Increase Of Undercooling Research Articles

Metastable phase separation and duplex metallic glass formation of liquid Zr<sub>35</sub>Al<sub>23</sub>Ni<sub>22</sub>Gd<sub>20</sub> alloy

Duplex metallic glass with two amorphous phases has been extensively investigated for desirable strength and plasticity. In this paper, the metastable phase separation and dual amorphous phase formation of liquid Zr<sub>35</sub>Al<sub>23</sub>Ni<sub>22</sub>Gd<sub>20</sub> alloy under substantial undercooling condition and rapid cooling condition are studied by drop tube technology. The equilibrium solidification structure consists of three crystalline phases, while the critical undercooling temperature of metastable phase separation is determined to be 516 K (0.37<i>T</i><sub>L</sub>). The separated Zr-rich liquid phase undergoes amorphous transition and becomes amorphous AM-Zr phase with the composition of Zr<sub>45</sub>Ni<sub>23</sub>Al<sub>23</sub>Gd<sub>9</sub> when alloy undercooling is increased to 624 K (0.45<i>T</i><sub>L</sub>). After that, the Gd-rich liquid phase forms amorphous AM-Gd phase with the composition of Gd<sub>39</sub>Al<sub>22</sub>Ni<sub>20</sub>Zr<sub>19</sub> at larger undercooling of 714 K (0.52<i>T</i><sub>L</sub>). With the increase of liquid undercooling and cooling rate, the kinetic mechanism of metastable phase separation changes from nucleation and growth type to spinodal decomposition type, and consequently the microstructure of dual amorphous phases transforms from a spherical morphology to a reticular structure. The average hardness and Young’s modulus, which are influenced by free volume, phase volume fraction and structure of dual amorphous phases, exhibit a complex variation of first increasing and then decreasing with the decrease of alloy droplet size. The formation of dual amorphous phases is in favor of the energy dissipation and the generation of multiple shear bands during mechanical compression, which improves the plasticity for this kind of amorphous alloy.

Simulation of single bubble dynamic process in pool boiling process under microgravity based on phase field method

We use the phase field method to track the gas–liquid interface based on the gas–liquid two-phase flow in the pool boiling process, and study the bubble nucleation, growth, deformation, departure and other dynamic behaviors on the heating surface under microgravity. By simulating the correlation between liquid undercooling and bubble dynamics, we find that the bubble growth time increases with the increase of liquid undercooling, but the effect of liquid undercooling on bubble height is not significant. Meanwhile, the gas–liquid–solid three-phase contact angle and the gravity level will also have an effect on the bubble growth time and bubble height. With the increase of the contact angle, the bubble growth time and bubble height when the bubble departs also increase. While the effect of gravity level is on the contrary, the smaller the gravity level is, the larger the bubble height and bubble growth time when the bubble separates.

A correlation of eutectic growth mechanism with resultant microhardness enhancement for undercooled Ni-47.7%Mo alloy

Liquid Ni-47.7 % Mo alloy was undercooled by 328 K (0.21 TE) and solidified under microgravity condition to investigate the correlation of microhardness enhancement with eutectic growth mechanism. The microstructural morphology transformed from (Ni) + NiMo lamellar eutectic to a duplex mixture of lamellar and anomalous eutectics, and finally into anomalous eutectic. With the increase of alloy undercooling, the eutectic interphase spacings decreased significantly, leading to a remarkable structural refinement and a conspicuous rise of Vickers hardness. The dependence of microhardness upon eutectic spacing was revealed for lamellar and anomalous eutectics, respectively.

Crystal nucleation in Au49Ag5.5Pd2.3Cu26.9Si16.3 glass and undercooled melt

Crystal nucleation is an indispensable step in the crystallization of metallic glasses. With the increase of undercooling, different nucleation kinetics from that predicted by the classical nucleation theory (CNT) could be revealed. A prerequisite to addressing the nucleation kinetics is building a well-identified amorphous structure, which is yet arduous. Moreover, due to the limited temporal and spatial resolution of the conventional experimental methods, nucleation during the crystallization of metallic glass cannot be well recognized. In this study, nanocalorimetry with ultrahigh sensitivity and ultrafast scanning rate is used to address the nucleation kinetics in glass and undercooled melt. With the increase in cooling rate, crystallization, homogeneous nucleation, and heterogeneous nucleation are successively distinguished. Accordingly, the critical cooling rates suppressing crystallization and nucleation are estimated, based on which a well-identified amorphous phase for nucleation studies is produced. By applying Tammann’s two-stage crystal nuclei development method, nucleation in glass and undercooled melt is investigated. According to the evolution of crystallization heat and overall heat on the reheating after the annealing, the underlying kinetic mechanism is revealed with the Johnson-Mehl-Avrami (JMA) method. The nucleation half-time indicates that homogeneous nucleation is dominant for the crystallization below Tg. For the annealing at and above Tg, crystallization completes by the inevitable heterogeneous nucleation followed by crystal growth.

