Proposition of a Growth Law as a Function of Solidification Parameters for Monotectic Alloy Systems

Abstract

Monotectic alloys show promising applications in wear-resistant automotive components, once these systems have remarkable self-lubricating properties that are of great interest for using in bearings. Much research has been devoted to better comprehend monotectic reactions. Some studies assume that the interphase spacing evolution in monotectic alloys follows the classical relationship used for eutectics or the dendritic growth relationship; however, some studies reported that the growth laws seem not to be valid for some cases. Because of that, obtaining single mathematical expressions that allow describing the development of solidification structures as a function of thermal parameters is very important. Based on the above, this chapter proposes a systematic analysis of the monotectic growth laws proposed in the literature and suggests exclusive growth laws as a function of solidification parameter for monotectic alloys solidified under different heat extracting configurations.

1. Introduction

Monotectic alloys have limited solubility in the liquid state as a determining characteristic. Some of these alloys are of unique importance, such as aluminum alloys dispersed with lead, bismuth, and indium, used as self-lubricating automotive components [1, 2, 3]. Bismuth, lead, and indium dispersed in the aluminum matrix have a low melting point and remain dispersed in the matrix, which reduces the hardness of the material but can improve the performance in service against adhesive and abrasive wear since they can flow easily in conditions of slipping, like on components that are in a relative motion [4, 5]. The sum of these characteristics results in a favorable tribological behavior. According to [6, 7], Al-Bi alloys are potential substitutes for materials containing lead additions.

Among the bearing alloys used today in the automotive industry, bronze with irregularly distributed lead stands out. The cars of the future require a bearing material with lower coefficients of friction and wear, besides being able to sustain higher dynamic pressures compared with those offered by bronze-lead alloys [8]. In addition, recent investigations have pointed to the possibility of manufacturing porous aluminum with deep pores using monotectic alloys and electrochemical attack, since with this method it is possible to produce anisotropic porous media with pore sizes between 5 μm and 20 μm, smaller than those obtained by classical procedures for manufacturing porous materials [9, 10].

In monotectic alloys, the immiscibility of the elements tends to give rise two very distinct phases during solidification. Normally the structure of this material is formed by a soft constituent (in smaller content) dispersed in the matrix of the constituent with greater resistance [11]. This low miscibility maybe caused by the atomic size mismatch, which makes that each element diffuses differently and leads to unique microstructure, and the great wear resistance property in monotectic alloys is seen as a characteristic of high-entropy alloys (HEAs). Recently, HEAs have drawn enormous attention in diverse fields because of their distinctive concept and unique properties [12, 13].

Due to their interesting peculiarities, important studies have been developed in the last three decades to improve understanding about the phenomena involved in the formation of the monotectic alloys [7, 8, 14, 15, 16, 17, 18, 19, 20, 21]. It is extremely important to understand the process of formation and development of these structures to control the properties of the final product, for example, recently, due to the potential application in tribological systems, some authors have developed studies with the purpose of verifying the influence of monotectic structures on micro-abrasive wear resistance [11, 22].

Interphase spacing is the commonly investigated microstructural parameter, with the attempt to obtain mathematical expressions that allow to describe the development of solidification structures as a function of solidification thermal parameters such temperature gradient or growth rate, but some other parameters, such particles’ diameters were also analyzed.

In analyzing the monotectic reaction, [14] found that the development of many monotectic alloys during directional solidification is in accordance with a relationship between interphase spacing (λ) and growth rate (v). This relationship, given by λ²v = C, where C is a constant, is the same that governs the development of regular eutectics, according to studies by [23]. The monotectic systems can be separated into two categories, i.e., those that are in accordance with the model cited with acceptable agreement and those that respect the power-function relationship but with exponents different from that proposed by [14]. Furthermore, in many studies where values for C have been determined, the solidification conditions are not transient [8, 14, 17, 24, 25], which tends to distance the conditions of these experiments from those occurring in industrial processes.

Calberg and Bergman [15] have found that the relationship of growth to some irregular monotectic structures is like the relationship governing dendritic growth, given by λ.v^a.G^b = C, where v is the growth rate, C is a constant for both cases, G is the temperature gradient, and a and b are constants.

Yang and Liu [17] analyzed the particle diameter of hypermonotectic alloys and found results that lead to power-type functions, however, with different exponents from those proposed ever, with different exponents from those proposed for the quantification of interphase spacings.

