1. Introduction

Surface tension is a phenomenon that occurs whenever a liquid is in contact with other liquids or even gases; then, an interface is established and it acts like a stretched elastic membrane. A surface is called wet if the contact angle is less than 90° and nonwet otherwise. A substrate, like dust or pollution, can contribute energetically to the membrane, decreasing its contact angle. Another feature is the magnitude of the surface tension <sup>σ</sup> (<sup>N</sup> ˜ <sup>m</sup>°1). An important effect is the creation of curved meniscus, leading to capillary rise or depression. Viscosity, on the other hand, is a measure of the certain fluid's resistance to flow due to its internal friction. A high viscosity fluid has a trend to resist its motion, such as engine oil, while a fluid with low viscosity flows easily, such as water. Viscosity is a function of fluid's shear stress and its velocity gradient, and its magnitude η is expressed in (Pa � s). Both viscosity and surface tension are thermodynamic properties of a fluid, and, consequently, can be derived by means of thermodynamic relations.

Many solutions can be found in the literature concerning the viscosity dependence on temperature for pure metals [1–7] and for alloys [8–15]. Budai et al. have reviewed the existing models used to predict dependence of viscosity on temperature of alloys, for cases where the viscosities of pure components are already known [10, 15–18] and those that are independent of experimental data [19]. Budai et al. extended the Kaptay unified equation [10] for the viscosity of alloys, which has been named BBK model [20]. The BKK model was shown to fail in the prediction of viscosity for alloy systems with components that melt congruently [21].

Solutions for surface tension as a function of temperature are generally based on: Butler formulation [22–25]; statistical thermodynamics surface density-functional theory [26–28]; semi-empirical thermodynamic model [29]; and thermodynamic models [30, 31]. All these models are normally specific for certain binary or ternary alloy systems, or they are general but considerably difficult to apply.

In 1992, Egry [32] derived a relation between surface tension and viscosity deduced from statistical mechanics for the melting temperature, based upon the expressions of Fowler [26] for surface tension and Born and Green [33] for viscosity. Both expressions are expressed as integrals over the product of interatomic forces and the pair distribution functions. The author extended this relation to a finite temperature range by using data available in the literature [34].

In this work, by using a straightforward solution for viscosity for molten pure metals [7] and alloys [10], a comparison between numerical simulations and experimental data for the surface tension and viscosity of pure liquid metals and liquid alloys is provided, in order to validate Egry's relation for pure molten metals [34]. An extension of this relation is derived for multicomponent alloys. The surface tension is calculated and plotted against temperature for ternary and quaternary aluminum alloys.

## 2. Modeling

The modeling section is divided into models dealing with the viscosity of pure metals and multicomponent alloys, and with the surface tension-viscosity relation equations for pure liquid metals and alloys.

#### 2.1 Model for the viscosity of pure liquid metals

Kaptay [7] derived a unified equation for the viscosity of pure liquid metals as a function of temperature, which encompasses the activation energy and the free volume concept. Based on the Andrade's equation [35] as a starting approach, the activation energy concept has been incorporated. By combining again with Andrade's formulation with free volume concept, an equation for the dependence of viscosity of pure metals on temperature has been derived. As this equation obeys both concepts, the authors named it as a unified equation for the viscosity of pure metals. The derived equation for the viscosity of pure metals as a function of temperature is given by,

$$\eta\_i = A \frac{\mathbf{M}\_i^{1/2} \cdot T^{1/2}}{V\_i^{2/3}} \cdot \exp\left(B \cdot \frac{T\_{m,i}}{T}\right) \tag{1}$$

where η<sup>i</sup> is the viscosity of the liquid metal i, A and B are temperatureindependent semi-empirical parameters being approximately identical for all liquid Dependence of Surface Tension and Viscosity on Temperature in Multicomponent Alloys DOI: http://dx.doi.org/10.5772/intechopen.82307

metals, Mi (kg � mol�<sup>1</sup> ) is the atomic weight of the metal, Vi is the molar volume (m<sup>3</sup> � mol�<sup>1</sup> ), Tm,i is the melting temperature of the pure liquid metal i (K), and T is the temperature above the melting point (K).

#### 2.2 Model for the viscosity of multicomponent alloys

An Arrhenius-type viscosity equation can be extended to deal with viscosity of multicomponent alloy, by applying Redlich-Kister polynomial to excess viscosity,

$$
\Delta \boldsymbol{\eta}^{\rm E} = \sum\_{\rm i} \sum\_{\rm j>i} \mathbf{x}\_{\rm i} \cdot \mathbf{x}\_{\rm j} \sum\_{\rm k=0}^{\rm m} \mathbf{A}\_{\rm i,j}^{\rm k} \left(\mathbf{x}\_{\rm i} - \mathbf{x}\_{\rm j}\right)^{\rm k} \tag{2}
$$

where xi and xj are the molar contents of the solute compounds ″ i ″ and ″ j, ″ respectively. Ak i,j are the polynomial parameters related to a binary ″ i j″ system. The ideal viscosity term can be expressed as

$$\eta^{ideal}\left(T,\mathbf{x}\_{i},\mathbf{x}\_{j}\right) = \sum\_{i=1}^{3} \mathbf{x}\_{i} \cdot \eta\_{0}^{i} \exp\left(\frac{E\_{A}^{i}}{\overline{R} \cdot T}\right) \tag{3}$$

where η<sup>0</sup> is the pre-exponential factor independent of the temperature, interpreted as an asymptotic viscosity at very high temperature, E<sup>i</sup> <sup>A</sup> is the activation energy of viscous flow of the component i. By combining Eqs. (2) and (3), we obtain,

