.Normalized values and depending on the Mach number for different models
\r\n\tMethadone maintenance treatment (MMT) has become the main pharmacological option for the treatment of opioid dependence. Methadone remains the gold standard in the substitution treatment, which is a harm reduction intervention, because the patient does not become abstinent, but there are a series of positive changes. Currently, the surveillance of methadone substitution treatment is considered an ongoing challenge, given the need for the individualization and the increasing of the therapy efficiency. Methadone has been also studied as an analgesic for the management of cancer pain and other chronic pain conditions.\r\n
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Action of super-power and ultra-short laser pulses on highly absorbing condensed media is investigated in the past two decades  - . The urgency of this problem is primarily determined by a variety of practical applications of pulsed laser irradiation. In a short period of time scientists have mastered such operations as ablation of elemental materials by femtosecond lasers (100 fs) , femtosecond nanostructuring , generation of metallic nanoparticles and nanostructures by laser ablation of massive targets by 40 fs and 2ps pulses , etc.
The use of super-power G~10121015 W/cm2 and ultra-short τL≈10-1210-15 s laser pulses for dimensional processing of materials, such as cutting of various materials by pico-femtosecond laser pulses , micro-drilling by a femtosecond laser [10 ], surface etching of metals (Al, Cu, Mo, Ni) and semiconductors (Si) , is accompanied by realization of unique physical conditions. In particular, the duration of action becomes comparable with the characteristic times of thermalization and phase transitions in matter. This leads to the need to address complex fundamental problems, including the heating of the material and the kinetics of phase transitions in a strong deviation from local thermodynamic equilibrium.
It should be noted that the majority of laser technologies is associated with the beginning of phase transformations in the material. In particular, the action of pico - femtosecond laser pulses of high intensity on solid targets is one of the ways to create individual particles with unique characteristics, or to form their streams, consisting of a cluster, liquid or solid fragments of the target. The formation of the particle flux in the pulsed laser ablation is observed for a wide range of materials: metals, semiconductors and insulators. The possibility of usage of laser ablation products in practical applications was the impetus for a number of experimental and theoretical studies , aimed at studying the conditions and mechanisms of formation of particles of nano - and micro- sizes during the laser exposure.
The accumulation of knowledge of the experimental nature in the first place in this dynamic field leads not only to the variety of practical applications of pulsed laser action, but also to a number of issues of independent physical interest.
The main features of ultrashort action on metals are associated with high speed and voluminous nature of the energy release of the laser pulse. The high rate of heating of a condensed medium is associated with rapid phase transformations of matter, characterized by the transfer of superheated phase boundaries of high-power fluxes of mass and energy. Overheated metastable states at the interface are characterized by temperatures, whose values can be hundreds of degrees higher than the equilibrium values of the melting point or boiling point. Removal of the energy by the flow of matter in conjunction with volume mechanism of energy release of laser radiation contribute to the formation of metastable superheated regions in the volume of solid and liquid phases with near-surface temperature maximum. Calculations  showed that the maximum speed of melting front are comparable to the speed of sound ~ (0.5 - 6) km / s, and the phase velocity of the solidification front are (10 - 200) m / s. Accordingly, the maximum superheating / undercooling can reach several thousand / hundred degrees. The achievement of such overheating and overcooling leads to large gradients of the Gibbs energy, which actually determine the driving force for high-speed phase transformations.
Physics of supercooled states in metallic and nonmetallic systems, because of their widespread use in the production of new materials technology, is relatively well-studied . Superheated states received little attention until recently, largely because of the difficulty of experimental investigation and non-obviousness of their application. The situation changed with the advent of femtosecond laser pulses and their application for the production of nanoparticles and nanomaterials  - .
The purpose of this chapter is the theoretical study of nonequilibrium pulsed laser heating of metals, the kinetics and dynamics of phase transitions in a deep deviation from the local thermodynamic equilibrium.
