Spherulitic Crystal Growth Velocity in Selenium Supercooled Liquid

Article pubs.acs.org/crystal

Spherulitic Crystal Growth Velocity in Selenium Supercooled Liquid Jiří Málek, Jaroslav Barták,* and Jana Shánělová Department of Physical Chemistry, University of Pardubice, Studentská 573, Pardubice 532 10, Czech Republic ABSTRACT: The spherulitic crystal growth velocity in selenium supercooled liquid has been measured by infrared microscopy in isothermal conditions for rapidly heated samples of a-Se from temperatures well below Tg. These data are compared and analyzed along with previously published data obtained on samples quenched from a temperature well above the melting point. The spherulites grew linearly over a course of time that corresponds to crystal growth controlled by crystal−liquid interface kinetics. The crystal growth velocity data obtained for these two different thermal histories can be described by the normal growth model for moderate supercoolings (ΔT 90 K) that is also consistent with morphological observations. However, the prediction based on this model still significantly deviates for intermediate supercoolings. It is shown that all experimental data in the whole temperature range (Tg < T < Tm) can be described by a combined approach including both these models, taking into account actual viscosity scaling of crystal growth η−1. The kinetic information captured in the DSC curve is analyzed, providing evidence that the non-isothermal crystallization process can be described by the Johnson-Mehl-Avrami model. Two distinct regions characterized by different values of apparent activation energy were found. The transition between these regions coincides with morphological changes of spherulitic crystals and other properties. These regions are characterized by distinct apparent activation energies whose values are consistent with those obtained from microscopic measurement of crystal growth velocity.

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INTRODUCTION Although chalcogenide glasses have been studied over seven decades, interest in these materials is not diminishing as they are finding new life in a growing range of various optoelectronic, photonics, photoconducting, sensing, and memory device applications.1,2 Elemental selenium is chemically the simplest, though not the easiest system among these materials. While photoconductive properties of amorphous selenium (a-Se) led to a real breakthrough in the photocopying industry 60 years ago,3 a quite new application field was found recently for this material in direct conversion flat panel X-ray imagers used in medicine, security, and industry.4 One of the key properties of the amorphous material is its long-term stability and resistance to crystallization. On the other hand, controlled crystallization is important for phasechange memory devices. Therefore, it is not surprising that there are quite numerous studies on selenium crystal growth kinetics.5−17 However, many of these studies lack necessary experimental details. A notable exception is the excellent and detailed study by Ryschenkow and Faivre18 describing peculiarities of spherulitic crystal growth in supercooled selenium liquid by direct microscopy observation. These authors later proposed a specific mechanism based on noncrystallographic branching explaining different spherulitic modes observed at medium and high supercoolings.19 Several calorimetric studies were published recently, describing a complex crystallization behavior of amorphous selenium.20−27 Nevertheless, as far as we know, no attempt to describe crystal © 2016 American Chemical Society

growth behavior quantitatively in a broad temperature range has been made yet. In this paper we extended previous isothermal crystal growth velocity measurement on samples quenched from a temperature well above the melting point.18,19 These data are compared with our isothermal growth data obtained in a different way by fast heating of a-Se samples from temperature below Tg. Subsequently, we present a consistent description of the temperature dependence of the crystal growth velocity obtained for different thermal histories. This description is based on a combination of normal growth and screw dislocation growth models involving temperature dependent selenium viscosity data28 as well as the Gibbs energy change between supercooled liquid and crystalline phase calculated from heat capacity measurements.29−31 This description comprehensively explains present measurement as well as existing crystal growth data from the literature being consistent also with recent calorimetric studies.24−27 A recently published study on crystal growth in Se−Te bulk glasses32 is also discussed.

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MATERIALS AND METHODS

Bulk selenium glass was prepared from the pure element (5 N, SigmaAldrich) by the conventional melt-quenching method. An appropriate Received: June 13, 2016 Revised: August 2, 2016 Published: August 29, 2016 5811