Silva et al. [6, 7, 18, 26, 27, 28, 29] tried to evaluate the microstructural behavior of hypomonotectic and hypermonotectic alloys solidified in devices with upward and downward configurations. They found that the alloys that showed cellular configuration presented mathematical expressions correlating cellular spacing with growth rate in agreement with the one found and merged in the literature, λ₁ = C.v^−1,1, where v is the growth rate and C is a constant [30, 31, 32, 33]. In addition, the alloys solidified in an upward device and exhibited particle morphology, presented mathematical expressions congruent to the one for regular eutectic alloys. On the other hand, some alloys solidified in a downward setup, and the ones that presented fibrous morphologies did not fit in any expression, as well as alloys' particle diameters. The relationship governing dendritic growth could represent all the upward solidification microstructures. The results of particle diameters found also obey power-type functions, as suggested by Yang et al., but with different exponents.

Freitas [34], when analyzing the behavior of monotectic and hypermonotectic alloys, found power-type functions to quantify the interphase spacing and particle diameter with growth velocity, all different from the laws proposed previously. Costa et al. [31], who studied ternary monotectic alloys, but with a microstructure of particles and pearl strands, presented mathematical expressions to correlate interphase spacings with the velocity of the type, λ = C.v^−1.1, for cell growth and potency-like functions, but with different exponents, to analyze the diameter of the particles.

There is no agreement as to the validity of these growth laws to characterize the growth of monotectic systems. The vast majority of these studies are based on the use of solidification devices of the Bridgman type that impose a stationary regime of heat extraction [35], with few evaluating solidification processes of monotectic alloys under transient conditions, considering the effects of the flow of particles rich in solute in the remaining liquid volume, and correlating the measured thermal solidification parameters with the monotectic structures got, which brings the experiences closer to the industrial reality [28, 36]. There are also few studies that analyze the influence of the growth direction on the interphase spacings and on the dimensions of bismuth, lead, and indium particles [37]. The use of the relationships of [23] may not be valid under certain conditions, as verified by [29], in the case of downward directional solidification. Moreover, the relationship proposed by [15] for dendritic growth cannot be suitable to be employed in the case of downward directional solidification where the heat transfer can occur not only by conduction but also by convection.

Researchers often aim to study whether there is some association between two observed variables and to estimate the strength of this relationship. These research objectives can be quantitatively addressed by correlation analysis, which provides information about not only the strength but also the direction of a relationship. The two most used correlation coefficients in medical research, the Pearson coefficient, and the Spearman coefficient. With Pearson coefficient, two typical properties of the bivariate normal distribution can be relatively easily assessed, and researchers should check approximate compliance of their data. The correlation coefficient is sometimes criticized as having no obvious intrinsic interpretation, and researchers sometimes report the square of the correlation coefficient. This R² is termed the “coefficient of determination.” It can be interpreted as the proportion of variance in one variable that is accounted for by the other [38, 39].

With so many results already found, both for the same and different alloys and under different solidification conditions, with all of them establishing power-type laws as a proposition, it is understood to be a good alternative to organizing the results and performing visual and statistical analysis. To evaluate a possible correlation between several experiments performed and get standardized laws of monotectic growth that can make physical sense and thus specifically characterize monotectic growth, with its particularities of solidification process and microstructural morphologies, besides being able to predict this growth and be able to set solidification parameters that result in certain microstructural behavior.

Based on this consideration, this research aims to analyze statistically and physically various results to get a monotectic growth law as a function of position and growth rate for directionally solidified systems under transient heat extraction conditions, with expectations to be extended for all types of solidification.

2. Experimental procedure

In the present work, some results were analyzed: the ones of the works by [40, 41] concerning the vertical, upward, and downward solidification of monotectic alloys of the Al-Bi, Al-Pb, and Al-In system, by [34] regarding the upward vertical solidification of monotectic alloys of the Al-Bi and Al-Pb system and by [42], whose summaries of materials and methods are described in this work; however, details are found in the former literature.

Directional solidification devices, in which heat is extracted only through a water-cooled side, promoting vertical upward and downward and horizontal solidification, were used in the experiments permitting a wide range of cooling rates to be attained along the length of the castings.

The alloys detailed in Table 1 were prepared using alloying elements in a fixed proportion, which were melted in a refractory crucible in a muffle furnace. The molten alloy was then poured into the casting chamber of the directional solidification apparatus. The melt temperature was parameterized in ΔT_V above the liquidus temperature of each alloy at the beginning of each experiment, according to Table 1. Approaching the parameterized melt temperature, the electric heaters were disconnected and at the same time the controlled water flow was initiated, permitting the onset of solidification. Continuous temperature measurements in the castings were monitored during solidification via the output of a bank of fine type K, placed in the geometrical center of the molds’ cavities along their length. All the thermocouples were connected by coaxial cables to a data logger interfaced with a computer, capable of automatically record temperature data.

