Development of a Partial Clustering Model of Alloy Viscosity

Abstract

The purpose of this paper is to obtain a partial clustering model of viscosity including the influence of clusters. This paper also establishes a quantitative correlation between the dynamic viscosity of alloys and temperature of liquidus in isotherms. The research methods are a theoretical substantiation of possibility of the isolated use of the Boltzmann distribution (energy spectrum) for the kinetic energy of the chaotic (thermal) motion and particle collisions as applied to a condensed state of matter. In this paper, the author’s concept of chaotic particles is applied to substantiate the existence of an energy class of particles present in the liquid in the form of clusters. The novelty of the paper is that it obtains a quantitative physical and mathematical model of temperature dependences of the dynamic viscosity based on destruction of clusters as the temperature increases. The mathematical model is compared with viscosity data from the state diagram, starting from the liquidus barrier. This approach was developed first and allows constructing viscosity isotherms based on the thermochemical initial data with extrapolation to the region of ultra-high temperatures. The proposed new model is verified in an example of a Cu-Sn alloy. The high correlation coefficient indicates the correctness of the derived equations and possibility of predicting the distribution of the viscosity of the alloy at high temperatures based on its state diagram. But the main fundamental novelty of the work is the discovery of the relationship between the activation energy of viscous flow and the barrier of randomization, which is present in the partial clustering model. The application of the new partial clustering viscosity model can be utilized across various fields involving fluid dynamics. In our study, the practical implementation of this novel partial clustering viscosity model will ensure the effective execution of metallurgical processes designed using these values at extremely high temperatures, determine optimal operating conditions, and provide more substantiated requirements for metal and alloy production technologies.

1. Introduction

The research relevance of various properties of melts was determined by their application in modern equipment and technology and by the scientific significance of the obtained results.

One of the significant physical and kinetic characteristics of melts is viscosity. It is used to analyze various physical and chemical processes in liquid melts. Viscosity is a characteristic sensitive to changes in the structure [1,2,3]. The correlation between the viscosity of the complex alloys and state diagrams, in which all state transitions are reflected, is an important issue in the theory of the liquid state [4,5,6].

It is known that the viscosity of a liquid melt depends on its temperature. The melt superheating leads to significant changes in its properties and structure [7,8,9]. The process itself and the exact mechanism of superheating have not been completely understood yet. The existing explanations have discrepancies [10], although the superheating method of metals and alloys itself can improve their mechanical properties [11,12].

Initial concepts of viscosity were based on the assumption that interlayer frictional forces acted to resist the motion of liquids. A more comprehensive perspective on this property emerged with the recognition of its manifestation even in the stationary state, which directly followed from the Boltzmann distribution for an ideal gas, based on the statistical justification of the energy spectrum of distinguishable particles dispersed throughout the volume. These particles, detectable through beam diagnostic methods, exhibit a discernible variety of energy levels, which can be defined with any arbitrary step of variation. This marked the beginning of research aimed at establishing correlations between the distribution of energy levels (spectrum) according to the Boltzmann model and various physicochemical properties (more than 60 known properties, including density, electrical conductivity, vaporization, etc.), with viscosity being one of the primary properties investigated.

The existing attempts to quantitatively express the temperature dependence of the viscosity of an alloy are based on the concept of its “superheating” for a composition with a lower liquidus temperature to the same temperature above liquidus. In fact, the viscosity of such alloy superheating turns out to be lower than for a composition with a higher liquidus temperature. The fundamental association must be between viscosity, liquidus temperature and higher temperature. It must be within equilibrium thermodynamics which describes all areas of the state diagram. The “superheating” concept means a deviation from equilibrium. The fact of its use in search for a proportional correlation between the equilibrium liquidus temperature and viscosity indicates the absence of a sufficiently complete theory of the liquid state [13,14,15].

More rigorous expression and connection with the Boltzmann distribution have been found using the well-known Frenkel–Andrade viscosity model $\left(\mathsf{\eta }={\mathsf{\eta }}_{0}exp\left({\displaystyle \frac{E_a}{RT}}\right)\right)$ [16]. This transformed equation can be used as an adequate concept of fluidity rather than viscosity. By these criteria, the cautious concept “relative superheating” of the melt is still not scientific. It turns out that the memory of the liquidus temperature persists during further heating of the thermodynamic system to the general temperature T. This is expressed by a single equation of the average integral value of the thermal energy RT of each alloy composition.

