than two parameters, usually three. The definitely most frequently used viscosity equation in the field of glass science and industry is well known Vogel-Fulcher-Tammann (VFT) equation [23–25].

(3) log   η   =   log   η 0   +   B   /   ( T   −   T 0 )  (3)

where η₀, B and T₀ are empirical parameters.

We can also find other viscosity equations in literature. Here we want to mention two of them. The first one is Avramov-Milchev (AM) equation [26] with parameters η₀, β, α.

(4) log   η   =   log   η 0   +   β   /   T α  (4)

And second one was introduced by Mauro et al. [27]. Their model has also three parameters (η₀, K, C) and is denoted as MYEGA equation.

(5) log   η   =   l o g   η 0   +   K   /   T   exp   ( C   /   T )  (5)

This viscosity model was deduced on the base of well known Adam-Gibbs theory [28] and according to Mauro and coworkers it eliminates physically unrealistic behavior of Avramov-Milchev model at high temperatures and also does not embody singularity of viscosity for finite temperature. This singularity of viscosity at T₀ is included in Vogel-Fulcher-Tammann model.

It was written previously in this text that most chalcogenides exhibit temperature dependence of viscosity corresponding to Arrhenius type equation in the region of undercooled melt. On the other hand when the melt region is included the viscosity dependence for chalcogenides is always non-Arrhenian. It is well apparent from Fig. 2 where the experimental data from this work are completed with data for melt region published by Perron et al. [16]. The VFT equation is the most suitable, from previously mentioned models, for description of viscosities depicted in Fig. 2. It is also apparent from the plot that viscosities of melt for all three compositions are almost same. It is very interesting and together with knowledge of selenium melt viscosity (which is also very close to depicted dependencies) it gives us the opportunity to estimate viscosity in melt region also for composition with 5 at.% of tellurium. The parameters of all VFT fits are summarized in Table 2. VFT parameters for pure selenium (log η = –4.03, B = 1214 K and T₀ = 226.1 K) was published in our previous work [17] for collection of data from several literature sources.

Very interesting interpretation of viscosity data is normalized Arrhenius plot [29] which is also known as Angell plot, referring the name of its well known propagator. It is dependence of logarithm of viscosity (or relaxation time) on reduced temperature T₁₂/T where T₁₂ is temperature corresponding to viscosity value 10¹² Pa.s (viscosity glass transition temperature). Kinetic fragility parameter m is defined according to Angell plot as slop of given dependence close to T₁₂ temperature.

(6) m   =   d   log   η   /   d ( T 12   /   T ) / T → T 12  (6)

Fig. 2

Viscosity dependencies for Se₉₀Te₁₀, Se₈₀Te₂₀ and Se₇₀Te₃₀ chalcogenides. The experimental data from this work are plotted together with literature data for melt region [16]. Lines represent VFT fits.

The definition of kinetic fragility can be used for transformation of previously mentioned viscosity equations (eq. 3–5) and express them as function of parameter log η₀, m and T₁₂. It was done by Mauro and coworkers [27] and it is useful because important parameters m and T₁₂ can be obtained directly from the fit. In the same way simple Arrhenius equation can be also rewritten as

(7) log   η   =   12 − m   +   ( m   T 12   /   T )  (7)

The definition of mentioned kinetic fragility is clear but its own determination is not easy even if accurate viscosity data are available. The determination of m is complicated due to the definition as slope of viscosity versus (T₁₂/T) dependence. For the viscosity dependencies which cannot be described by Arrhenius model, the fragility parameter can significantly vary with given viscosity model and also with change of region which is included into the fit. In Table 3 the fragility parameters of studied Se–Te compositions including pure selenium are calculated in different ways. It is seen that fragility determined by VFT model in region of undercooled melt and glass is very different from value determined in same way from whole viscosity dependence including melt. This difference is largest for pure selenium. It is caused by the fact that experimental data for selenium exhibit divergence from VFT curve at high viscosities. Divergence is not so high if only region of undercooled melt is included into fit. Empirical VFT fit is then capable to fit better the experimental data but m calculated in this way is higher. It is apparent from the Table 3 that different values of fragility parameter are obtained also for different viscosity models applied for the same viscosity interval (undercooled melt and glass). It is also possible to calculate fragility parameter only from narrow viscosity interval around T₁₂ where every viscosity dependence embodies behavior corresponding to Arrhenius model. Here we calculated m in viscosity interval from 10¹¹ Pa.s to higher values. Problem with this approach is that usually only few data points are considered to the fit and fitting of three or four points is very sensitive to any deviation of each point. Hence obtained kinetic fragility can be influenced by high error in its determination. It is also well apparent from the Table 3. As was said in the previous text most chalcogenide materials do not embody behavior similar to here mentioned Se, Se₉₅Te₅ and Se₉₀Te₁₀. Most of them can be fitted by Arrhenius model in whole region of undercooled melt similarly to Se₈₀Te₂₀ and Se₇₀Te₃₀ studied in this work. There is of course no problem with different values of fragility parameter determined by different models in region of undercooled melt in the case of such materials. Nevertheless the different values of m are obtained if melt region is considered to fit (Table 3). It is apparent from the difference among fragility values determined by different ways that everybody should be very wary if fragility values are compared. Especially when there is no information about fragility determination way and values are collected from different sources. It is apparent from Table 3 that T₁₂ temperature is less sensitive to way of determination than fragility. That is not surprising regarding to T₁₂ definition. Determined values for studied system are compared in Fig. 3 with glass transition temperatures determined in Se–Te system and published by different authors [10–12]. Viscosity glass transition temperatures are lower than other values but embody same compositional trend. It is in agreement with expectation and it was published for other systems e.g., [21].