DOI: 10.1021/acs.cgd.6b00897 Cryst. Growth Des. 2016, 16, 5811−5821

Crystal Growth & Design

Article

Figure 1. Morphology of spherulitical crystals grown in supercooled selenium liquid. IR microscopy, polarized light: (a) T = 106 °C, t = 180 min, sample fracture; (b) T = 98 °C, t = 180 °C, polished sample; (c) T = 128.7 °C, t = 60 min, sample fracture; (d) T = 128 °C, t = 30 min, polished sample. SEM microscopy: (e,f) T = 123 °C, t = 30 min; sample fracture. amount (10−15 g) of polycrystalline selenium pellets was placed into a carefully washed and dried silica ampule (6 mm diameter, 1 mm wall thickness). The ampule was then evacuated to the pressure 10−3 Pa and sealed. The sealed ampule was annealed in a rocking furnace at 600 °C for 20 h. The glass was then prepared by quenching the melt in ampule in cold water. Then the ampule was opened and the a-Se ingot sectioned by a diamond saw to small cylindrical specimens (ca 4.3 mm diameter, 2 mm height). These specimens were used for crystal growth studies. The amorphous nature of the as-prepared material, as well as identification of crystallized phase, was verified by X-ray diffraction analysis using a Bruker D8 Advance AXS diffractometer equipped with a horizontal goniometer and scintillation counter utilizing CuKα radiation. The crystal growth kinetics was studied by an Olympus BX51 microscope in reflection mode, equipped with the infrared XM10 camera. The bulk specimens were placed in a preheated computercontrolled furnace (central hot zone was constant within ±0.5 °C). The time needed for thermal equilibration of the specimen was estimated to be less than 2 min. The samples were heat-treated for

different times (the shortest time was 5 min) at selected temperatures (76−219 °C) in the air atmosphere. The temperature was recorded during the entire time of heat treatment. Immediately after thermal treatment the specimens were quickly cooled down in cold ethanol (ca −3 °C) and thoroughly analyzed by IR microscopy. Because of the quite fast surface crystallization, the samples were polished or sectioned to remove the surface layer and to clearly observe the bulk crystallization. The sizes of well developed spherulites were recorded and measured. The time evolution of diameter of at least 15−30 different spherulites was measured at a selected temperature and the crystal growth velocity was calculated as a mean of all these data. The calorimetric experiments were performed using a conventional differential scanning calorimeter DSC 822e (Mettler-Toledo) equipped with a cooling accessory. Dry nitrogen was used as the purge gas at a rate of 20 cm3/min. Hermetically sealed aluminum sample pans were used. For temperature, heat flow and τ-lag calibrations of the calorimeter high purity melting standards were used (In, Zn, and Ga). Small pieces of bulk sample (about 20 mg) were used for the 5812



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measurement. Further details of the measurement procedure and data analysis are described elsewhere.24

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RESULTS The selenium supercooled liquid obtained by controlled heating from glassy state exhibited a relatively low heterogeneous nucleation center density. Well-defined close spherulitic aggregates grew from these primary centers at the sample surface as well as within the volume, as shown by IR micrographs in Figure 1. The fine structure composed by lamellar tips is shown in an SEM photograph (Figure 1f). The spherulites are completely crystallized and the growth front is quite compact. No single crystals were observed in the studied temperature range (75−219 °C). The contrast banded spherulites were observed in polarized light (Figure 1a,b) at temperatures below 120 °C. These spherulites are banded, exhibiting fine and close morphology. They are composed of fine lamellae radiating from a spherulitic center and show concentric rings of optical contrast between crossed polarizers. Figure 2 shows the band

Figure 3. Time dependence of spherulitic diameter of crystals grown in a-Se at chosen temperature.

separated crystals. Typical standard deviations are shown as error bars for all data points. The linear behavior of this dependence is typical for crystal growth controlled by interface kinetics. Crystal growth velocity u was determined as the slope of the linear fit of this dependence. Experimental crystal growth velocity data for selenium are summarized in Table 1. Table 1. Crystal Growth Velocities in a-Se T (°C) 76.0 99.6 107.1 113.0 119.0 123.2 128.7 140.9 151.3 158.0 158.3 173.4 179.0 187.3 218.5

Figure 2. Temperature dependence of band spacing in Se-crystals.

spacing as a function of temperature combining data of Ryschenkow and Faivre18 obtained by isothermal crystal growth experiments in melt quenched samples and our isothermal crystal growth data recorded after heating of glassy samples. The fact that the band spacing has nearly the same temperature dependence for quite different sample treatment implies that the underlying mechanism is not affected by experimental conditions, such as thermal history, nuclei distribution, and so forth. As shown earlier by Ryschenkow and Faivre18 there are two distinct types of spherulites (A and B) composed of thin lamellae (∼50 nm) differing in the lamellae twist as a consequence of small-angle (noncrystallographic) branching. These authors observed that the spherulites of both types are banded below 120 or 130 °C, respectively (Figure 2). The radial growth for both spherulitic types corresponds to crystalographic dense axis a (see the Spherulitic Morphology section in the Discussion). The crystal growth velocity of spherulites A and B is similar within the experimental uncertainty. Figure 3 shows the time dependence of crystal size measured as a spherulitic diameter. Every point in this figure corresponds to a mean value of independent measurements of 15−30

u (μm/min) 0.0286 0.53 0.798 1.72 3.16 3.84 5.6 8.9 13.9 15.6 14.8 24.7 29.2 28.0 12.1

± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

0.0015 0.05 0.017 0.14 0.15 0.05 0.5 0.7 0.6 1.9 0.5 0.3 1.5 1.4 1.1

Temperature dependent data for crystal growth velocity reported by Ryschenkow and Faivre18 are compared with our experimental results in Figure 4 on a logarithmic scale. These authors quenched the selenium liquid from a temperature above the melting point (Tm) to the temperature where the growth velocity was measured.18 In contrast, the crystal growth experiments described in this paper took place in a supercooled liquid obtained by fast heating of a-Se. Despite the different experimental procedure there is quite good agreement between these measurements. The crystal growth velocity has been measured in a broad range, from 76 °C to the melting point, over 5 orders of magnitude. Maximum growth velocity of selenium spherulites was found from combined data and its value is about 43 μm/min (Figure 4). Another significant point is a bend visible on the u(T) curve between 110 and 130 °C that coincides with transition between unbanded type A and banded type B spherulites. This transition is not accompanied 5813



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growing crystal ukin and the probability that the newly formed crystalline layer is retained within the solid phase. This probability is temperature dependent and usually expressed as [1 − exp(−ΔG/RT)], where ΔG is the Gibbs energy difference between the crystalline phase and the supercooled liquid. For classical crystal growth mechanisms the atoms of the newly formed phase can be added or removed from any site on the crystal−liquid interface (normal growth) or attached preferentially to screw dislocation sites on the surface of the growing crystal (screw dislocation growth).35,36 In both cases, the transport process is similar to self-diffusion. The temperature dependence of the self-diffusion coefficient is not always available, so it is usually replaced by the inverse shear viscosity η−1 according to the Stokes−Einstein equation37 that is implemented in both models of crystal growth. Ediger et al.38 analyzed ukin scaling with viscosity for a wide range of organic and inorganic materials, showing that in fact it scales as η−ξ with the exponent ξ smaller than unity u ukin = ∝ η− ξ 1 − exp( −ΔG /RT ) (1)

Figure 4. Temperature dependence of crystal growth velocity together with non-isothermal DSC curves measured at different heating rates.

by a significant change in spherulitic growth velocity. A similar effect was observed for the unbanded to banded spherulitic transition in liquid crystalline material 4-cyano-4′-decyloxybiphenyl.33 The heat flow evolved during the crystallization process can easily be detected by a sensitive DSC instrument. Figure 4 also shows DSC data in the whole temperature range (10 °C/min) as well as in the crystallization range (15, 20, and 30 °C/min). The enthalpy change of the crystallization ΔHc can be obtained by integration of the measured heat flow over the whole crystallization peak assuming a linear baseline. For DSC data shown in Figure 4 it was found to be ΔHc = −67.3 ± 0.8 J/g. The XRD analysis of fully crystallized sample (Figure 5)

Figure 6 shows the temperature dependence of viscosity for liquid selenium summarizing previously published data

Figure 6. Temperature dependence of viscosity and heat capacity (inset) of selenium.

collected by Bernatz et al.39 and Koštál28 ranging from Tg to temperatures well above Tm. All data can be approximated by Vogel−Fulcher−Tamman (VFT) equation40−42 B ln(η /Pa·s) = log η0 + T − T0 (2)

Figure 5. XRD pattern of amorphous and fully crystalline Se samples.

where log η0 = −9.27 ± 0.17, B = 2795 ± 50 K, and T0 = 226.1 ± 1.1 K. The inset in Figure 6 shows the temperature dependences of heat capacity of selenium crystal, liquid, and supercooled liquid.29−31 The Gibbs free energy between supercooled liquid and crystalline phase (ΔG) can easily be calculated from the heat capacity difference between supercooled melt and crystalline phase ΔCp = Cpcr − Cpm

confirmed trigonal crystal lattice with parameters a = 0.4369 nm and c = 0.4952 nm that is in good agreement with the literature.18,34 The polymeric Se molecules form helical chains along the trigonal a-axis, that coincides with radial axis of spherulitic growth.18

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DISCUSSION Viscosity Scaling of Crystal Growth Velocity. The macroscopic isothermal crystal growth velocity of a crystal in the supercooled liquid of the same composition can be expressed as the product of the rate at which molecules from liquid adjacent to the growth front are incorporated into