It is another matter when it comes to development of any theory with establishment of “zero” approximations, principles, and ideal states. In this case, simplifications, ease of understanding, and analytical expressions will be absolutely necessary and should be rigorously justified with transitions to a more accurate level of application.

Therefor, the concept of “superheating” will take its position as a private solution in the principal theory [17]. This approach was successfully applied by us earlier on the existing problems of thermodynamics, and on the entropy-information mathematical theory of evolution of self-organizing hierarchical systems.

A similar transition for the fluid theory can take place if it leads to a principal theory of all aggregating states of matter, and “superheating” turns out to be a fragment of a more general pattern. It is not excluded that in this state of the fluid theory, it plays the positive role of the catalyst and even the activation energy of the above transition due to its redundant nature.

2. Materials and Methods

2.1. Concept of Chaotic Particles

In summary, the concept of chaotic particles can be presented as follows.

Our recently proposed (2004 [18]) concept has been published in articles and monographs, and has become the basis for modernization of the large-scale metallurgical production of copper wire rods based on a new view of plastic deformation.

Leontovich M.A.’s monograph [19] is devoted to the introduction of thermodynamics and statistical physics, and describes that the Boltzmann distribution of the kinetic energy of an ideal gas can be applied to the condensed state of matter. At first, this description caused bewilderment because of the too bold extrapolation in the academician’s scientific work. Then, we understood that it is kinetic energy of chaotic motion of colliding particles, in fact, a thermal energy with a common measure of temperature T, K, and a value of this energy RT (or kT) for any aggregate state of matter.

The Boltzmann distribution [20] showed that the energy spectrum of this distribution covers the kinetic energy levels from zero to infinity, including the obligatory equilibrium self-adjustment in subordination to equality:

$$P_i={\displaystyle \frac{N_i}{N}}=e^{-{\displaystyle \frac{{\epsilon }_i}{kT}}}/{{\displaystyle \sum }}_{i=1}^{\infty }{e}^{-{\displaystyle \frac{{\epsilon }_i}{kT}}},$$

(1)

where P_i—share and number of N_i particles with energy ε_i = iΔε (Δε—energy variation step); N—total number of particles; ${\displaystyle \sum }_{i=1}^{\infty }{e}^{-{\displaystyle \frac{{\epsilon }_i}{kT}}}$—statistical sum. The ratio N_i/N allows for interpreting it as the probability of existence and detection of particles with ε_i energy level.

It should be stated that the decoding of the energy level ε_i by a step of its variation Δε differs from all known distribution functions in its focus on its own internal structure. It turns out that it is able to manifest itself as a cause of real impacts on the thermodynamic system.

A very useful property of this distribution is an ability to distinguish in it practically any set of energy levels and express it as a common share.

For instance, the fraction of particles with a higher energy can be found by setting an energy barrier ε_a = aΔε using the following formula:

$$P_{>a}={\displaystyle \frac{{{\displaystyle \sum }}_{i=a}^{\infty }{e}^{-\frac{i\Delta \epsilon }{kT}}}{{{\displaystyle \sum }}_{i=1}^{\infty }{e}^{-\frac{i\Delta \epsilon }{kT}}}}={\displaystyle \frac{{{\displaystyle \int }}_{{\epsilon }_a}^{\infty }{e}^{-\frac{\epsilon }{kT}}d\epsilon }{{{\displaystyle \int }}_0^{\infty }{e}^{-\frac{\epsilon }{kT}}d\epsilon }}={\displaystyle \frac{kT{e}^{-\frac{{\epsilon }_a}{kT}}}{kT}}=e^{-{\displaystyle \frac{{\epsilon }_a}{kT}}}=e^{-{\displaystyle \frac{E_a}{RT}}}.$$

(2)

In this case, the opposite energy package will be equal to $P_{<a}=1-P_{>a}=1-e^{-{\displaystyle \frac{E_a}{RT}}}$.