The main tool for studying the processes initiated by laser pulses of picosecond and femtosecond duration are the methods of mathematical modeling and computational experiment (CE). The possibilities of experimental approaches in this area are very limited due to the large transience of the processes. Computational experiments are preferred in the cases where the natural experiment is not possible, is very difficult or very expensive. The statement of CE is especially convincing in the studies of the kinetics and dynamics of fast processes. For its statement, the computational experiment, which is an important link connecting theory and field experiments, requires the development of appropriate models, determining the properties of all substances studied, the development of computational algorithms and the creation of program codes.
Construction of a theory that covers most of the features of pulsed laser action on materials is very difficult. The methods of mathematical modeling in theoretical constructs have that advantage that you can use the phenomenological and the experimental data.
The description of the kinetics of fast phase transitions of the first kind is carried out in two classes of mathematical models: continuum and atomistic. Continuous models are based on the equations of continuum mechanics, as a rule, are represented as partial differential equations with appropriate boundary conditions and equations of state. Continuum model is used to describe the macro-level processes, heterogeneous kinetics and dynamics of phase transitions of the 1st kind . Atomistic approach is based on a model of molecular dynamics is used to describe the kinetics of homogeneous phase transitions .
The basis of first-order phase transformations - melting-solidification and evaporation-condensation are two qualitatively different mechanisms: heterogeneous and homogeneous ones. The heterogeneous mechanism is characterized by a sharp interface between the phases (phase front of zero thickness) and determines the dynamics of phase transformations of the 1st type. The homogeneous mechanism of phase transitions that arise usually under the influence of the volume heating or cooling is associated with the processes of volume melting and boiling or spontaneous crystallization and condensation.
Historically, the theory of phase transformations of the 1st type of melting is based on two fundamental sections of classical physics, thermodynamics and kinetics.
Thermodynamics is a macroscopic theory , which from the energy point of view, considers the properties of macroscopic bodies in equilibrium. This allows to have a great community for the conclusions of thermodynamics. However, thermodynamics does not take into account the internal structure of the considered bodies and some of its conclusions and regulations do not have physical clarity. One way to describe the equilibrium processes in the equilibrium thermodynamics is a theory of thermodynamic potentials.
The properties of a thermodynamic system are determined by thermodynamic parameters. The energy state of a thermodynamic system in equilibrium is uniquely determined by the parameters of the system. There is a unique relationship between the parameters of the system, which is mathematically represented by the state function. From a mathematical point of view, this means that the function has a total differential.
The basis of the method of thermodynamic potentials is just the possibility of introducing the state functions for the equilibrium processes, with total differentials describing the change in the state of a thermodynamic system.
The main identity of thermodynamics of equilibrium processes is usually represented as
Depending on the choice of the two independent parameters, one can introduce thermodynamic potentials, which differentiation allows determining the other unknown parameters of the state. In general, a thermodynamic potential may be a function of various parameters. In this notation, the internal energy is given as a function of entropy and volume:. It is the function of the state and it has a total differential with respect to its variables. The total differential can be used to determine the temperature and pressure. However, the usage of entropy and volume as two independent variables is inconvenient because they are difficult to control in experiment. It is more convenient to use pressure and temperature as two independent variables.
After consecutive transformation of internal energy first into enthalpy
which total differential, taking into account the basic identity of thermodynamics has the form
and then into Gibbs energy (Gibbs thermodynamic potential)
one can obtain the total differential of Gibbs energy, which with account of the basic identity of thermodynamics (1), takes the form
It is convenient in the fact that the independent variables and are easily to modify and to control in experiment.
After reaching the equilibrium state of the system, Gibbs potential takes its minimum value and becomes constant:. This allows to use the condition of minimum of the Gibbs potential for the description of equilibrium states in which and.
Thermodynamic systems in equilibrium state do not necessarily have to be a homogeneous medium. A system in equilibrium may be composed of several phases, different in their physical and chemical properties, separated by the phase boundaries not changing over time.
The multiphase thermodynamic systems are most simply described, components of which are in equilibrium states, and there is no transfer of matter, energy and momentum through interphase boundaries. In this case, such thermodynamic system is in equilibrium and methods of equilibrium thermodynamics apply to describe it.