ΔG = ΔHm

ΔT + Tm

∫T

Tm

ΔCp dT − T

ΔCp = wT 3 + xT 2 + yT + z 5814

∫T

Tm

ΔCp

dT T

(3a) (3b)



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the kinetic exponent ξ can hardly be obtained from the fragility for selenium. Similar results has been reported also for the Se100‑xTex system (x = 10, 20, and 30) where the kinetic exponent was found to be significantly higher than the prediction from fragility.32 Description of Crystal Growth Kinetics. It seems that relaxation of the supercooled selenium liquid to its equilibrium composition is quite fast, not significantly affecting growth velocity18 and, therefore, standard methods of description of crystal growth kinetics can be applied. The general trends of crystal growth processes taking place in supercooled liquid can be assessed by the method proposed by Jackson, Uhlmann, and Hunt46 that is based on inspection of the plot of reduced growth rate UR versus supercooling ΔT u·η UR = 1 − exp( −ΔG /RT ) (4)

where ΔHm = 6.16 kJ/mol is the enthalpy of melting23 and w = 1.331 × 10−7 J/mol·K4, x = −2.644 × 10−4 J/mol·K3, y = 0.183 J/mol·K2, z = −50.457 J/mol·K. Figure 7 shows ukin plot in log−log format calculated from the crystal growth velocity by eqs 1 and 3, as a function of selenium liquid viscosity (eq 2).

If the UR(ΔT) plot results in a horizontal line that does not depend on supercooling, then the normal crystal growth model should be used. For the screw dislocation growth model, the plot in the form of a straight line with positive slope is obtained. The plot of UR vs ΔT calculated by eq 4 for experimental crystal growth velocities on rapidly heated samples from temperature well below Tg (our data) and on samples quenched from a temperature well above melting point18 is shown in Figure 8. These two data sets are very similar within

Figure 7. Kinetic part of crystal growth velocity dependence on viscosity in a-Se. The marked region corresponds to the temperature region of spherulite morphology change.

In fact, eq 1 is defined for the normal growth model (eq 5). Nascimento and Zanotto43 discussed its applicability for the screw dislocation (eq 6) model and 2D surface nucleated model, proposing normalized kinetic coefficients. We tested the normalized kinetic coefficient for the screw dislocation model at large departures from the selenium melting point (80 < ΔT < 130 K), confirming its negligible influence on selenium data. Figure 7 shows that a linear dependence corresponding to a power law anticipated by eq 1 is observed in the low and high viscosity regions. The slopes corresponding to the kinetic exponent ξ = 0.94 ± 0.05 are similar and close to 1 within the limit of experimental errors, though the intercepts are different. Therefore, the Stokes−Einstein equation is fulfilled near the melting point Tm as well as in highly supercooled liquid down to Tg. It seems that one of important preconditions of the applicability of the normal growth and screw dislocation models is fulfilled for spherulitic crystal growth velocity in selenium liquid in a relatively wide viscosity range. However, the ukin vs η plot on log−log scale exhibits quite different behavior in the intermediate viscosity region (101.5 < η < 103 Pa·s) . Here the kinetic exponent is considerably lower, ξ = 0.66 ± 0.05. It should be pointed out that the onset of this region for crystal growth from supercoled liquid of higher viscosity coincides with a spherulite morphology change and the extinction of concentric bands reported by Ryschenkow and Faivre18 (indicated in Figure 6). Ediger et al.38 demonstrated that for numerous organic and inorganic glasses, the exponent ξ depends linearly on the fragility m of superccoled liquid near the glass transition (ξ = 1.1−0.005·mη). The fragility of selenium melt reported by Málek et al.,44 mη ≅ 61, is very close to mη ≅ 62 published by Roland et al.45 According to Ediger’s suggestion it would provide an estimation of ξ ≅ 0.79−0.80 that is an intermediate value of both limits obtained from the log ukin − log η plot shown in Figure 7. Nevertheless, it seems that the real value of

Figure 8. Reduced crystal growth rate dependence on supercooling. The marked regions correspond to the temperature region of spherulite morphology change (green line) and applicability of NG and SDG models (black arrows).

the limits of experimental errors. This indicates that the effect of nucleation is negligible and does not interfere in crystal growth kinetics. Figure 8 clearly shows that for moderate supercoolings (down to about ΔT ≅ 60 K) the UR plot is practically independent of supercooling. It seems, therefore, that this temperature region can be described by the normal growth (NG) model. This model assumes a rough crystal−liquid interface on atomic scale, where the growth units are oriented to the sites with the lowest energy.36,46 The crystal growth velocity can be expressed as u= 5815

kBT 3πa02η

·[1 − exp( −ΔG /RT )] (5) DOI: 10.1021/acs.cgd.6b00897 Cryst. Growth Des. 2016, 16, 5811−5821