Our proposed concept describes the separation of three energy classes of the chaotic particles. These particles are always present in three aggregating states at any temperature, with a constant mutual exchange of a thermal energy and with the corresponding fractional distribution [20], and therefore named as the crystal-mobile particles (crm) with their fractional distribution according to the non-exceeding of the RT_m barrier:

P_crm = 1 − exp[−RT_m/(RT)] = 1 − exp(−T_m/T),

(3)

named as the liquid-mobile particles (lqm) with their fraction in the interval between RT_m and RT_b

P_lqm = exp(−T_m/T) − exp(−T_b/T),

(4)

named as the vapor-mobile particles (vm) exceeding the RT_b barrier:

P_vm = exp(−T_b/T),

(5)

and collectively covering the entire Boltzmann energy spectrum

P_crm + P_lqm + P_vm = 1.

(6)

This paper will be limited to the aspect of analyzing the proposed concept on the correlation between the liquid and solid states from the point of view of the presence of crystal-mobile particles in the liquid. These particles extend into a gaseous state of matter, and also the liquid- and vapor-mobile particles into a solid state. The crystal-mobile particles being the least energy-consuming in the liquid are present in some aggregating state without separation into their own phase, i.e., they are unstable and thus claiming to be clusters.

Therefore the absolute invariant has been established for the crystal-moving particles, i.e., for any substances having melting temperature: at T = T_m according to (3) the fraction of P_crm particles is equal to it was possible to establish almost immediately an absolute invariant for crystal- mobile particles, i.e., for any substances having a melting point: at T = T_m according to (3) the fraction of P_crm particles is equal to

$$P_{crm,m}=1-\left(-T_m/T_m\right)=1-e^{-1}\approx 0.63.$$

(7)

This result clearly claimed to be of fundamental significance. It referred to a condition of loss by a solid state of its stability. It was almost no different from a proportion of the golden section for the critical share of many natural and artificial objects [21,22,23] in its structural component

$$P_{gs,crm}={\displaystyle \frac{\sqrt{5}-1}{2}}=0.618\dots \approx 0.62.$$

Invariant (7) follows directly from the Boltzmann distribution (1). It contains, in its mathematical structure, the possibility of aggregating energy levels (2) and classifying them as the natural barriers to the stability of the solid and liquid states of matter, T_m and T_b (3)–(5), in their inseparable unity (6). The interpretation is important as it characterizes the Bose–Einstein temperature as loss of stability of a solid state at superiority of average amplitude of vibrations of atoms in nodes of a crystal lattice over average distances between them. All this predetermined our attempts to focus on using the fundamental result P_crm,m = 1 − e⁻¹ to quantitatively describe the emergence and instability of clusters in the fluid composition. It was suggested that the instability of clusters originates from their adaptive nature in the form of aggregations of an arbitrary shape and content of particles (atoms, molecules) with the equal probabilities of mutual transitions of (n − 1)- and n-particle aggregations.

In mathematical terms, a series that provides the maximum variety and diversity of its members with integrity, i.e., functional convergence and sum expression, is suitable for this purpose. Such series can be formed from a known distribution of a function of the form $\left|{\displaystyle \frac{1}{1-x}}\right|$ if the condition $\left|x\right|<1$ is satisfied. This condition is satisfied, i.e., P_crm < 1. Then, it applies to any temperature:

$$\left|{\displaystyle \frac{1}{1-x}}\right|=1+x+x^2+\dots +x^n\dots ⟹{\displaystyle \frac{1}{1-P_{crm}}}=1+P_{crm}+P_{crm}^2+\dots P_{crm}^n+\dots .$$

(8)

Transferring the unit of the series to the left, processing the fractional expression and transferring the denominator to the right lead to identical transformations to the required solution:

$$P_{crm}=\left(1-P_{crm}\right)\left(P_{crm}+P_{crm}^2+P_{crm}^3+\dots +P_{crm}^n+\dots \right).$$

(9)

From here, we obtain the distribution of aggregation by the number n of particles (atoms, molecules) included in them:

$$P_{crm,n}=\left(1-P_{crm}\right)P_{crm}^n$$

(10)

and, as follows from Equations of (10) and (11),

$${{\displaystyle \sum }}_{n=1}^{\infty }{P}_{crm,n}=P_{crm}.$$

(11)

This series formally includes single constituents P_crm,1 which cannot be considered as clusters. They belong to the energy class of crystal-mobile particles. They immediately unite with matrix single liquid-mobile and vapor-mobile particles, leaving the fraction of aggregating particles of the thermodynamic system as clusters by difference

$$P_{cl}=P_{crm}-P_{crm,1}=P_{crm}-\left(1-P_{crm}\right)P_{crm}=P_{crm}^2={\left(1-e^{-1}\right)}^2\approx 0.400.$$

(12)

That is, they also belong to the proportion of the golden ratio. They repeat the regular sequential changes of the golden proportion when it is squared with reduction to a fraction. It is equal to the adaptive part in the original proportion, which is transferred into the structural part, but is already in a new version of the golden ratio [22,23].

