Internal melting and coarsening of liquid droplets in an Al–Cu alloy: a 4-D experimental study

Abstract

In conventional melting experiments of pure monocrystalline metals, the phase transformation starts at the sample surface and progresses inwards according to thermal gradients. In solutionized alloys, traces of internal melting are usually observed after reheating and quenching from the semi-solid state. The formation and development of these liquid pockets are not fully understood despite their significance in semi-solid processing, where the formability is greatly influenced by the distribution of liquid within the feedstock material. In situ X-ray microtomography was performed in this study to shed light on this phenomenon. We report in detail the melting and isothermal holding of a model binary alloy where a remarkable number of liquid droplets were observed to develop and coalesce. Various computational tools have been used to study their statistical evolution as well as the local ripening mechanisms involved. We analysed an interesting case of particle coarsening which differs from classical case studies by the fact that the fast-diffusing liquid phase is entrapped within the slow-diffusing solid medium.

Appendix

Correction of the droplet distribution

When two droplets are relatively close (i.e. at a distance between each other of less than about one or two voxels) it becomes difficult, in some cases to locally segment them by simple thresholding without affecting the whole sample. Considering the great number of objects and their small size, classical 3-D segmentation treatments do not provide convincing results. Numerous non-spherical objects can thus be noticed after thresholding of 3-D images. These objects are non-segmented droplets that are, however, clearly distinct in the greyscale 3-D images. It therefore induces a bias in the measurement of the equivalent radius R _equi as well as the number N of droplets per unit volume. N thus appears smaller while R _equi is overestimated. A post-treatment was applied in order to correct this bias. This treatment is based on the analysis of the DSDV projected in a sphericity versus equivalent radius map (Fig. 8). Note that a marching cubes technique was used to calculate the volume and surface of objects. It was observed that the spheroidization of small droplets (i.e. with an equivalent radius smaller than about 10 μm) occurs very quickly with a characteristic time smaller than the time resolution $\Updelta t_{{\rm FTS}}$. Therefore the great majority of small droplets should necessarily have a sphericity ψ _obj close to one. Consequently, a small object with an equivalent radius R ^obj_equi having a sphericity much lower than one should correspond to n liquid droplets with an average equivalent radius R ^cor_equi that have not been segmented. Thus, by considering the volume of an object, it gives

$$ R^{\text{obj}}_{{\rm equi}} = n^{(1/3)}R^{\text{cor}}_{{\rm equi}} $$

(7)

Combining Eqs. 6 and 7, it can be shown that

$$ {\psi}_{{\rm obj}} = n^{(-1/2)} = {\left(\frac{R^{\text{obj}}_{{\rm equi}}}{R^{\text{cor}}_{{\rm equi}}}\right)}^{(-3/2)} $$

(8)

It implies that these objects should be aligned on a curve given by Eq. 8 with a higher density of objects observed for ψ _obj = n ^(−1/2) where n is an integer. Figure 8 shows that these trends is confirmed. A criterion based on observations has thus been arbitrarily defined to identify the objects requiring treatment. All objects on the left hand side of the dashed curve in Fig. 8 were corrected by considering those with a sphericity

$$ (n +0.5)^{(-1/2)}\le {\psi}_{{\rm obj}} < (n-0.5)^{(-1/2)} $$

(9)

as n droplets with an average equivalent radius R ^obj_equi . The effect on the DSDV is shown on Fig. 9. In this calculation we assume that the distances between droplets are negligible, however, it is worth reminding that most of them are actually not in contact but as mentioned previously they are too close to be segmented by regular image treatments.

Fig. 8

3-D diagram showing the liquid distribution into a sphericity ψ versus equivalent radius R _equi map from raw data at t _FL + 38.3 min. The dotted curve corresponds to the criterion for objects treatment

Fig. 9

Effect of the correction treatment on the droplet size distribution in volume at the time step t _FL + 38.3 min. The distribution of objects (non-processed) with sphericity superior or equal to 0.85 is also shown on the graph

Evolution of liquid arms

The solute concentration in liquid at equilibrium with an interface depends on the interface mean curvature γ _mean. When two regions A and B of different curvatures are close, it gives rise to a flux of solute ${\mathop{J}\limits^{\rightarrow}}_{s}$ between the two regions:

$$ {\mathop{J}\limits^{\rightarrow}}_{s}= \frac{2\sigma_{{\rm SL}} TD_{{\rm L}}}{\rho_{{\rm L}}m_{{\rm L}}L_{{\rm f}}}\frac{1}{\alpha a} \left( \gamma_{{\rm mean}}^{B} - \gamma_{{\rm mean}}^{A} \right) {\mathop{u}\limits^{\rightarrow}}_{A\rightarrow B} $$

(10)

where σ _SL is the solid–liquid interface energy, T is temperature, αa the diffusion distance, D _L the diffusion coefficient in the liquid, m _L the slope of the liquidus line, ρ _L the liquid density and L _f the volumetric heat of fusion. In the IGA mechanism [7] this flux of solute induces an advance of the interdendritic groove by transforming liquid into solid at the bottom of the liquid pool. The flux consumed by this process is given by:

$$ {\mathop{J}\limits^{\rightarrow}}_{r} =\frac{C_{{\rm L}}(1-k)}{\rho_{{\rm L}}(1-\beta)}\frac{1}{S}\frac{\text{d}V}{\text{d}t} {\mathop{u}\limits^{\rightarrow}} $$

(11)

where β is the solidification shrinkage, C _L is liquid concentration, k is the partition coefficient, dV the solidifying element volume and S the surface of the growing interface. The resulting advance of the liquid pool is then:

$$ \frac{\text{d}p}{\text{d}t} = \xi \frac{1}{a^{2}} $$

(12)

where ξ is $\frac{\pi\sigma_{{\rm SL}} T(1-\beta)D_{\rm L}}{2 \alpha m_{\rm L} C_{\rm L}(1-k)L_{\rm f}}$. Using σ _SL = 0.16 J m⁻² [26, 27], T = 843 K, β = 0.06,  D _L = 3.55 10⁻⁹ m² s⁻¹ [26], m _L =  −3.394 K/wt%Cu, C _L = 26.6 wt%Cu, k = 0.171, α = 3 [7], L _f =  −1.05 10⁻⁹ J m⁻³ [28] it gives a value of 3.01⁻¹⁸ m³ s⁻¹ for ξ at T = 843 K and 5.19⁻¹⁸ m³ s⁻¹ at T = 883 K with D _L = 4.14 10⁻⁹ m² s⁻¹ and C _L = 17.8 wt%Cu. It is worth mentioning that if we consider an axisymmetric geometry for the liquid arms, the region A can be modelled by half a sphere and the interface in region B has an overall positive curvature with principal curvatures of opposite sign. It is interesting to note that if region B has a mean curvature of 1/(2a) we retrieve a similar equation.