Printability and microstructure of directed energy deposited SS316l-IN718 multi-material: numerical modeling and experimental analysis

In the present paper, the interrelated aspects of additive manufacturing-microstructure-property in directed energy deposition of SS316L-IN718 multi-material were studied through numerical modeling and experimental evaluation. The printability concept and solidification principles were used for this purpose. The printability analysis showed that the SS316L section is more susceptible to composition change and lack of fusion, respectively due to the high equilibrium vapor pressure of manganese and the more efficient heat loss in the initial layers. However, the IN718 section is more prone to distortion due to the formation of a larger melt pool, with a maximum thermal strain of 3.95 × 10−3 in the last layer. As the process continues, due to heat accumulation and extension of the melt pool, the cooling rate decreases and the undercooling level increases, which respectively result in coarser microstructure and more instability of solidification front in the build direction, as also observed in the experimental results. The difference is that the dendritic microstructure of the IN718 section, due to the eutectic reaction L → γ + Laves, is formed on a smaller scale compared to the cellular microstructure of the SS316L section. Also, the decrease in cooling rate caused the secondary phase fraction in each section (delta ferrite in SS316L and Laves in IN718) to increase almost linearly. However, the hardness calculation and measurement showed similarly, even though with the transition from SS316L to IN718 the hardness is significantly increased due to higher yield strength of the matrix and the presence of Laves intermetallic phase (~ 260 HV0.3), the hardness in each section decreases slightly due to the coarsening of the microstructure from the initial layer to the final.

Anomalous kinetics, patterns formation in recalescence, and final microstructure of rapidly solidified Al-rich Al-Ni alloys

From thermodynamical consideration, rather a monotonically increasing crystal growth velocity with increasing undercooling is expected in the crystallization of liquids, mixtures, and alloys [P.K. Galenko and D. Jou, Physics Reports 818 (2019) 1]. By contrast to this general theoretical statement, Al-rich Al-Ni alloys show an anomalous solidification behavior: the solid-liquid interface velocity slows down as the undercooling increases [R. Lengsdorf, D. Holland-Moritz, D. M. Herlach, Scripta Materialia 62 (2010) 365]. It is also found that besides the anomalous growth behaviour, changes in the shape of the recalescence front as the growth front morphology occur. In the light of recent measurements in microgravity with an Al-25at.% Ni alloy sample onboard the International Space Station (ISS) results confirming this anomalous behavior as an unexpected trend in solidification kinetics are presented. The measurements show multiple nucleation events forming the growth front, a mechanism that has been observed for the first time in Al-Ni alloys [D. Herlach et al., Physical Review Materials 3 (2019) 073402; M. Reinartz et al. JOM 74 (2022) 2420] and summarized with detailed analysis in the present publication over a wider range of concentrations. Particularly, the experimental measurements and obtained data directly demonstrate that the growth front does thus not consist of dendrite tips (as in usual rapid solidifying samples), but of newly forming nuclei propagating along the sample surface in a coordinated manner. Theoretical analysis on intensive nucleation ahead of crystal growth front is made using the previously developed model [D.V. Alexandrov, Journal of Physics A: Mathematical and Theoretical 50 (2017) 345101]. Using equations of this model, quantitative calculations confirm the interpretation of experimentally observed propagation of the recalescence front and obtained data on the microstructure of droplets solidified in electromagnetic levitation facility (EML) on the Ground, under reduced gravity during parabolic flights, and in microgravity conditions onboard the ISS.