Thus, the structure of the liquid state within the chaotic particles is determined by the coexistence of clusters with single crystal-mobile, liquid-mobile and vapor-mobile particles. They occupy the entire energy spectrum of the Boltzmann distribution: the “coldest” part—clusters, the “cool” part—single crystal-mobile, the “warm” part—single liquid-mobile, and the “hot” part—single vapor-mobile.

2.2. Viscosity of Alloy and Its Partial Clustering Model

The previously proposed and verified partial clustering model of viscosity of alloys [18] is based on the Boltzmann distribution (energy spectrum) and the concept of chaotic particles. It only includes the kinetic energy of the thermal motion in a thermodynamic system at the rest state. In this case, the viscosity is determined by a material substrate, clusters, as the lowest-energy unstable aggregations occurring during the melting from a solid phase. They are present in the liquid as random aggregations that transform into a crystal lattice on cooling. It ensures a harmonious combination of the solid and liquid states in subordination to the system-wide criteria of stability.

The temperature dependence of the viscosity of a liquid alloy of pure components of A and B is called a partial clustering model because it obeys Raoult’s law. Thus, it is an equilibrium characteristic and depends on a molar fraction of components. A similar function is the liquidus line which separates the solid and liquid states of the alloy at the liquidus temperature T_liq. This line, being an equilibrium line on state diagrams, displays the equilibrium deviation from the ideal linear distribution. This deviation is expressed as a line of complex configuration, containing extreme values and kinks. Such line is observed as the mole fraction of any of the components continuously changes. The analytical form of this dependence is unknown. However, certain regularities have been stated for viscosity isotherms [24] constructed on the same fractional distribution of components of A and B; namely, as the melt temperature increases over the same interval, the viscosity decreases, the isotherms smooth out, and they converge and become less similar to the liquidus line. Our research [18] has described this before using the example of a model dependence of viscosity for pure substances of A and B for discussion. The transition was realized from mapping models to experimental conditions. The fundamental Boltzmann distribution was used by us as the basis of responsibility for the deviation of viscosity from the ideal viscosity according to Raoul’s law to express the correlation of these deviations to the cluster content and to represent the quantitative form of viscosity η_T = f(T_liq).

The partial clustering model is consistent with the concept of viscosity as internal friction of liquid layers, in which clusters of different sizes create additional roughness of the friction surface. This model does not contradict the “hole” theory of viscosity due to the exchange presence and participation of vapor-mobile particles that transmit their impulses to formation of holes. The best similarity was found with the quasi-polycrystalline fluid theory [25], in which two forms of exchange appear as aggregations and single particles.

The connection of the partial clustering model of viscosity with the state diagram of alloys is based on representation of the liquidus line as a thermal barrier RT_liq similar to the thermal barrier RT_m during the melting of pure substances. Other thermal barriers associated with the formation and destruction of unstable intermetallic A_xB_y aggregations are also taken into account. These aggregations are analogous to clusters for pure substances. An important role is played by consequences of mixing the alloy components, expressed as the total thermal barrier of RT_ch chaotization. This barrier is opposed to the thermal barrier RT when heating the melt to a temperature T.

The fraction of the intermetallic clusters is expressed through the proportion of crystal-mobile particles, also the same for pure components, based on the same Boltzmann distribution, chaotic particles and cluster size distribution. The combined effect of clusters on viscosity of the alloy is determined by Raoult’s law in the form of partial contributions from the pure components. Therefore, the calculation of viscosity of the alloy can be subordinated to a proportion. If the fraction of clusters at the alloy temperature T is greater or less than the fraction of clusters for the pure components at the same temperature, then the viscosity of the alloy of this composition will be correspondingly greater or less than the known viscosity for the pure components. And this applies in all cases as $P_{cl}=P_{crm}^2$ (12).