Type of solidification device—author	Alloy composition	Temperature (°C)
		TL ou TM	TP	∆TP (%)
Upward-Silva et al. [41]	Al-3.2wt.%Bi	658	690	5
Upward-Freitas et al. [5]	Al-3.2wt.%Bi	658	690.9	5
Downward-Silva et al. [41]	Al-3.2wt.%Bi	658	690	5
Upward-Silva et al. [7]	Al-5.0wt.%Bi	708.6	744	5
Upward-Silva et al. [7]	Al-7.0wt.%Bi	750	788	5
Upward-Silva et al. [19]	Al-0.9wt.%Pb	659.6	692	≈5
Downward-Silva et al. [28]	Al-0.9wt.%Pb	659.6	692	≈5
Upward-Silva et al. [19]	Al-1.2wt.%Pb	659	692	≈5
Downward-Silva et al. [28]	Al-1.2wt.%Pb	659	692	≈5
Horizontal-Konno [40]	Al-1.2wt.%Pb	659	692	≈5
Upward-Silva et al. [34]	Al-2.1wt.%Pb	720.6	798	10
Upward-Freitas [33]	Al-2.5wt.%Pb	745	782.25	5
Upward-Silva et al. [27]	Al-5.5wt.%In	652.2	671	3

Table 1.

Experiments performed with alloys of the Al-Bi, Al-Pb, and Al-In systems: T_M, monotectic temperature at equilibrium; T_L, liquidus line temperature; T_P, pouring temperature; ΔT_p, percentage of overheating of the liquidus line.

The ingots were subsequently sectioned along their vertical axis, ground, and etched with an acid solution to reveal the macrostructure (Poulton' s reagent: 5 mL H2O; 5 mL of hydrofluoric acid (HF) 48%; 30 mL of nitric acid (HNO3) 65%; 60 mL of hydrochloric acid (HCl) 37%). The microstructural characterizations of the directionally solidified (DS) alloy castings were performed by extracting samples at different sections along the length of the castings. The samples were polished with SiC papers, finely polished with diamond paste, and then etched with an acid solution to reveal the microstructure (hydrofluoric acid (HF)). The examinations of the microstructures were performed using an optical metallurgical microscope. The interphase spacings (λ) were measured from the optical images of the solidified samples. They were measured along the casting longitudinal section by averaging the horizontal distance between the droplets, fibers, and/or strings of pearls, adopting as reference the center of each morphology. The evolution of the particles size (droplet diameter: d) along the casting lengths was also determined. A scanning electron microscope equipped with an energy-dispersive spectrometer complemented the microstructural characterization.

Table 1 presents: the list of directional solidification experiments performed with the device used, alloy composition, liquidus temperatures (hypomonotectic and hypermonotectic alloys) or monotectic temperature at equilibrium (T_L or T_M, respectively), and pouring temperature (T_P) along with the percentage of liquidus temperature super-heat (ΔT_V%). Figure 1 shows some typical microstructures obtained from the monotectic solidification processes: droplets, fibers, and strings of pearls.

Figure 1.

Macrostructure and longitudinal microstructure of the Al-2.5wt.%Pb and Al-3.2wt.%Bi alloys, respectively, solidified in a vertical upward device [14].

To get a result as a function of position, all points from the experimental results by [34, 40, 41] were used for interphase spacings and particle diameters for each position, and the results were dimensionless by dividing each point by the value of spacing or average diameter for each structure observed, i.e., for an ingot in which a structure of particles from 0 mm to 40 mm and fibers from 60 mm to 90 mm was found, and calling λ_i the value of the spacing for a position i in this structure, it is seen that:

• λ_MED for droplets morphology = λ₂₀ [μm]

• value of dimensionlessization for a position i in this structure = λ_i/λ₂₀ dimensionless

• λ_MED for fibrous morphology = λ₇₅ [μm]

• value of dimensionlessization for a position i in this structure = λ_i/λ₇₅ dimensionless

Dimensionlessization is of fundamental importance in physical analyses because of the significant reduction in experimental effort to establish the relationship between two variables, besides producing a better approximation than the variables themselves.

After dimensionlessization, all points were plotted as a function of position, also calculating the coefficient of determination (R²), which is one way to assess the quality of the model’s fit.