Three-dimensional microstructure evolution of Ti–6Al–4V during multi-layer printing: a phase-field simulation

Remelting and grain growth in deposited layers and the columnar-to-equiaxed transition (CET) have a significant impact on microstructural evolution during the multilayer printing process. A three-dimensional phase-field model incorporating a heterogeneous nucleation model was developed to study the microstructural evolution under different scanning strategies and scanning velocities during directed energy deposition (DED) additive manufacturing. An experiment was performed to validate the predicted grain morphologies using the proposed model. The results indicate that undercooling determines the extent of heterogeneous nucleation and controls the CET. In comparison with constitutional undercooling and curvature undercooling, thermal undercooling contributes more to the DED of Ti–6Al–4V. The maximum undercooling occurs on the top layer leading to a higher ratio of equiaxed grain formation. The undercooling increases with the build height, which is beneficial for the CET. The growth direction of the columnar grains is controlled by the size and shape of the melt pool. The bidirectional scanning strategy leads to bent columnar grains as a result of the changing growth direction between adjacent deposited layers. The decrease in the scanning velocity leads to coarsening of the grains.

Dual-modification effect of Ca element on the hypereutectic Al-40wt% Cu alloy

The influence of Ca element on the primary θ-Al2Cu and eutectic (θ-Al2Cu+α-Al) microstructure in Al-40wt% Cu hypereutectic alloy was investigated. The results showed that the Ca can significantly improve the size and morphology of primary θ-Al2Cu, which was mainly attributed to the enrichment of Ca atoms in the front of solid-liquid interface during solidification. The enrichment of Ca can hinder the stacking of solute atoms, which impeded the radial growth, so as to make the θ-Al2Cu transform from angular planar grains into equiaxed grains with fine size. In addition, the enrichment of Ca also prevented the stacking of solute atoms at spiral dislocation steps during the axial growth. The decrease of eutectic spacing was mainly attributed to the increase in undercooling. Compared with the tensile strength of unmodified Al-40Cu alloy, the tensile strength of Al-40Cu alloy was enhanced by adding 0.6 wt% Ca which increased from 81.21 MPa to 144.80 MPa. In addition, the fracture elongation increased from 2.3% to 4.48%. The improvement of morphology of primary θ-Al2Cu phase was the basic reason for the enhancement of mechanical properties of Al-40Cu alloy.

Rapid Eutectic Growth Kinetics of Undercooled Nb-Si Alloys at Electrostatic Levitation State

The rapid eutectic growth kinetics of undercooled Nb-Si alloys was investigated by using electrostatic levitation (ESL) technique combined with triggering method. For eutectic Nb-17.3 at.% Si alloy, the rapidly solidified microstructure was characterized by (Nb) dendrites and (Nb)+Nb3Si eutectics, which was different from the master alloy composed of complete eutectic structures. The transition of primary phase from the faceted Nb3Si phase to non-faceted (Nb) phase and the formation of complete eutectic microstructure were observed within hypereutectic Nb-18.7 at.% Si alloy with the increase of undercooling. Three distinct solidification paths leading to different microstructures and constituent phases were revealed by controlling the melt undercooling. Furthermore, the phase selection mechanism was clarified from the perspective of nucleation kinetics. Based on the experimental and analytical results, the coupled zone of (Nb)+Nb3Si eutectic was qualitatively predicted. The final microstructures solidified at different undercoolings were all composed of the (Nb) phase and Nb3Si phase, whose crystaline structures were both confirmed by transmission electron microscopy analysis. Besides, it was found that the volume fractions of (Nb) and Nb3Si phases could be undercooling control adjusted by. According to the nano-indentation and Vickers micro-indentation tests, the mechanical properties varied significantly with solidification paths. Compared with the master alloy, the hardness of hypereutectic Nb-18.7 at.% Si alloy at 157 and 473 K undercoolings was decreased by 42.7 % and 9.9 %, while the fracture toughness was enhanced remarkably by 593.1 % and 144.8 %, respectively. This may provide a desirable approach for designing Nb-Si alloys with improved performances.