Article

where a0 is the mean interatomic distance in the interface layer and kB is the Boltzmann constant. Figure 9 shows the NG

Se polymeric chains are attached to the lamellar tips in the radial direction of spherulitic growth with spherulitic lamellar packing via noncrystallographic branching where dislocations and stresses applied on the growing crystal lamellae play a significant role (see next section). The SDG model describes data reasonably well from Tg to about 130 °C (i.e., for supercooling ΔT > 90 K) and then in the vicinity of the melting point (Figure 9). This model provides a considerably better description of experimental growth velocity data in comparison to the NG model, similarly as reported32 for Se90Te10. However, there is still a significant discrepancy in intermediate temperature range and the predicted maximum crystal growth velocity is shifted to lower temperatures in contrast to experimental evidence. Taking into account the important influence of the viscosity on the crystal growth velocity, we can assume that the crystal growth velocity in the entire temperature interval can be expressed by eq 1 in a somewhat rewritten form, i.e. u = k 0·η−ξ [1 − exp( −ΔG /RT )]

Figure 9. Temperature dependence of crystal growth velocity in a-Se. The marked region corresponds to the temperature region of spherulite morphology change. The inset shows the growth velocity fitted by the OCG model in linear scale.

where k0 is a constant characteristic for a given supercooled system. It has been shown that except the intermediate viscosity region [1.5 < log(η/Pa.s) < 3] the kinetic exponent ξ is within the limit of experimental error close to 1 (Figure 7) for the whole viscosity range (8 orders of magnitude). Then the overall crystal growth (OCG) velocity can be expressed as a combination of growth velocities given by eqs 5, 6, and 7

model prediction of crystal growth velocity calculated (dashed line) by fitting of eq 5 to all experimental data using the temperature dependence of viscosity and ΔG described by eqs 2 and 3. There is just one fitting parameter which was found to be a0 = 2.81 ± 0.04 nm. This value is about 6 times higher than the trigonal lattice parameter of crystalline selenium (a = 0.437 nm, c = 0.495 nm). Thus, it seems to be quite far from the expected mean interatomic distance of the crystal−liquid interface layer. A very similar behavior has recently been found also for Se100‑xTex supercooled liquids.32 The reason for a such physically unrealistic value of parameter a0 is unknown. However, with respect to the prediction based on the UR(ΔT) plot for selenium, there is fair agreement between the NG model and experimental growth data down to about 160 °C (i.e., for supercooling ΔT < 60 K). Nevertheless, the NG model significantly and systematically underestimates experimental data for higher supercooling. Figure 8 shows that the UR plot is linear for large supercoolings (90 < ΔT < 160 K). Thus, we can assume that the crystal growth data could be described by the screw dislocation growth (SDG) model. This model is based on a lattice distortion in which the attachment of growth units to the crystal surface results in the development of a spiral growth pattern or a screw dislocation.35 The crystal growth velocity is then expressed as u=

kBT 3πa02η

·

ΔT [1 − exp( −ΔG /RT )] 2πTm

(7)

u = [k 0 + k1T + k 2T 2]·

1 − exp( −ΔG /RT ) η

(8)

where k1 = kB(2π+1)/6π a0 and k2 = −kB/6π Figure 9 shows the OCG velocity prediction (solid line, log-scale) calculated by fitting of eq 8 to experimental data in the whole temperature range, using log η and ΔG data described by eqs 2 and 3. The inset shows the same plot for crystal growth velocity in the linear scale. The fitting parameters were found to be k0 = (8.4 ± 0.6)·10−3 N/m, k1 = (−3.6 ± 0.3)·10−5 N/m·K, and k2 = (3.9 ± 0.3)·10−8 N/m·K2. It is clearly seen that there is a satisfactory agreement between model prediction and experimental data reported in this paper as well as for the data reported by Ryschenkow and Faivre.18 This combined approach provides much better results than standard NG and SDG models. Spherulitic Morphology. The sizes, shapes, and mutual arrangement in selenium spherulites seem to be similar to those observed in liquid crystals33 and molecular crystals,47,48 as well as in polymers.49−51 All these materials and many others have been shown to form banded spherulites composed of twisted lamellae.52 Twisting is characterized by a full twist period (pitch), the length required to achieve to 2π rotation. The contrast bands in selenium spherulites shown in Figure 2 are then a consequence of in-phase rotation of the tangential axis c along each fiber.18 Recently, it has been shown52 that the full twist period P and lamellar thickness h are strongly correlated, according to a power law (P ∼ hn). The exponent n is quite large for polymers (14.2−9.2) but significantly smaller for molecular systems (2.9−1.0). Selenium spherulites are between these limits with the exponent 1.8 < n < 5.4.53 Selenium crystallite thickness decreases exponentially with increasing supercooling.18,19 A similar power law relationship P ∝ (1/ ΔT)q between the full twist period and supercooling for spherulites as well as for isolated crystals such as hippuric acid 2