$$\eta\left(T,\varkappa\_{i},\varkappa\_{\hat{\jmath}}\right) = \eta^{ideal}\left(T,\varkappa\_{i},\varkappa\_{\hat{\jmath}}\right) + \Delta\eta^{E} \tag{4}$$

then, we have,

$$\eta\left(T, \mathbf{x}\_i, \mathbf{x}\_j, \dots, \mathbf{x}\_n\right) = \sum\_{i=1}^3 \mathbf{x}\_i \cdot \eta\_0^i \exp\left(\frac{E\_A^i}{\overline{R} \cdot T}\right) + \sum\_i \sum\_{j>i} \mathbf{x}\_i \cdot \mathbf{x}\_j \sum\_{k=0}^m A\_{i,j}^k \left(\mathbf{x}\_i - \mathbf{x}\_j\right)^k \tag{5}$$

where A<sup>k</sup> i,j are parameters for the viscosity of binary systems, according to Zhang et al. [36].

Kaptay [10], based on the Seetharaman-Du Sichen equation, regarding the theoretical relationship between the cohesion energy of the alloy and the activation of viscous flow, proposed the following equation:

$$\eta\left(T, \mathbf{x}\_i, \mathbf{x}\_j, \dots, \mathbf{x}\_n\right) = \frac{h \cdot N\_{Av}}{\sum\_i \mathbf{x}\_i \cdot \mathbf{V}\_i + \Delta \mathbf{V}^E} \cdot \exp\left[\frac{\sum\_i \mathbf{x}\_i \cdot \Delta G\_i^\* - a \cdot \Delta H}{\overline{R} \cdot T}\right] \tag{6}$$

where <sup>h</sup> is the Planck constant (6:<sup>626</sup> � <sup>10</sup>�<sup>34</sup> <sup>J</sup>:s), NAv is the Avogadro number (mol�<sup>1</sup> ), <sup>Δ</sup>G<sup>∗</sup> is the Gibbs energy of activation of the viscous flow (<sup>J</sup> � mol�<sup>1</sup> ), <sup>i</sup> defined as

$$
\Delta G\_i^\* = \overline{R} \cdot T \cdot \ln \left[ \frac{\eta\_i \cdot M\_i}{h \cdot N\_{Av} \cdot \rho\_i} \right] \tag{7}
$$

<sup>Δ</sup>V<sup>E</sup> is the excess molar volume on alloy formation (m<sup>3</sup> � mol�<sup>1</sup> ) and α is a ratio of a two properties ratios related to the melting temperature. The first ratio is between the measured activation energy and the melting point of pure liquid metals (38:<sup>4</sup> � <sup>2</sup>:<sup>7</sup> <sup>J</sup> � mol�<sup>1</sup> � <sup>K</sup>�<sup>1</sup> ), and the second ratio is the cohesion energy of pure liquid metals and the melting points (248 � <sup>17</sup> <sup>J</sup> � mol�<sup>1</sup> � <sup>K</sup>�<sup>1</sup> ), providing α ffi 0:155 � 0:015 [10].

#### 2.3 Relation between surface tension and viscosity for pure liquid metals

Egry [32] derived a relation between surface tension and viscosity deduced from statistical mechanics for a finite temperature range, based upon the expressions of Fowler [26] for surface tension (σÞ and Born and Green [33] for viscosity. Both expressions are expressed as integrals over the product of interatomic forces and the pair distribution functions. The Fowler expression is

$$
\sigma = \frac{\pi}{8} \frac{n^2}{8} \int\_0^\infty dR \cdot R^4 \, \frac{d\rho(R)}{dR} \cdot \mathbf{g}(R) \tag{8}
$$

where n is the particle number density, φ is the pair potential, and g Rð Þ is the correlation function. All these functions depend on the temperature.

In a very similar way, Born and Green [33] derived an expression for the viscosity (ηÞ of a fluid using a kinetic theory, which can be expressed as

$$\eta = \sqrt{\frac{m}{k \cdot T}} \cdot \frac{2\pi}{15} \int\_0^\infty dR \cdot R^4 \left. \frac{d\rho(R)}{dR} \cdot \mathbf{g}(R) \tag{9}$$

where m is the atomic mass, k is the Boltzmann's constant, and T is the absolute temperature. As the integral terms of both integrals cancel each other, Egry [34] deduced the following relationship between density and viscosity for pure metals for a finite temperature above the melting point:

$$\frac{\sigma\_i}{\eta\_i} = \frac{15}{16} \sqrt{\frac{k \ T}{m}} \tag{10}$$

where <sup>1</sup> <sup>σ</sup><sup>i</sup> is the surface tension in <sup>ð</sup>N:m� <sup>Þ</sup>, and <sup>η</sup><sup>i</sup> is the viscosity in <sup>ð</sup>Pa:s<sup>Þ</sup> of <sup>a</sup> pure liquid metal i.

In 2005, Kaptay [7] derived a surface tension-viscosity relation, as

$$\frac{\sigma\_i}{\eta\_i} = \frac{0.182 \cdot \left(211 + C\_{p,i}\right) \cdot T\_{m,i} - \left(2 + 0.182 \cdot C\_{p,i}\right) \cdot T}{1.61 \cdot M\_i^{1/2} \cdot \exp\left(2.34 \cdot T\_{m,i}/T\right)}\tag{11}$$

� <sup>1</sup> K�<sup>1</sup> � where Cp,i is the heat capacity of the pure liquid metal i in J:mol� .