Exploration of crystal growth behavior in Au-based metallic glass by nanocalorimetry

The Au49Ag5.5Pd2.3Cu26.9Si16.3 (at.%) bulk metallic glass is among the promising Au-based metallic glasses with the enhanced glass-forming ability and thermal stability. However, crystal growth kinetics during the crystallization of this metallic glass remains ambiguous. In this study, the rapid crystallization of a Au49Ag5.5Pd2.3Cu26.9Si16.3 bulk metallic glass in a wide temperature range was realized by nanocalorimetry. The underlying thermodynamic and kinetic features of crystal growth were ascertained based on the Cohen and Grest (C&G), Mauro-Yue-Ellison-Gupta-Allan (MYEGA) and Schmelzer models, and the adaptability of these three models was demonstrated. The maximum crystal growth rate (0.037 m/s) and corresponding temperature (595 K) estimated by these three models were similar but different crystal growth behaviors in the deeply undercooled melt were distinguished. With an increase in undercooling, the breakdown of the Stokes-Einstein relation was discovered, and a temperature-dependent decoupling exponent was demonstrated by the Schmelzer model, causing a divergence of crystal growth rate from that estimated by C&G and MYEGA models. Moreover, screw-dislocation-mediated growth was stated as the dominant crystal growth mechanism during the crystallization of the Au49Ag5.5Pd2.3Cu26.9Si16.3 bulk metallic glass.

Simulation on in-situ crystal growth of lead-free solder Sn-57Bi alloy

Lead-free solder Sn-57Bi alloy is a good candidate material for electronic packaging. However, it is difficult to observe in-situ crystal growth process of Sn-57Bi alloy at high temperature and pressure environment. In this paper, based on the Monte Carlo method with the effect of volumetric and interfacial free energy taken into consideration quantitatively, a probability model for eutectic growth is established to simulate the in-situ crystal growth of Sn-57Bi alloy. The in-situ crystal growth of Sn-57Bi alloy at different undercoolings is studied. The simulation result shows the coupled growth of two phases in eutectic alloys, along with that lamellar spacing decreases and growth velocity increases with the increase of undercooling, agreeing with the reported experimental works on the eutectic growth. This work presents a good method to study the in-situ crystal growth process of alloys.

A temperature-dependent atomistic-informed phase-field model to study dendritic growth

The effects of solid–liquid interface anisotropy and kinetics on crystallization of highly undercooled titanium melts are studied by coupling molecular dynamics and phase-field simulations. Unlike previous models and to consider the actual physics of crystal growth, the phase-field parameters, representing interface mobility, solid–liquid transformation barrier, and interfacial energy gradient, are temperature dependent. The parameters are determined by a combination of molecular dynamics simulations and classical thermodynamic calculations based on the temperature-dependent solid–liquid interface properties and kinetic coefficient. We investigated Ti dendritic growth as a benchmark example to demonstrate that the phase-field model presented in this work is more compatible with the experimental data and theoretical models in comparison to the earlier models with constant model parameters. The capillary fluctuation method is used to determine the solid–liquid interface energy and its anisotropy for undercoolings up to 400 K. Similar to theoretical models, the average solid–liquid interface energy decreases with temperature, and the preferred dendrite growth direction shifts from 〈100〉 to 〈110〉 direction as the undercooling increases. Phase-field simulations also show other favorite growth directions, implying that there is a competition between the interface anisotropy and kinetics of the solid–liquid interface.

A Model for the Anomalous Velocity-Undercooling Behaviour of Levitated Al-Ni Alloys On-board the International Space Station

Al-Ni alloys (for Ni < 45 at.%) show a unique property in that, over at least part of the accessible undercooling range, the recalescence velocity measured in electromagnetically levitated samples is observed to decrease as the undercooling increases. This result has been subject to careful validation, including microgravity experiments utilising the TEMPUS levitation facility on-board the International Space Station (ISS). In these experiments, anomalous growth is observed to coincide with a recalescence morphology comprising multiple circular growth fronts [Herlach et al. Phys. Rev. Mat. 3, 073,402 (2019)], termed “scales”. In this paper we present an analysis of high speed video data from the ISS experiments in which we show that such scale-like growth is consistent with a recalescence front that is initially confined to a thin layer on the surface of the sample. This then nucleates a slower, radial inward growth, which is consistent with microstructures observed in Al-Ni droplets. We show that such surface recalescence would be favoured for samples which were surface enriched in Ni, wherein the recalescence velocity (at fixed nucleation temperature) increases rapidly with Ni-concentration. Moreover, it is shown that the anomalous velocity behaviour can be matched in all compositions studied if the surface enhancement in Ni is a linear function of the nucleation temperature with a gradient of 0.03 at.% K−1. Analysis of historical results from the literature indicates that such surface Ni-enhancement may have been present, but overlooked, in other experiments on Al-rich Al-Ni droplets.