(6)

where ΔT and parameter a0 have the same meaning as in eqs 4 and 5, respectively. Figure 8 shows the SDG model prediction of crystal growth velocity (dashed−dotted line) calculated by fitting eq 6 to all experimental data using log η and ΔG data described by eqs 2 and 3. The fitting parameter was found to be a0 = 0.29 ± 0.01 nm. This value is nearly one-half of the lattice parameter of crystalline selenium, which means that it is physically more plausible and can be considered as the mean interatomic distance in the interface layer at the growth front. This is consistent with the assumed growth mechanism where 5816

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shown that the phase field theory is able to describe quite a wide range of observed spherulite morphologies. What is Revealed by Calorimetric Experiments. The crystallization enthalpy change obtained by integration of the measured heat flow over the whole crystallization peak (Figure 4) slightly depends on temperature due to the corresponding heat capacity change. However, its average value can be estimated as ΔHc = −67.3 ± 0.8 J/g for heating rates in the range 10 ≤ β ≤ 30 °C/min. Similarly, the melting enthalpy was found to be ΔHm = 77.2 ± 0.8 J/g for the same experimental conditions. Both enthalpic changes ΔHc and ΔHm are consistent, taking into account the selenium heat capacity change with temperature. Within the limit of experimental error, the melting enthalpy is in agreement with the value obtained using a high-precision adiabatic calorimeter by Grönwold21 (ΔHm = 78.00 ± 0.05 J/g). Taking this reliable value and melting temperature of crystalline selenium (Tm = 221.15 ± 0.02 °C),21 we can estimate the entropy of melting as ΔSm = 12.46 J/mol·K (∼1.5R). According to the Jackson theory,46 the isotropic and nonfaceted crystallization should be observed for such materials exhibiting a low entropy of melting (ΔSm < 2R). This prediction is, in fact, fulfilled for type A spherulites where the growing crystal edge exhibits nearly smooth, rounded shape. On the other hand, faceted hexagonal vertices of microcrystalline lamellae were observed for type B spherulites.19 Characteristic crystallization peaks shown in Figure 4 reflect the whole crystallization process, possibly involving nucleation as well as surface and bulk crystal growth. More detailed calorimetric studies24−27 indicate such complex behavior where the surface crystallization is present. However, it is negligible in comparison with volume crystallization for the bulk specimen. Therefore, DSC curves can be converted to kinetic data that might subsequently be analyzed. It is assumed that the rate of the crystallization process is proportional to the measured heat flow, ϕ, normalized per sample mass. The fraction crystallized, α, then can be obtained by partial integration of nonisothermal heat flow after baseline subtraction:59