In statistics, simple linear regression is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable (conventionally, the x and y coordinates in a Cartesian coordinate system) and finds a linear function (a non-vertical straight line) that, as accurately as possible, predicts the dependent variable values as a function of the independent variables. The adjective simple refers to the fact that the outcome variable is related to a single predictor [43, 44].

It is common to make the additional stipulation that the ordinary least squares (OLS) method should be used: the accuracy of each predicted value is measured by its squared residual (vertical distance between the point of the dataset and the fitted line), and the goal is to make the sum of these squared deviations as small as possible. Other regression methods that can be used in place of ordinary least squares include least absolute deviations (minimizing the sum of absolute values of residuals) and the Theil-Sen estimator (which chooses a line whose slope is the median of the slopes determined by pairs of sample points).

Consider the model function

$$y=\alpha +\mathit{\beta x}$$E1

which describes a line with slope β and y-intercept α. In general, such a relationship may not hold exactly for the largely unobserved population of values of the independent and dependent variables; it can be called the unobserved deviations from the above equation the errors. Suppose it is observed n data pairs and call them {(x_i, y_i), i = 1, …, n}. The underlying relationship between y_i and x_i involving this error term ε_i can be described by

$$y_i=\alpha +\beta x_i+{\epsilon }_i$$E2

This relationship between the true (but unobserved) underlying parameters α and β and the data points is called a linear regression model.

The goal is to find estimated values $\widehat{\alpha }$ and $\widehat{\beta }$ for the parameters α and β, which would provide the “best” fit in some sense for the data points. On the case of this work, the “best” fit will be understood as in the least-squares approach: a line that minimizes the sum of squared residuals ${\widehat{\epsilon }}_i$ (differences between actual and predicted values of the dependent variable y), each of which is given by, for any candidate parameter values α and β,

$$\widehat{{\epsilon }_i}=\underset{¯}{y}-\widehat{\beta }\underset{¯}{x}$$E3

In other words, $\widehat{\alpha }$ and $\widehat{\beta }$ solve the following minimization problem:

$$\mathit{\text{Find}}{\mathit{min}}_{\alpha ,\beta }Q\left(\alpha ,\beta \right)\mathit{\text{for}}Q\left(\alpha ,\beta \right)=\sum _{i=1}^n{\widehat{\epsilon }}_i^2+\sum _{i=1}^n{\left(y_i-\alpha -\beta x_i\right)}^2$$E4

By expanding to get a quadratic expression in α and β, α and β values can be derived to minimize the objective function Q (these minimizing values are denoted $\widehat{\alpha }$ and $\widehat{\beta }$ [6]:

$$\widehat{\alpha }=y_i-\alpha -\beta x_i$$E5

$$\widehat{\beta }=\frac{{\sum }_{i=1}^n\left(x_i-\underset{¯}{x}\right)\left(y_i-\underset{¯}{y}\right)}{{\sum }_{i=1}^n{\left(x_i-\underset{¯}{x}\right)}^2}=\frac{\mathit{Cov}\left(x,y\right)}{\mathit{Var}\left(x\right)}=r_{\mathit{xy}}\frac{{\delta }_y}{{\delta }_x}$$E6

where x and y are the average of the x_i and y_i, respectively; r_xy is the sample correlation coefficient between x and y; δ_x and δ_y are the uncorrected sample standard deviations of x and y; and Var and Cov are the sample variance and sample covariance, respectively.

Substituting the above expressions for $\widehat{\alpha }$ and $\widehat{\beta }$ into

$$f=\widehat{\alpha }+\widehat{\beta }x$$E7

yields

$$\frac{f-\underset{¯}{y}}{{\delta }_y}=r_{\mathit{xy}}\frac{x-\underset{¯}{x}}{{\delta }_x}$$E8

This shows that r_xy is the slope of the regression line of the standardized data points (and that this line passes through the origin).

Generalizing the $\underset{¯}{x}$ notation, a horizontal bar can be written over an expression to indicate the average value of that expression over the set of samples. For example:

$$\underset{¯}{\mathit{xy}}=\frac{1}{n}{\sum }_{i=1}^n{x}_i{y}_i$$E9

This notation allows a concise formula for r_xy:

$$r_{\mathit{xy}}=\frac{\underset{¯}{\mathit{xy}}-\underset{¯}{x}\underset{¯}{y}}{\sqrt{\left(\underset{¯}{x^2}-{\underset{¯}{x}}^2\right)\left(\underset{¯}{y^2}-{\underset{¯}{y}}^2\right)}}$$E10

The coefficient of determination (Pearson product-moment correlation coefficient, or “Pearson’s correlation coefficient,” commonly called simply “the correlation coefficient” – “R squared”) is equal to ${r^2}_{\mathit{xy}}$ when the model is linear with a single independent variable, and it is the most familiar measure of dependence between two quantities [45].