The Morphology and Solute Segregation of Dendrite Growth in Ti-4.5% Al Alloy: A Phase-Field Study

Ti-Al alloys have excellent high-temperature performance and are often used in the manufacture of high-pressure compressors and low-pressure turbine blades for military aircraft engines. However, solute segregation is easy to occur in the solidification process of Ti-Al alloys, which will affect their properties. In this study, we used the quantitative phase-field model developed by Karma to study the equiaxed dendrite growth of Ti-4.5% Al alloy. The effects of supersaturation, undercooling and thermal disturbance on the dendrite morphology and solute segregation were studied. The results showed that the increase of supersaturation and undercooling will promote the growth of secondary dendrite arms and aggravate the solute segregation. When the undercooling is large, the solute in the root of the primary dendrite arms is seriously enriched, and when the supersaturation is large, the time for the dendrite tips to reach a steady-state will be shortened. The thermal disturbance mainly affects the morphology and distribution of the secondary dendrite arms but has almost no effect on the steady-state of the primary dendrite tips. This is helpful to understand the cause of solute segregation in Ti-Al alloy theoretically.

Study on Microstructure Refinement Mechanism in Deeply Supercooled Solidification of Ni-25at.%Cu Alloys

Maximum undercooling degrees were obtained by glass purification and superheating. Rapid solidification in an undercooled alloy was systematically studied. As the undercooling increases, the solidification front appears randomly. The microstructure undergoes a transformation from coarse dendrite grains to granular grains to oriented dendrite to equiaxed crystal. The results show two grain refinement under high and low undercooling. No high strength textures are found under high undercooling conditions, and many grain boundaries of high angle nature are observed, which confirms recrystallization. In the TEM characterization of high undercooling, high density dislocation strain indicates that accumulated stress in the rapid solidification process induces microstructure plastic strain and promotes recrystallization. The results show that the grain boundary of refined grain is coarser at low undercooling, but narrower at high undercooling. Twins and subgrains observed at large undercooling indicate that the grain refinement mechanisms are very different.

In situ investigation of the Columnar-to-Equiaxed Transition during directional solidification of Al–20 wt.%Cu alloys on Earth and in microgravity

Solidification experiments on Al–20 wt.%Cu alloys with grain refiner were performed in a Bridgman-type furnace to investigate the impact of gravity-driven phenomena on the Columnar-to-Equiaxed Transition (CET) during directional solidification (i) in microgravity, on board the sounding rocket MASER-14 and (ii) on Earth in three growth directions, namely horizontal, vertical upward (counter-gravity direction) and vertical downward (in-gravity direction). The CET was provoked by a step increase of the cooling rate and was visualised in situ and in real-time using X-radiography. This paper reports direct observations of dendritic columnar growth, CET and the subsequent equiaxed regime, and quantitative characterisation of the microstructures. The increase in constitutional undercooling that induces the nucleation of the first grains ahead of the columnar front can be related to the rapid thermal change following the increase in cooling rate. The nucleation distance of equiaxed grains ahead of the columnar front is larger in microgravity than on Earth and possible origins for this difference are discussed. Both mechanical and solutal blocking mechanisms were observed at CET. Mechanical blocking is attributed to the occurrence of solidification-induced shrinkage flow. The relative importance of each blocking mechanism is detailed for each growth configuration. In particular, mechanical blocking is shown to be the principal type of impingement in the microgravity experiment. The final equiaxed grain structure was characterised, and the results confirm that gravity effects become less significant at high growth rates.

Solidification microstructure in a supercooled binary alloy

Abstract The maximum undercooling that has been achieved for Ni-Cu alloy, by using molten glass purification and cyclic super-heating technology, is 270 K. With the help of high-speed photography, the solidification front images of Ni-Cu alloy at various typical undercooling were obtained. Two grain refinements occurred in the range of 60 K&lt; ΔT &lt; 100 K and ΔT &gt; 170 K, the solidification front became smoother, and the solidification position appeared randomly. With the increase of undercooling, the transition from solute diffusion to thermal diffusion leads to the transition from coarse dendrite to directional fine dendrite. At large undercooling, considerable stress is accumulated and some dislocations exist in the microstructure. However, the proportion of high-angle grain boundaries is as high as 89%, with twin boundaries of 13.6% and most strain-free structures, and the microhardness decreases sharply. This indicates that the accumulated stress at large undercooling causes the plastic strains in the microstructure, and in the later stage of recalescence, part of the plastic strains is dissipated by the system and acts as the driving force to promote the recrystallization of the microstructure.