Fig. 3

Viscosity glass transition temperatures T₁₂ determined in this work (VFT-undercooled melt and glass region) depicted against Te content together with glass transition temperature determined by different author and in different ways (Kasap et al. [10] – DSC; cooling 5 K/min, Reddy and Bhatnager [11] – DSC, Svoboda et al. [12] – DSC; heating 10 K/min after cooling same rate from equilibrium). The straight line through T₁₂ data is plotted only to guide the eyes and has no physical meaning.

Table 3

Kinetic fragility parameters m and T₁₂ temperatures determined for four studied compositions from Se–Te system and for pure selenium in different ways.

Viscosity models are marked according to viscosity regions which were used for fitting and calculation of mentioned parameters. For clear comparison, data for pure selenium for region of undercooled melt and glass are our data published earlier [17]. Melt region for pure selenium is represented by data published in same work [16] as Se–Te melt viscosities. Errors correspond only to errors of fitting (± mentioned value).

[umg – undercooled melt and glass region, hvr – high viscosity region, wvi – whole viscosity interval (including melt, undercooled melt and glass)].

The determination of kinetic fragility value is not the only problem of normalized Arrhenius plot concept. The concept itself is a simplification of materials behavior. It is well apparent from Fig. 4 where experimental data published in this work together with data published by Perron et al. [16] are plotted in coordinates of Angell plot. Viscosities of Se–Te compositions are very close to well-known fragile material o-terphenyl in the range of undercooled melt but they significantly differ from o-terphenyl in melt region. Viscosities of Se–Te compositions also differ in melt from simulated VFT curve with fragility parameter 85. They approximately follow simulated curve with m = 55. This is result of Angell plot simplifications. First simplification of this plot is that viscosities of all materials at infinite temperature are 10^–5 Pa.s. Nemilov [30] estimated that viscosities at extremely high temperatures could lie within the range log (η/Pa.s) = –(4.5±1). Second simplification of Angell plot was mentioned in previous text. The information about shape of viscosity curve can be hardly characterized by only one parameter (kinetic fragility). The viscosity curves are more individual and can cross each other in the Angell plot. It is also necessary to consider fact that viscosity models are usually expressed by use of empirical parameters or by use of parameters with not clear physical meaning and there is no generally correct and exact viscosity equation. Nevertheless authors do not claim that concept of kinetic fragility is incorrect. Just conversely the concept is very useful and can help to explain many observed features of materials behavior. Authors only stressed that the concept is simplification of real viscosity behavior and also determination of fragility parameter value itself can be ambiguous.

Fig. 4

Normalized Arrhenius plot (Angell plot) depicts data for studied Se–Te materials. Only several viscosity points for each composition were plotted in undercooled melt region to keep clarity. Data for SiO₂ and o-terphenyl were reproduced from Angell [29]. Simulated VFT curves with fragility parameter m = 85 and 55 are also depicted.

Conclusion

Viscous behavior of four compositions from selenium-tellurium system was studied by penetration and parallel-plate methods in region of undercooled melt and glass. The obtained data points with combination of literature data for melt region are well described by empirical Vogel-Fulcher-Tammann equation. Other tested viscosity models (Avramov-Milchev and MYEGA) do not fit experimental data as well as VFT model. The concept of kinetic fragility was discussed and its simplifications were pointed out. Ambiguity of kinetic fragility parameter m was shown on measured Se–Te data. Necessity of same way of calculation resulting from this ambiguity was pointed out.