There is a corollary of the Cauchy-Schwarz inequality that the absolute value of the Pearson correlation coefficient is not bigger than 1. The correlation coefficient is +1 in the case of a perfect direct (increasing) linear relationship (correlation), −1 in the case of a perfect decreasing (inverse) linear relationship (anticorrelation), and some value in the open interval (−1,1) in all other cases, indicating the degree of linear dependence between the variables. As it approaches zero there is less of a relationship (closer to uncorrelated). The closer the coefficient is to either −1 or 1, the stronger the correlation between the variables, although there is a detailed table of the representativeness of the values, in module, as shown in Table 2 [46].

Value	Representativeness
0.9 and beyond	Very strong correlation
Between 0.7 and 0.9	Strong correlation
Between 0.3 and 0.7	Moderate correlation
Between 0 and 0.3	Weak correlation
0	Negligible correlation

Table 2.

Interpretation of the correlation coefficient values, in module.

The R² is, therefore, a descriptive measure of the goodness of fit got. It refers to how the amount of variability in the data that is explained by the fitted regression model. However, the value of the coefficient of determination depends on the number of observations (n), increasing when n decreases. If n = 2, always R² = 1.

R² should be used with caution, as it is always possible to make it larger by adding enough terms to the model. Thus, if, for example, there are no repeated data (over one y value for the same x a polynomial of degree (n − 1)), it will give a perfect fit (R² = 1) for n data. When there are repeated values, o will never be equal to 1, as the model cannot explain the variability, because of pure error.

Although R² increases with the addition of terms to the model, this does not mean that the new model is superior to the previous one. Unless the new model’s residual sum of squares is reduced to an amount equal to the original residual mean square, the new model will have a larger residual mean square than the original because of the loss of 1 degree of freedom. This new model could be worse than the previous one.

The magnitude of R² also depends on the amplitude of variation of the regressor variable (x). R² will increase with greater amplitude of variation of the x’s and decrease otherwise. It can be shown that

$$E\left[R^2\right]\cong \frac{{\widehat{\beta }}_1^2{\sum }_{i=1}^n{\left(x_i-\overline{x}\right)}^2}{{\widehat{\beta }}_1^2{\sum }_{i=1}^n{\left(x_i-\overline{x}\right)}^2+{\sigma }^2}$$E11

Thus, a big value of R² may be large simply because x has varied by a very large amplitude. R² may be small because the amplitude of the x's was too small to allow a relationship with y to be detected. In general, R² does not measure the magnitude of the slope of the line. A big value of R² does not mean a steeper line. It does not consider the model’s lack of fit; it may be large, even if y and x are nonlinearly related [47, 48]. Hence, although R² values were used as an adjustment basis in this work, which is widely accepted in the literature [34, 36, 40, 49], this should not be considered alone, but always combined with other diagnoses of the model.

Although the establishment of a single morphology versus position relationship for aluminum-based monotectic alloys directionally solidified is of great importance, it is known the thermal parameters have a considerable influence on the development of the solidification structures. Therefore, it is interesting that a relationship in function, e.g., of the growth rate could be established for these alloys. In addition, if the solidification processes were performed again and the growth rate values at each ingot position were different from those resulting from the original experiment, the interphase spacing values at each position would also be different compared to the results of the original experiment, which emphasizes the importance of obtaining the growth laws of microstructural parameters as a function of the solidification thermal parameters.

In this way, following the same methodology used for the analysis of the interphase spacings and droplets diameters with positions from metal/mold interface, the ordered pairs of the curves that relate the interphase spacing and the droplet diameters with the corresponding growth rates obtained in the studies were used [6, 7, 20, 28, 29, 34, 36, 40]. Again, the values of interphase spacing, and droplet diameters used in the present analysis were dimensionless.

With the dimensionless points of all the data collected and plotted, it was observed that the points of a single experiment were quite parallel to the points of the other ones, which led to the proposition of a function that translated and could represent the growth of each microstructure. Once again, a statistical analysis by simple linear regression was used since this is the most suitable tool for continuous input-output values and represents the situation in question [50].