Rapid solidification of undercooled FeCoNiCu multi-principal element alloy: Mechanical and tribological properties

A FeCoNiCu multi-principal element alloy (MPEA) was undercooled and its microstructures, mechanical and tribological performance were investigated for different solidification conditions. All the microstructures were consisted of two phases, i.e., the primary Cu-depleted phase and the secondary Cu-rich phase. With the increase of undercooling, a transition of the primary phase from coarse dendrites to fine equiaxed grains happened and the volume fraction of the secondary phase decreased considerably. The primary phase was supersaturated by Cu at high undercooling and segregation of Cu was inhibited considerably by rapid solidification. Consequently, the hardness and yield strength were improved remarkably due to grain refinement strengthening and solution strengthening, and the wear volume was decreased significantly due to both improved mechanical properties and oxidation induced by friction. For the largest undercooling obtained in this study, (i.e., ΔT = 226 K), the yield strength was increased by about 50% and the wear volume was decreased by about 45% as compared with the as-cast case. The current work shows rapid solidification could be an effective way to modulate both mechanical and tribological properties of MPEAs and hence is helpful for their potential applications as engineering materials.

Multiphase Field Modeling of Dendritic Solidification of Low-Carbon Steel with Peritectic Phase Transition

In the present study, an improved multiphase field model with taking into consideration the S/L interface anisotropy is proposed to simulate the dendritic growth with peritectic transition of low-carbon steel, and a new random heterogeneous nucleation method is proposed to treat the γ phase nucleation during the peritectic solidification process. The γ phase growth around the dendrite (the single dendrite and polycrystalline ferrite) and the subsequent peritectic transformation during the peritectic solidification are simulated. The results show that when the temperature reaches the γ phase nucleation temperature, γ nuclei suddenly nucleate at the δ/L interface, and laterally grow with the movement of L/γ/δ triple point around the δ/L interface of δ dendrite. After the δ dendrite is fully wrapped by the continuous intervening γ phase, the γ phase continues to grow with direct phase transformation from both δ phase and liquid phase. With the increase of melt undercooling, more δ phase and liquid phase are consumed to form γ phase, and the intervening γ phase between the δ dendrite and liquid phase becomes more and more thick, but the δ dendrite arm becomes more and more thin. During the dendritic solidification process, the solute would be rejected from dendrite trunk with the dendritic growth and enriched at the dendrite boundary, especially at the dendrite root. The solute enrichment at the dendrite root would inhibit the γ phase growth at the dendrite root and thus the thickness of intervening γ phase at the dendrite root is thinnest around the dendrite. Moreover, the solute concentration in γ phase is among the δ phase and liquid phase, because the carbon solubility in γ phase is higher than that in δ phase, but lower than that in liquid phase.

Quantitative phase-field simulations of globulitic solidification from the critical nucleus with the effects of interface curvature and attachment kinetics

A one-dimensional quantitative phase-field model (thin-interface approximation) considering the effects of interface curvature and attachment kinetics was implemented to simulate the solidification of a solid sphere (globulitic solidification) of a pure substance in an undercooled infinite melt, beginning from the critical nucleus. The classical sharp interface model for the same problem was also solved and the growth velocity and interface temperature were calculated and compared with the results of the phase-field model. Numerical conditions of the phase-field model, namely interface thickness, numerical mesh spacing, and critical nucleus model, that give close agreement with the sharp interface model (quantitative modeling) were determined. The possibility of a unique controlling step of the overall solidification kinetics by heat transfer or interface attachment kinetics or even of a mixed control was analyzed by defining relative undercoolings at the solid–liquid interface. For both metallic and non-metallic materials, growth is initially controlled by interface attachment kinetics, but, as the solid sphere grows and the melt heats up, it changes to the mixed control regime with heat transfer before spherical growth becomes unstable. The spherical radius for the change to mixed control increases when the kinetic coefficient or the magnitude of melt undercooling increases, because the heat transfer step is facilitated relative to that of attachment kinetics.