Abstract
Here we show theoretically that the history of solid growth during “rapid” solidification must be Sshaped, in accord with the constructal law of design in nature. In the beginning the rate of solidification increases and after reaching a maximum it decreases monotonically as the volume of solid tends toward a plateau. The Shistory is a consequence of four configurations for the flow of heat from the solidification front to the subcooled surroundings, in this chronological order: solid spheres centered at nucleation sites, needles that invade longitudinally, radial growth by conduction and finally radial lateral conduction to interstices that are warming up. The solid volume (B_{s}) vs time (t) is an Scurve because it is a power law of type B_{s} ~ t^{n} where the exponent n first increases and then decreases in time (n = 3/2, 2, 1, …). The initial portion of the S curve is not an exponential.
Introduction
An extremely common phenomenon in nature is the Sshaped history of areas and volumes swept by spreading flows and collecting flows. Examples of Scurve spreading histories are the growth of populations^{1}, the spreading of technologies^{2} and the spreading of news and information^{3}, Fig. 1. The histories of collecting flows also exhibit S curves: examples are mining and the extraction of minerals, such as the Hubbert peak of oil extraction^{4}.
Rapid solidification is another common phenomenon, where the Scurve history is about the growth of solid in a subcooled liquid or gas. This phenomenon is classical textbook material in materials science^{5,6,7} and has generated a voluminous body of research dedicated to explaining the Sshaped history. The classical explanation is based on the JohnsonMehlAvrami (JMA) model of solidification^{8,9,10,11,12,13}, which consists of postulating solidification as a swarm of uniformly distributed spheres of solid that grow from initial nucleation sites, such that initially the solid volume increases exponentially. Although this model leads to a formula with two empirical constants that can be used to correlate experimental measurements, the Scurve remains a theoretical puzzle, while the theoretical basis of the sphere model is being questioned^{8}: the physical meaning of the two empirical constants is not known.
In this paper we go back to the state of knowledge that existed before the JMA model and show how to predict the Scurve of solidification purely theoretically. The starting idea is to see solidification as a “spreading flow” like the flows exhibited in Fig. 1 and to recognize that this spreading flow has the same natural tendency as all the inanimate and animate flow systems that morph freely into configurations that provide greater access to what flows, over time. This natural tendency is summarized as the constructal law^{14,15}. Here, we ask two questions:
First, what flow is spreading during solidification? It is not the solid, because the solid and its subcooled surroundings are motionless. The spreading flow is the flow of heat, which emanates from the solid surface and flows in all directions into the surroundings.
Second, what is the configuration of the heat flow system? We do not postulate the configuration (e.g. spheres in the JMA model). Instead, we rely on the physics principle^{14,15} that the flow system generates and evolves its flow architecture in order for the solidification process to be the more “rapid” at every stage in its history. We show that although the solidification is triggered as small spheres around nucleation sites, at longer times the greater solidification rate is associated with needles and dendrites, in accord with common observations (e.g. Fig. 2^{16}).
Results
Solidification begins at pointsize nucleation sites around which the solid grows as tiny spheres. As shown later in the Discussion section, the spherical growth slows down and is replaced by needleshaped growth, which is faster. The transition from spherical to needleshaped growth is in accord with the constructal law and defines the early part of the S. Because most of the Sshape of the solidification curve is due to needle solidification, we start by examining the needle configuration.
Needle invasion
Consider the solidification of a volume of pure substance that is initially in a subcooled liquid state of temperature T_{f}, which is lower than the solidification temperature T_{s}. Along the axis of this volume grows a solid needle with a speed (V) that increases monotonically with the degree of subcooling (T_{s} − T_{f}) and which has been documented extensively (e.g. Refs. 17,18,19,20,21). The instantaneous length of the needle is x and the length of the volume is L (Fig. 3).
Because of solidification, the needle grows in length and its older portions become thicker. In every stationary crosssection, the volume of solid is isothermal at T_{s} and is proportional to the volume of liquid heated by the latent heat of solidification released on the solidification front,
In other words, the latent heat of solidification (h_{sf}) that was released on the periphery of the disc of solid is now the sensible heat of the annulus of heated liquid,
The scaling relation (2) is based on the assumption that D_{f} ≫ D_{s} and it yields the constant factor in the proportionality of Eq. (1), namely
The objective of this analysis is to predict the evolution of the solidification process, which means to predict the history of the volume of solid . Because of Eq. (3), the solid volume is proportional to the liquid volume heated by the heat of solidification, . Consequently, the shape of the history function B_{s}(t) is the same as the shape of B_{f}(t), where in accord with Eq. (3),
where b is constant. The time during which the liquid and solid thicknesses grow to the length scales D_{f} (t) and D_{s} (t) is
During this time, the liquid thickness grows by thermal diffusion,
where α_{f} is the liquid thermal diffusivity. From Eqs. (5) and (6) follows the history of the volume of heated liquid,
The first conclusion is that B_{f} and B_{s} increase in proportion with t^{2}, along the needle “invasion” curve shown in Fig. 4. The volume increase is accelerated in time, but it is not exponential. This behavior lasts until the needle invades the entire liquid volume, x ~ L, which happens during the invasion time
when the liquid volume length scale is with the corresponding solid volume B_{si} derived from Eq. (4).