The main idea in constructing a new model of viscosity is to look for its connection with clusters, not with temperature as a determining argument, not with the mechanics of motion and rest, not with the kinetics of viscous flow! The connection with clusters in the form of a certain material substrate is subject to the rigorous energy balance of its invisible existence. The quantitative model of viscosity of the A-B alloy is presented by a combination of the fractional content of clusters in the alloy and pure components of the alloy including the subordination of clusters to their probability distribution (10) as part of the concept of chaotic particles (3–6) based on the Boltzmann distribution (energy spectrum) (1):

$$\begin{gathered} {\mathsf{\eta }}_T=X_A{\stackrel{.}{\mathsf{\eta }}}_{T,A}{\left[{\displaystyle \frac{1-exp\left(-{\Delta }_{ch}H/\left(RT\right)\right)}{1-exp\left(-T_{m,A}/T\right)}}\right]}^2 \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+X_B{\stackrel{.}{\mathsf{\eta }}}_{T,B}{\left[{\displaystyle \frac{1-exp\left(-{\Delta }_{ch}H/\left(RT\right)\right)}{1-exp\left(-T_{m,B}/T\right)}}\right]}^2 \end{gathered}$$

(13)

where X_A and X_B are the partial shares of substances A and B; ${\stackrel{.}{\mathsf{\eta }}}_{T,A}$ and ${\stackrel{.}{\mathsf{\eta }}}_{T,B}$ are the viscosity values of pure substances A and B, mPa·s, at temperature T ≥ T_liq, K; Δ_chH = RT_liq + Δ_mixH + Δ_fH + … is a total thermal barrier to destruction (chaotization) of clusters when the alloy is heated in opposition to its increasing thermal content RT, J∙mol⁻¹.

In this model, alongside temperature, the heat of chaos emerges as a crucial factor, which includes the temperature and heat of the liquidus, heat of mixing, and other chaos barriers. The sum of these components is denoted as the generalized heat of chaos across the entire composition range of the alloy (0–100%, w, Sn). Moreover, a correlation was identified between the activation energy of viscous flow and the heat of chaos of the melt during heating.

To use the model (13), it is necessary to know the liquidus temperature in the full range of the A-B alloy state diagram, preferably at the melting points of the formed compounds; to have experimental or approximated (according to Arrhenius or the cluster-associate model) temperature dependences of the viscosity of the pure components; to use the values of the heat of mixing Δ_mixH and the formation Δ_fH of A_xB_y intermetallic compounds as a function of temperature to account for the heat of chaotization Δ_chH.

Literature analysis has shown the need to take into account various factors affecting the viscosity of melts [26,27,28,29,30,31]. So, the authors in [26,27,28] do not take into account the presence of associates in the melt. Or the proposed models are developed only for a specific alloy system without possibility of further prediction [29]. The authors in [30] proposed a model to estimate the viscosity related to associates. But data on the thermodynamic characteristics of the alloys, the molar volume of the alloys and the activation energy of the associates are required to use this model. Our study aims to fill this gap.

This paper aims to fill this gap in the theory of the liquid state and to develop a new model based on a fundamentally new approach including the energy component of matter.

Previously, testing of the partial clustering model of viscosity on known data on viscosity of Cu-Al and Cu-Sn alloys showed the reliability of this model and its high demands on the volume and accuracy of the initial thermal and physical parameters, but the possibility of testing significantly reduces.

To test our model, we used Cu-Sn alloys [31]. Their composition varied with increasing tin content. The compositions are shown below.

N	Cu-w, % Sn
	1	2	3	4	5	6	7	8	9
Cu-Sn alloy	0	10	25	30.6	38	58.6	92.4	99.3	100

3. Results

Verification of the proposed partial clustering model will be shown to calculate the dynamic viscosity of alloys on the experimental data of Cu-Sn alloy described in [31]. The authors of [31] presented the results of an experiment to determine the dynamic viscosity in the vicinity of the liquidus line for the Cu-Sn alloy on a high-frequency viscometer. A close correlation of the obtained equations for nine selected alloy samples was provided.