has been reported,54 indicating that isolated fibers are more strongly twisted than closely packed ones embedded in spherulitic structure. This type of relationship is also found for selenium (for data in Figure 2) where the power law exponent q ≈ 3.5 is very similar to the value reported by Keith and Padden for polyethylene spherulitic growth.55 As mentioned already, selenium crystallizes into two distinct spherulitic forms (types A and B).18 At medium temperatures the spherulites are not so compact, and individual thin subcrystal tips can easily be observed (see Figure 1f). Bisault et al.19 showed that both these spherulitic forms are composed by microlamellae. In type A, the growing crystal edge exhibits a nearly smooth, rounded shape. On tangential cross-section this growth mode shows a crescent shape that probably provides natural explanation of the lamellae twist and characteristic banded spherulites mentioned earlier.19 The characteristic feature in type B is a prismatic faceting of microlamellae stacks parallel to the basal plane.18,19 The general features of spherulitic growth are wellknown52,53 and selenium is no exception from these rules. Spherulites are usually formed via noncrystallographic branching where dislocations52 and stresses19 applied on the growing crystal lamellae play a significant role. This is probably why the SDG model provides a better approximation of experimental growth velocity data and a physically more plausible value of the a0 parameter in comparison with the NG model (also observed for Se100‑xTex supercooled liquids32). The morphology change region (marked by green bar in Figures 7, 8, and 9) corresponds to the transition zone between two spherulitic types (Figure 2). However, these two forms of growth can be simultaneously active in a wider temperature range,18 though one of them can be destabilized under the given experimental conditions causing a less stable form to be gradually transformed into the stable one.18,19 This morphology change can influence kinetic behavior, though its effect might be quite complex. The transition between NG and SDG model depicted in Figure 8 is rather gradual (see arrows). Therefore, it is not surprising that the microscopically observed morphological transition did not fit exactly with the expected NG-SDG crossover. In crystal growing from glasses and viscous liquids, stress can be induced by specific volume change due to crystal formation. Caroli et al.56 estimated such stress for selenium as approximately 30 GPa, that is smaller than the theoretical fracture limit for of the crystal, but larger than its critical shear stress. These authors56 also demonstrated that the very high viscosity of highly supercooled liquid is possibly stabilizing spherulitic symmetry by slowing down the growth velocity. Another effect related to high viscosity comes in combination with crystal distortions on a much smaller scale that are manifested by stresses induced at growing lamellar tips. The presence of plastic deformation has been found in banded selenium spherulites.18 The large complexity of spherulitic morphology, strong influence of highly viscous supercooled liquid, and considerable stresses created at lamellar tips of spherulitic crystals, at least partially, explain why the overall temperature dependence of selenium crystal growth velocity cannot be described by simple kinetic model (such as NG or SDG) and a combination of several kinetic equations has to be used instead (eq 8). For a more detailed description of this complex crystallization process, one could possibly use the phase field theory incorporating noncrystallographic branching and other features of spherulitic growth. Granasy et al.57,58 have

α=

1 ΔHcβ

∫T

T

ϕ · dT

(9)

on

where Ton corresponds to the beginning of the baseline approximation. The heat flow ϕ corresponding to the crystallization process then can be written as59 ϕ = ΔHcA exp( −Ea /RT ) ·f (α)

(10)

where A is the preexponential factor and Ea is apparent activation energy describing the crystallization process. The f(α) function expresses the Johnson-Mehl-Avrami nucleation− growth model (JMA)60−65 f (α) = m(1 − α)[− ln(1 − α)]1 − 1/ m

(11)

Any direct comparison of non-isothermal DSC curves represented as ϕ(T,α) plots is complicated by the fact that they strongly depend on temperature and heating rate. However, it has been shown59,66 that if DSC data are multiplied by T2 and plotted as a function of the fraction crystallized, they are proportional to the z(α) function that is characteristic for particular kinetic model as expressed by the following equation59,67 5817


Crystal Growth & Design z(α ) = f (α )

∫0

α

dα ≈ ϕ·T 2 f (α )

Article

The Kissinger plot is depicted in Figure 11 for DSC experimental data presented here (see Figure 4) as well as for

(12)

This function for the JMA model can be expressed as z(α) = m(1 − α)[− ln(1 − α)]

(13)

Figure 10 shows the z(α) functions (points) obtained by transformation of all DSC data shown in Figure 4 by eq 12

Figure 11. Determination of activation energy of crystallization from DSC data in a-Se by eq 14.

lower heating rates reported earlier by Svoboda24 (points). Clearly, there are two distinct temperature regions characterized by different values of apparent activation energy. The high temperature region corresponding to DSC data taken at higher heating rate (see Figure 4) is characterized by a low value of apparent activation energy (Ea1 = 38 kJ/mol). A considerably higher value of apparent activation energy (Ea2 = 117 kJ/mol) is found in the low temperature region reached at lower heating rates. The latter one was estimated using just three experimental points, so its uncertainty is higher. As mentioned above, the calorimetric data captured in the DSC curve may reflect a quite complex crystallization process involving surface and volume crystal growth, secondary nucleation, as well as the heat transfer phenomena.65,70 For this reason not too much physical meaning should be attached to the value of apparent activation energy. Nevertheless, in some cases where the crystallization process can be described by a simple model such as JMA it is possible to draw a more detailed conclusion. Here we can expect that apparent activation energy could be associated with the crystal growth kinetics in the relatively narrow temperature range of DSC crystallization experiments. This hypothesis can easily be tested. We can assume that in a narrow temperature range the crystal growth velocity can be described by a simple exponential dependence on temperature that can be written in logarithmic form as follows

Figure 10. Dependence of z(α) function on crystallized fraction. Points were calculated from DSC data by eq 12. Pull line was obtained by eq 13.