Konno [42] studied the behavior of an Al-1.2wt.%Pb alloy solidified in a horizontal directional device. It is the only study performed in horizontal direction of these type of alloys, and Konno et al. [21], who analyzed the influence of the directionality of heat extraction on the morphology of the Al-1.2wt.% Pb alloy, found that horizontal solidification has sufficient convective movements to break the fiber formation and generate a morphology of only irregularly distributed particles; however, it has higher cooling rates than the solidification performed in a descending device, resulting in an interesting and differentiated behavior of the solidification process. Thus, a comparison of the growth law as a function of speed for this alloy was carried out separately, to ratify the accuracy of the proposed expressions.

3. Results and discussion

The graph in Figure 2 presents the values of interphase spacing and particle diameter (when applicable), respectively, as a function of position for all monotectic structures analyzed. Where different morphologies occurred in the same ingot, this was divided according to the type of morphology found (globular or fibrous). In these cases, the position adopted as a reference for nondimensionalization was the average position of each structure. As it can be seen, there was a collapse between all the results got for both cases, which means that a single law of growth can be established, λ = k.(P)^0,5. It is observed, besides the collapsed points, that the curves tend to a certain parallelism between them, a fact used to find the value of k that best represents each of the experimental situations got. The collapse between the points is best visualized in a log-log scale, which can reduce data dispersion and, therefore, present better quality in relation to the linear scale for this case.

Figure 2.

Globular and fibrous interphase spacings and particle diameter (in which particles occurred) as a function of position for all studied alloys.

Table 3 presents the condensed k values got for the best determination coefficients found for each case, separated by alloy. By observing the tables, it is noted that no coefficient of determination was in the range considered being of weak intensity (0–0.3), with most being between moderate intensity (0.3–0.7) and some between strong intensity (above 0.7), which is considered a satisfactory result considering the number of experiments analyzed. The lower values of determination coefficients were found for the downward experiments, which could be justified because of the intense convective flow that occurs on these cases. Even so, the results are acceptable both by visual analysis and by the values of the coefficient of determination.

Alloy	Type of solidification device	Structure, parameter	k	R^2
Al-3.2wt.%Bi	Upward and Downward	Droplet, interphase spacing, diameter	0.14–0.16	0.48–0.81
Al-5.0wt.%Bi	Upward	Droplet, interphase spacing, diameter	0.15–0.17	0.60–0.70
Al-7.0wt.%Bi	Upward	Droplet, interphase spacing, diameter	0.16	0.67–0.83
Al-0.9wt.%Pb	Upward and Downward	Droplet, fiber, interphase spacing, diameter	0.10–0.33	0.35–0.71
Al-1.2wt.%Pb	Upward and Downward	Droplet, fiber, interphase spacing, diameter	0.11–0.33	0.36–0.77
Al-2.1wt.%Pb	Upward-Silva et al. [34]	Fiber, interphase spacing	0.19	0.84
Al-2.5wt.%Pb	Upward-Freitas [33]	Fiber, interphase spacing	0.15	0.62
Al-5.5wt.%In	Upward-Silva et al. [27]	Droplet, interphase spacing, diameter	0.12–0.13	0.50–0.66

Table 3.

Values of the constant k got and respective correlation coefficients for the spacing/diameter versus position function found for the experiments carried out with the alloys of the Al-Bi, Al-Pb, and Al-In systems.

It was also noted that the k values, both for fibers and for particles, are between 0.1 and 0.19, except for lead particles, which all found k values beyond those mentioned above, which are even higher when considering the results got in experiments with upward configuration (k ranging from 0.32 to 0.33). In addition, it could be seen that the values of k, when observed in experiments with the occurrence of particles, are quite close in certain situations, which leads to the elaboration of Table 4, which shows the new values of k got for each situation, evidencing that for these cases a single function can be established to represent the evolution of the spacings and diameters of the particles as a function of the position from the metal/mold interface in each system, except for the Al-Pb alloy, where the upward solidification cases differ from downward.

Alloy	Type of solidification device	Function	R^2
Al-Bi	Upward and Downward	λ = 0.15.(P)^0.5	0.48–0.82
Al-Pb	Upward	λ = 0.33.(P)^0.5	0.54–0.67
	Downward	λ = 0.23.(P)^0.5	0.37–0.71
Al-In	Upward	λ = 0.13.(P)^0.5	0.33–0.65

Table 4.

Adjusted values of the function got and respective correlation coefficients for the spacing/diameter versus position function found for the experiments carried out with the alloys of the Al-Bi, Al-Pb, and Al-In systems.