It was of interest that the purpose of this experiment in [31] was to check a interpretation of superheating of the alloy above the liquidus line by the contrary principle. The repeated uniform heating of the alloy was set for the selected nine samples over the entire liquidus line every 20 degrees, starting from the line itself. As a result, viscosity polytherms close to the liquidus line were obtained with convergence and leveling at the increasing temperature and corresponding decrease in viscosity. However, the gradient of viscosity increase over the same temperature intervals turned out to be significantly different, although the superheating was equal. Therefore, the influence of such superheating is apparent. It has no relation to the equilibrium state of the thermodynamic system, which is independent of the path and the initial temperature of the alloy. It can be repeatedly heated and cooled and remain as the laws of thermodynamics dictate at a constant given temperature.

Table 1. Parameters of the Arrhenius equation of the Cu-Sn alloy. N—sample number of the alloy. T_liq according to our measurements from the diagram [31].

N	Cu-Sn Alloy, w, %	A, mPa·s	Eν, J/mole	Tliq, K	ΔmixH, J/mole	ΔchH = RTliq + ΔmixH
1	Cu	1.769	10,833.7	1356	0	11,274
2	Cu-10% Sn	1.219	14,116.8	1273	2300	12,884
3	Cu-25% Sn	0.874	17,080.8	1104	3900	13,079
4	Cu-30.6% Sn	1.187	13,373.8	1019	3950	12,422
5	Cu-38% Sn	0.988	14,151.8	1004	3200	11,548
6	Cu-58.6% Sn	1.436	6830.2	1004	900	9248
7	Cu-92.4% Sn	1.036	3953.8	684	30	5717
8	Cu-99.3% Sn	0.979	2963.0	490	10	4084
9	Sn	0.841	3488.8	505	0	4199

presents the results of these experiments using the Arrhenius equation. The equation includes a pre-exponential factor A (mPa·s), activation energy of viscous flow E_ν (J/mol) and heat of mixing Δ_mixH (J/mol) [32]. Also, our calculations of viscosity polytherms using the Arrhenius equation for each intermediate numbered sample are presented. The liquidus temperature T_liq, K obtained from our measurements on the state diagram of the Cu-Sn alloy in [31] is pointed out.

It was found that, from the seemingly disparate data [31], a regular dependence of the activation energy of viscous flow on the alloy composition emerged. Remarkably, the same regularity was observed in the relationship between the total heat of chaos and the same alloy compositions of Cu-Sn. Moreover, it was found that the values of these regularities were strikingly similar in absolute magnitude. Based on this, Figure 1 is constructed.

Figure 1 shows the dependence of the activation energy of the viscous flow on the composition of the Cu-Sn alloy. In general, its shape repeats the shape of the viscosity change curve at the liquidus temperature, but it is much more closely correlated with the total barrier of randomization in the model (13). It should also be noted that when T = T_m = T_liq and RT_liq = RT_m.

The results of calculating the data using model (13) are shown in Figure 2 and

Table 2. Viscosity of Cu-Sn alloy according to model (13), top row; by Arrhenius equation for the experimental data [31], bottom row. The interval of viscosity measurements for each sample is presented (according to Figure 3 in [31]). N—sample number.

N	${\mathit{\eta }}_{{\mathit{T}}_{\mathit{l}\mathit{i}\mathit{q}}}$, mPa·s	ηT, mPa·s, at T, K											Measurement Interval η, mPa·s
		800	900	1000	1100	1200	1300	1400	1500	1600	1700	1800
1	4.624	–	–	–	–	–	–	4.489 4.449	4.129 4.093	3.776 3.994	3.492 3.807	3.237 3.648	4.0–4.7
2	4.626	–	–	–	–	–	4.507 4.500	4.123 4.099	3.782 3.781	3.480 3.523	3.212 3.309	2.971 3.131	3.8–4.7
3	5.619	–	–	–	–	5.081 4.842	4.599 4.244	4.180 3.791	3.812 3.438	3.489 3.156	3.204 2.926	2.952 2.736	4.5–6.3
4	5.754	–	–	–	5.245 5.123	4.708 4.585	4.245 4.091	3.844 3.743	3.485 3.469	3.190 3.244	2.923 3.058	2.687 2.901	4.3–5.8
5	5.383	–	–	–	5.036 4.831	4.516 4.232	4.069 3.784	3.682 3.438	3.346 3.164	3.053 2.942	2.796 2.759	2.569 2.606	4.1–5.8
6	3.255	–	–	–	2.916 3.050	2.615 2.865	2.356 2.716	2.132 2.586	1.938 2.495	1768 2410	1.619 2.358	1.487 2.276	3.0–3.7
7	2.076	1.803 1.968	1.548 1.842	1.341 1.743	1.171 1.673	1.031 1.614	0.914 1.566	0.816 1.525	0.733 1.491	0.661 1.468	0.600 1.496	0.546 1.414	1.8–2.1
8	2.026	1.059 1.528	0.890 1.485	0.758 1.398	0.652 1.354	0.568 1.318	0.498 1.288	0.440 1.263	0.392 1.242	0.351 1.223	0.307 1.207	0.287 1.193	1.65–2.0
9	4.624	1.057 1.421	0.890 1.340	0.798 1.279	0.654 1.232	0.569 1.193	0.500 1.161	0.442 1.135	0.394 1.112	0.354 1.093	0.319 1.076	0.289 1.062	1.5–1.9