(points). These plots are normalized within the (0,1) range to facilitate the comparisons of different data sets. The z(α) curve calculated by eq 13 (full line) fits the experimental data very well, clearly showing that the JMA model provides an adequate description of the crystallization process under non-isothermal conditions. Another way to confirm the applicability of the JMA model is based on the shape analysis of the DSC curve.68 The JMA model was originally derived in isothermal conditions.64 Henderson65 has shown that its validity can be extended to non-isothermal conditions if the entire nucleation process takes place during the early stages of crystallization and it becomes negligible afterward. The crystal growth velocity is defined just by temperature and does not depend on the previous thermal history. These conditions are consistent with our experimental observation, so it is not surprising that the JMA model is valid in this case. Svoboda24−27 recently reported more detailed experimental studies for selenium crystallization in powdered and thin-film samples under isothermal and nonisothermal conditions. These studies reveal quite complex crystallization kinetics. Observed overall heat flow is a superposition of several subprocesses reflecting distinct mechanisms that can be elucidated by deconvolution and subsequent detailed analysis. From these experimental results it follows that for the bulk specimen, the surface crystallization is present but negligible in comparison with volume crystallization in DSC data. The apparent activation energy characterizing the crystallization process can be determined by the Kissinger method69 from the temperature shift of the maximum of DSC peak (Tp) with heating rate (β) ⎛ T 2 ⎞ −E p a + constant ln⎜⎜ ⎟⎟ = β RT ⎝ ⎠

log(u) = −

EG + constant RT

(15)

where EG is the activation energy of crystal growth. Figure 12 shows our experimental data for selenium growth velocity as well as data reported by Ryschenkow and Faivre.18 It is evident that there are two distinct temperature regions corresponding to high temperature and low temperature data. An intermediate temperature region (marked by dashed lines) coincides with spherulite morphology change, extinction of concentric bands reported by Ryschenkow and Faivre18 (indicated in Figure 1), as well as with the change in log η − log u plot (Figure 7). Activation energies of crystal growth in the high temperature region (EG1 = 38 ± 1 kJ/mol) and low

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ogy change, extinction of concentric bands, as well as with the change in the log η − log u plot. The high temperature region is characterized by the low value of apparent activation energy (Ea1 = 38 kJ/mol). A considerably higher value of apparent activation energy (Ea2 = 117 kJ/mol) is found in the low temperature region. As expected, these values are identical with activation energies of crystal growth within the limit of experimental errors.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: 00420466037346. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work has been supported by the Czech Science Foundation under grant No. 16-10562S. The authors would like to express their thanks to Dr. Roman Svoboda for DSC measurements reported in this paper.

Figure 12. Determination of activation energy of crystal growth by eq 15. The marked region corresponds to the temperature region of spherulite morphology change.

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temperature region (EG1 = 137 ± 4 kJ/mol) correspond to the values determined from DSC data taking into account experimental uncertainties and different types of measurement. Therefore, it seems that our previous assumptions concerning the description of calorimetric data are consistent.

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CONCLUSIONS The spherulitic crystal growth velocity in selenium supercooled liquid has been measured in isothermal conditions for samples obtained by fast heating of a-Se from temperatures well below Tg. These data are compared and analyzed along with previously published data of Ryschenkow and Faivre obtained by isothermal crystal growth experiments on samples quenched from a temperature well above the melting point. In both experimental setups the crystals grew linearly over the course of time, which is typical for crystal growth controlled by crystal− liquid interface kinetics with negligible influence of nucleation. The crystallization is quite complex involving two distinct spherulitic forms composed of radially twisted lamellae and characteristic concentric bands. Despite this complexity, the growth velocity data for both different thermal histories can be described by the normal growth model for moderate supercoolings (ΔT < 60 K). The screw dislocation growth provides a better description of crystal growth velocity description for larger supercoolings (ΔT > 90 K) that is also consistent with morphological observations. Nevertheless, there is still a significant discrepancy for intermediate supercoolings (10 < ΔT < 60 K). It is shown that all experimental data in the entire temperature (Tg < T < Tm) range can be described by a combined approach including both these models, taking into account the actual viscosity scaling of crystal growth η−1. It is shown that the kinetic information captured in the DSC curve can easily be analyzed providing evidence that the nonisothermal crystallization process can be described by the Johnson-Mehl-Avrami model. Under these circumstances the apparent activation energy determined from DSC peak shift with heating rate could be associated with the activation energy of crystal growth in the temperature range of DSC experiments. Two distinct regions characterized by different values of apparent activation energy were found. The transition between these temperature regions coincides with spherulite morphol5819



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Spherulitic Crystal Growth Velocity in Selenium Supercooled Liquid

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