Figure 3 highlights the behavior of the solidification processes of the ingots, highlighting the morphology got in each alloy system and emphasizing the behavior of the spacing and particle diameters that resulted in the proposed law.

Figure 3.

Schematic representation of the behavior of solidified alloys, which resulted in the proposed law.

The graph in Figure 4 shows the values of the interphase spacings and the droplet diameters, respectively, as a function of the growth rate, for all monotectic structures analyzed. A parallel behavior is observed between the resulting curves of the experiments. Thus, the growth law λ = k (v)^−2.1 could be established.

Figure 4.

Evolution of droplet diameters and interphase spacings of the droplet and fiber structures as a function of growth rate for all analyzed alloys.

Table 5 presents the condensed values of k obtained for the best coefficient of determination found in each case, separated by alloy and morphology. Observing the tables, it could be noted that no coefficient of determination was in the range considered of weak effect (under 0.3), being the majority in the range of moderate effect (0.3–0.7) and some in the strong effect range (above 0.7), which is considered a satisfactory result considering the number of experiments analyzed. Once again, the lower values of determination coefficients were found for the downward experiments, which could be justified because of the intense convective flow that occurs on these cases and for fibrous morphologies, which are more irregular.

Alloy	Type of solidification device	Structure, parameter	k	R^2
Al-3.2wt.%Bi	Upward and downward	Droplet, interphase spacing, diameter	0.07–1.89	0.30–0.78
Al-5.0wt.%Bi	Upward	Droplet, interphase spacing, diameter	0.92–1.10	0.74–0.75
Al-7.0wt.%Bi	Upward	Droplet, interphase spacing, diameter	3.10–3.20	0.56–0.76
Al-0.9wt.%Pb	Upward and downward	Droplet, fiber, interphase spacing, diameter	0.20–1.40	0.33–0.73
Al-1.2wt.%Pb	Upward and downward	Droplet, fiber, interphase spacing, diameter	0.02–1.60	0.34–0.85
Al-2.1wt.%Pb	Upward	Fiber, interphase spacing	1.80	0.33
Al-2.5wt.%Pb	Upward	Fiber, interphase spacing	0.54	0.30
Al-5.5wt.%In	Upward	Droplet, interphase spacing, diameter	0.72–0.80	0.56–0.63

Table 5.

Values of the constant k obtained and the respective coefficients of determination for the interphase spacing/droplet diameter as a function of the growth rate found for the experiments performed with the alloys of the Al-Bi, Al-Pb, and Al-In systems.

Considering that the results were more acceptable for the formulation of a growth law as a function of position for monotectic alloys and the position from the metal/mold interface of the ingots is directly related to the cooling rate of the alloys, a growth law as a function of the cooling rate can be quite adequate, a parameter that has not yet been directly considered in the formulations of monotectic growth laws. Even so, the results are acceptable both by visual analysis and by the values of the coefficient of determination.

When droplets occur, the values of k obtained for both spacing and diameter are remarkably close (except for the Al-3.2 wt.%Bi alloy solidified in the upward vertical device). This indicates that they can be represented by a single function.

Table 6 shows the new values of k obtained for each situation, including for the situation where more discrepant k values were obtained, showing that, when droplet structure occur, a single function can be established in each experiment to represent the correlation between spacing and diameter with growth rate. No other correlation between k values was observed.

Alloy	Type of solidification device	k	R^2 (λ)	R^2 (d)
Al-3.2wt.%Bi	Upward	0.65	0.39	0.39
Al-3.2wt.%Bi	Downward	0.07	0.34	0.30
Al-5.0wt.%Bi	Upward	1.01	0.72	0.74
Al-7.0wt.%Bi	Upward	3.15	0.75	0.55
Al-0.9wt.%Pb	Upward	1.40	0.57	0.73
Al-0.9wt.%Pb	Downward	0.40	0.65	0.49
Al-1.2wt.%Pb	Upward	1.45	0.46	0.72
Al-1.2wt.%Pb	Downward	0.04	0.57	0.46
Al-5.5wt.%In	Upward	0.76	0.52	0.47

Table 6.

Adjusted values of the constant k obtained and the respective coefficients of determination for the interphase spacing/droplet diameter as a function of the growth rate found for the experiments performed with Al-Bi, Al-Pb, and Al-In system alloys when the droplet morphology occurs.

As well as in the previous analysis, Figure 5 highlights the behavior of the solidification processes of the ingots, highlighting the morphology got in each alloy system and emphasizing the behavior of the spacing and particle diameters that resulted in the proposed law.

Figure 5.

Schematic representation of the behavior of solidified alloys, which resulted in the second proposed law.