.

Analysis of the data shows that the viscosity of the alloy decreases from copper to tin and from low to high temperatures, in full conformity with the expected influence of the Arrhenius equation and model (13). But the differences between the calculated and experimental (by the Arrhenius equation) viscosity values turn into their incomparability in the absolute values. It indicates a significant impact of some unaccounted factor.

The factor in question is the relationship between the activation energy of viscous flow and the total chaos factor of melts, expressed as E_ν ≈ Δ_chH.

The evident correlation between E_a and Δ_chH in this case should be reinforced by statistical quality and quantity criteria. These criteria are as follows:

-

A systematic deviation of one dependence from the other, with no prolonged overestimations or underestimations;

-

The closest alignment of data points at the boundaries of the comparable dependencies;

-

The largest divergence of points at corresponding extrema;

-

The absence of unpaired extrema;

-

Selection of “reference” dependencies based on their closest proximity to experimental definitions (in this case, thermal definitions are measured directly, while activation energy is determined only through a calculated linearization from the inverse absolute temperature);

-

Calculation of the nonlinear multiple correlation coefficient yields R = 0.729, with statistical significance according to Student’s t-test, tR = 6.05 > 2, for three factors (T, w, T_liq) with t being a probability of 0.9;

-

Full coverage of the Cu-Sn composition range (from 0 to 100% w Sn).

All these correlation indicators are strictly adhered to, and thus, they manifest regularly and systematically, particularly in the study of the activation energy of viscosity in alloys with energy expenditure measurements.

The left part of Table 2 is separated by the zone of results related to the temperature below the liquidus line. The right part of Table 2 presents the results of calculations in the extrapolation region of the Arrhenius equation. In the intervals of the experiment conducted and the direct action of this equation, viscosities were recorded. They were distinguished by a high adequacy of the partial clustering equation (13). The statistical processing of data showed that a correlation coefficient R = 0.9804, and its significance t_R = 84 >> 2, R₂ = 0.9613. It can be concluded that the verification of the new viscosity model (13) on the existing rigorous results of an independent experiment [31] has been successfully conducted. Further development of this model is to be found in a more detailed representation of the nature and proportion of clusters [33,34,35]. The results of calculating the data using model (13) and the Arrhenius equation are shown in Figure 3.

As can be seen in Figure 3, the viscosity line at the liquidus temperature (η (T_liq)), depending on the alloy composition, generally follows the shape of the curves at different temperatures. For a better perception of this line (η (T_liq)), it was raised above the rest of the curves by two points (2 + (η (T_liq)). In this case, the isotherms according to Equation (13) repeat the specific features of the Cu-Sn alloy, which has a smoothed maximum at 25–35 atm. % Sn, as well as a change in the temperature gradient of the liquidus according to the composition of the alloy at 92.4 and 99.3 atm. % Sn.

4. Discussion

The absence of a principal theory of the liquid state forces researchers of the viscosity of alloys to apply such concepts as “superheating” of alloys with low liquidus temperatures in comparison with higher values of T_liq within the general isotherm of viscosity.