Consolidation
Beyond the invasion time t_{i}, the solid and liquid volumes continue to grow in proportion with each other, but they grow radially laterally until D_{f} spreads over the entire subcooled liquid. This is the “consolidation” process. The growth of the liquid thickness is in accord with the solution to the problem of thermal diffusion around a line heat source of uniform and constant temperature (the needle, T_{s}) embedded in an infinite conducting medium (T_{f})^{22}:
where R(t) is the radius of the solid cylinder, t = 0 is the start of the needle solidification process and T(r, t) is the temperature in the liquid. The cylindrical geometry to which Eqs. (9) and (10) refer is visualized in the x = constant cut through the solid shown in Fig. 3. Ei is the exponential integral function
while λ is a monotonic function of the liquid subcooling Stefan number [Ste = c_{f} (T_{s} − T_{f})/h_{sf}], as shown in Fig. 5:
The temperature distribution in the subcooled liquid is plotted in Fig. 5 for the case Ste = 1. The abscissa shows that the radial length scale (r) of the heated liquid increases as 2(α_{f}t)^{1/2}. The volume of liquid heated during the consolidation increases in proportion with the crosssectional area of the annulus of heated liquid, namely π (r^{2} − R^{2}), which has the scale α_{f}t.
The second conclusion is that the growth of B_{f} (or B_{s}) during consolidation process is proportional to t, which is slower than the t^{2} growth during the invasion process. This accounts for the inflexion of the S curve, which is the beginning of the slowdown of solidification, Fig. 4.
Plateau
The liquid volume has two dimensions, the longitudinal length L aligned with the needle and the lateral (radial) dimension Y, across the interstices. The time when the radial dimension of the heated liquid reaches Y is the consolidation time t_{c},
which follows from writing r ~ Y, where r ~ (α_{f}t)^{1/2}. After this event, the heated liquid (already Y thick) becomes warmer at a progressively smaller rate, as its average temperature rises to match the solidification temperature T_{s}. This final arc of the Scurve of solidification can be predicted as follows.
The heat current flows from the solidification front across the Ythick liquid of average temperature T_{l}(t) and has the scale k_{f}πD_{s}(T_{s} − T_{l})/Y, where it is assumed that Y ≫ D_{s}. The heat current is equal to the rate at which the energy of the liquid layer increases, namely ρ_{f}c_{f}πY^{2}dT_{l}/dt. Integrating the resulting equation and invoking the initial condition at , we obtain the decelerating rise of the liquid temperature,
where t_{p} is the time scale of reaching the T_{s} plateau:
The heat current released at the solidification front is proportional to the temperature gradient across the Ylayer, therefore it has the same exponential decay as in Eq. (14). The same behavior belongs to the rate of solidification dB_{s}/dt and consequently the history of the solid volume B_{s}(t) has the flattening tendency sketched in Fig. 4.
Discussion
In summary of the preceding analysis, the history of needle solid volume is composed of three successive periods (invasion, consolidation, plateau), each with its own time scale: t_{i}, t_{c} and t_{p}. The analysis began with the needle invasion process and it was based on the assumption that the solid is needle shaped, i.e. it is slender, not spherical. Why and under what conditions this assumption is correct is the foundation of the Scurve of solidification and it is demanded by the constructal law.
Solidification begins at pointsize nucleation sites and for this reason the earliest shape of the solid is spherical, of diameter D_{s}(t). The sphere is surrounded by an annulus of heated liquid of outer diameter D_{f}(t), which increases in proportion with (α_{f}t)^{1/2}. Equation (2) is replaced by
and leads to the conclusion that D_{s}(t) and D_{f}(t) increase proportionally and
The solid volume B_{s,sphere} (of order ) increases as
The needle is the alternative configuration, which competes with the sphere. During the same time interval (0 – t), the needle would acquire the volume [cf. Eq. (7)]
The intersection of Eqs. (18) and (19) reveals the time scale of needle onset,
where as shown in Fig. 6,
In conclusion, in the evolution of configuration the sphere is first and the needle next. Not the other way around. In accord with the constructal law, this sequence of heat flow designs facilitates more effectively the flow of heat toward equilibrium, i.e. this is the evolutionary design in which the solidification is more rapid.
The succession of four heatflow configurations (Fig. 6) is why the history of the solidified volume is S shaped. The S curve is a power law B_{s} ~ t^{n}, where the exponent varies over time in this sequence: n = 3/2, 2, 1 and finally n < 1. The early part of the Scurve is not an exponential.
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Acknowledgements
The authors acknowledge the Deanship of Scientific Research, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, for funding the project IN111042 during the course of this work.
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A.B. did the research work in collaboration with the other three coauthors, wrote the manuscript and made Figures 1, 3, 4 and 6. S.L. did the research work in collaboration with the other three coauthors, wrote the manuscript and made Figures 1, 3, 4 and 6. B.S.Y. did the research work in collaboration with the other three coauthors, wrote the manuscript and made Figure 2. A.Z.S. did the research work in collaboration with the other three coauthors, wrote the manuscript and made Figure 5.
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Bejan, A., Lorente, S., Yilbas, B. et al. Why solidification has an Sshaped history. Sci Rep 3, 1711 (2013). https://doi.org/10.1038/srep01711
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