In their results, [20] was able to correlate the results obtained for the evolution of the interphase spacing to the growth rate with the relation proposed by [23], λ = C. (v)^−0.5, only for occurrence of the droplet structure. The relationship proposed by [15], λ = C.G^a.v^b, was valid for cases of occurrence of fiber and droplet morphologies; however, that study is limited to cases of solidification in the upward vertical device.

In the other cases, different power functions were obtained λ = a.(V)^−b, with b reaching the following values: −2.2 for the occurrence of fibers in the upward vertical solidification [20]; −2 and −6.5 for the occurrence of droplets and fibers in the downward vertical solidification, respectively [29].

The results show that, contrary to what some authors have assumed [4, 14, 17, 24, 51] the growth of a monotectic alloy is not like eutectic, because although two phases grow side by side simultaneously, the thermocapillary effect acts in some cases, making the law proposed by [23] not valid. In their studies, [14] had already shown this fact when they developed their model for monotectic behavior, as well as [52], also in the development of their model.

Silva et al. [6] found power-type functions to evaluate the evolution of the particle diameter as a function of the solidification growth rate but with different exponent values as following: −0.4 for the solidification of Al-3.2% Bi alloy in a downward vertical device and −1.5 for the same alloy solidified in an upward vertical device. Silva et al. [7, 20, 36] also found power-type functions to express the particle diameter as a function of growth rate of many alloys from monotectic systems and a single exponent value, −1.5, was found for all alloys solidified in an upward vertical device.

According to the obtained results in this study, it is possible to establish a single law for growth for monotectic structures, which ratifies the growth mechanism proposed by [52]. These authors have developed a model based on the approximation of Hunt and Jackson for eutectics systems, which emphasize that as one of the phases concomitantly growing is a liquid, convection must have prominent role in total mass transport above the monotectic interface. Thus, the shape of the interface is particularly important for the flow field and the concentration field in front of the phase boundaries and, therefore, cannot assume a planar solidification front without surface convection.

The growth law proposed in this study (λ = k.(V) ^−2.1) was applied to the data obtained by [42]. The value found for k was 25. The coefficient of determination, R² = 0.94, was in the very strong intensity range (above 0.9), showing that the law proposed in this work can also be applied to the growth of binary monotectic alloys solidified in horizontal devices (Figure 6). The K value extremely higher than the ones found for other heat extraction configurations ratifies the evidence of a strong flow convection on this type of device.

Figure 6.

Evolution of interphase spacing of the droplet structure as a function of growth rate for Al −1.2 wt.% Pb alloy solidified under transient conditions in a horizontal directional device.

From the results of this analysis, it was observed that growth based on the solidification growth rate leads to a single growth law, λ = k. v^−2.1, and it is supposed that the analysis of the growth for monotectic systems can be extended for different situations, as for solidification experiments in steady state, for example.

4. Conclusions

Based on the results of the investigations carried out throughout this work, added to the comparisons made, and having as reference the other studies in the literature on the subject, the following major conclusions are derived from the present study:

1. When taking the values of spacing versus position and diameter versus position for the alloys studied, a single experimental law could be proposed, as λ = k.(P)^0.5, with no coefficient of determination found in the range considered as “weak intensity,” which is satisfactory, given the number of experiments analyzed.

2. When the values of λ and d versus v for the alloys studied were taken, a single experimental law could be established, in the form λ = k (v)^−2.1. Again, no coefficient of determination was found in the range considered as “weak effect,” which is considered satisfactory considering the number of experiments analyzed. For this case, it has been found that both spacings and droplet diameters can be represented by a single function.

3. The growth of a monotectic alloy is not analogous to the eutectic one, because although two phases grow side by side simultaneously, the thermocapillary effect acts in some cases, making the law proposed by Hunt and Jackson not valid for monotectic growth.

4. The proposed law was found to be applicable on monotectic horizontal transient solidification allowing to suppose that the function may be valid for the analysis of the growth for monotectic alloys for different situations, such as solidification experiments in steady state, for example.

5. Obtaining a single law of growth for all cases shows that the mechanism of formation and growth of structures in monotectic systems appears to be the set between the effects of Marangoni thermosolutal convection and coupled repulsive diffusion, regardless of morphology and/or of the solidification front.

Acknowledgments

The authors acknowledge the financial support provided by UFPA (Federal University of Pará), CAPES (Coordination of Superior Level Staff Improvement), and CNPq (The Brazilian Research Council), Brazil, grant 400634/2016-3.