The authors suggest that such simplifications can only be overcome by developing a principal theory of the solid, liquid and gaseous states of matter. This theory should be based on the Boltzmann distribution (energy spectrum) of kinetic energy of chaotic (thermal) motion and collision of particles. It is important to add this distribution with the concept of chaotic particles for three energy classes separated by the thermal barriers of the melting and boiling as proposed earlier by the authors.

A more detailed development of this concept allows us to define and quantify the fraction of clusters as the smallest energetic part of chaotic particles. It is characterized by a low kinetic mobility and creates obstacles to the overall mobility and thus the viscous consistency of alloys.

A new approach to the widely used structurally sensitive property of matter, dynamic viscosity, makes it possible to discover unexpected connections, for example, between the activation energy of the viscous flow E_ν (within the framework of the Arranius approximations) and the total thermal barrier of melt chaotization Δ_chH in the proposed partial-cluster viscosity model. The Arrhenius hypothesis about the activation energy is confirmed by the need to overcome the thermal barriers of melt chaotization using special experiments to measure and overcome mixing barriers during the formation of intermetallic structures, etc. At the same time, the temperature of any substance itself is reduced to the measurable accounting of such barriers, that is, it becomes evidence-based.

In connection with the above, the obtained probabilistic interpretation of the cluster presence in the liquid can be evaluated as a contribution to the theory of the liquid state itself. This contribution significantly complements structural studies of clusters [33,34,35] with their thermal component. It fills a gap in these studies and strengthens a unified probabilistic–deterministic mapping for any entity. Such mapping is essential and opens up the possibility of entropy-informed quantitative analysis of complex physicochemical systems [16]. Clusters have been found to be entangled with a material substrate which requires evaluating carefully selected experimental data on the manifestation and recording of any property. This justified our approach with the most reliable experimental data [31], from which we excluded the extrapolated values related to directly measured quantities. For our part, we used the fundamental distribution (energy spectrum) of Boltzmann only for the thermal component. In this case, we focused on the information limit of entropy in the region of T → ∞ and continued research into the Bose–Einstein concept at T → 0. Particular attention is paid to the clearly dominant role of the “coldest” part of the Boltzmann thermal spectrum, clusters.

It should be pointed out that obtaining results on the probabilistic interpretation of clusters allows for indirectly abandoning the concept of “superheating” of the alloy in relation to viscosity.

The application of the new partial clustering viscosity model can be employed across various domains of fluid dynamics.

The direct practical application of the partial clustering viscosity model is essential for the following:

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Developing an optimal melting regime to prevent “freezing” of the melt in the ladle during technological transport;

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Addressing “washout” of furnace linings during overheating;

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Managing emergency situations related to casting speeds in continuous rolling mill lines;

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Understanding lava flow dynamics during volcanic eruptions.

As a result, a fundamental and well-supported finding has been obtained regarding the correlation between the activation energy of fluidity, E_a, and the sum of the thermal chaos barriers, Δ_chH, during the heating of liquid alloys.

5. Conclusions

• A partial clustering model was obtained that includes the contribution of alloy components to its viscosity. In this model, the impact of clusters was compared with the energy barriers that can be calculated directly in formulas. This assumption can be considered as the fundamental significance of clusters for expressing viscosity, one of the most important properties of the liquid state.
• The new partial clustering model of viscosity was tested on the experimental data as an example of Cu-Sn alloy. The liquidus temperature was applied as a thermal barrier for the alloys. The high adequacy of the obtained results indicates the correctness of the proposed equation and possibility of its use based on the state diagram.
• The new partial clustering model of viscosity can be used for different alloys (two- and three-component) and to predict the behavior of this characteristic at the high temperatures, i.e., they face problems of difficulty in accurate measurement.
• A probabilistic interpretation of the cluster presence in a liquid can be appreciated as a contribution to the theory of the liquid state.
• The proposed model for the temperature dependence of dynamic viscosity is already suitable for practical application, particularly for extrapolation to temperatures above 1800 K. This temperature range is challenging to achieve experimentally, yet it is reliably represented in the model (13). Additionally, technological verification of the proposed model is feasible for viscosity-based control in the ultra-high temperature regime.
• The relationship of the concept of viscosity in the approximation of the Arrhenius physico-chemical model is revealed when comparing the activation energy of a viscous flow with the chaotic energy according to the partial clustering model, taking into account the thermal barrier along the liquidus line. This gives the proposed model fundamental importance.