Given that during phase transformations, each of the phases is a system with variable mass, the notion of chemical potential is introduced into the thermodynamic description. It is used to take into account not only exchange of energy but also the exchange of mass (particles). To determine the chemical potential, the term which takes into account the possibility of changing the number of particles in a homogeneous system (the same can be done with the other potentials) is formally introduced into the expression for the thermodynamic Gibbs potential:
If the thermodynamic potential is given as a function of temperature and pressure, then the value of the chemical potential is written as:
The chemical potential can be written through other thermodynamical functions but in this case it will be written in terms of other state parameters:
From the relation (6) it follows, that the chemical potential is a physical quantity that is equal to the value of some thermodynamical potential (with constant certain parameters) that is required to add to the system to change its number of particles by unity.
If macroscopic transport does not occur through the interphase boundaries, and the phases themselves are in a state of thermodynamic equilibrium, such thermodynamic system, in spite of its heterogeneity, will be in a state of thermodynamic equilibrium.
For the phase equilibrium in the one-component two-phase system, the following three conditions must be fulfilled:
the condition of thermal equilibrium, that means the equality of temperatures at both sides of the interphase boundary:,
the condition of mechanical equilibrium, consisting of the equality of pressure at the both sides of the interphase boundary:,
the condition of the equality of the Gibbs energy per particle, consisting of the requirement of the absence of the macroscopic transfer of molecules (atoms) of this material from one phase to another:
In principle, we can use not the Gibbs energy, but any thermodynamic potential, which has a minimum in equilibrium. It is not difficult to show the validity of these conditions. We shall use the total differential for the internal energy, taking into account changes in the number of particles:
We write this expression for each phase of the closed-loop system:
The closedness of the system automatically gives the following equation:
Consider an equilibrium two-phase system under some simplifying assumptions. Assume that the phases do not change the volume and do not exchange particles, i.e.,. Combining the equations (7) for this case, we obtain the expression:, which gives the condition of thermal equilibrium. Assuming the constancy of the entropy and the number of particles in phases, i.e. we obtain the condition of mechanical equilibrium:.
Given the conditions of thermal and mechanical equilibrium of (7) we obtain the equality of chemical potentials in the different phases. This equation can be solved for the variables and and may represent the equilibrium curves of the two phases in the form or. If we consider the boundary between liquid and solid, the equilibrium melting curve is obtained.
When describing the interface between liquid and gas, the equilibrium vaporization curve is obtained.
It should be noted that the processes at the interface are static in nature as in the case of equilibrium of different phases and also during phase transitions. There is a constant process of transition of particles from one phase to another at the interface. In equilibrium, these opposing processes compensate each other, and during supply or withdrawal of heat to one of the phases one of these processes begins to dominate leading to a change in the amount of matter in various states of aggregation.
If the components of the thermodynamic system are not in equilibrium with each other, then there are thermodynamic flows through their interface. This will be a process of transformation of matter from one state to another, i.e. phase transformation. Assuming that the occurring processes are quasi-static and the flows are small, one can use the methods of equilibrium thermodynamics to describe such non-equilibrium system. In this case we assume an infinitely small difference between the thermodynamic parameters in different parts of the system.
In the process of phase transitions of the 1st type, a number of quantities undergo abrupt changes at the interface, so in the following text in the thermodynamic equations, the sign of the differential will be replace by the corresponding value of the difference for the temperature of the phase transition.
The driving force of phase transitions of the first type is determined by the difference of Gibbs energy (or the magnitude of overheating/overcooling) for two phases at the interface, defined in (4) and (5) and can be written in two forms
Equilibrium. In equilibrium, and the equality (8) takes the form:
where the difference of enthalpy, is known as the equilibrium latent heat of transformation
From equation (9) we can obtain the dependence of the equilibrium pressure on the temperature, that is known as the curve of Clausius-Clapeyron:
If one take as the difference of the volumes of vapor and condensed phases, then since for ideal gas one obtain the expression
After integration, we find the temperature dependence of the equilibrium vapor pressure that is widely used for many materials:
The vapor pressure of the material in equilibrium with solid or liquid phase is called the saturated vapor pressure and with the notation, is usually written as
where is the temperature of the surface of the condensed phase, are the equilibrium values of pressure and the boiling point under normal conditions, is the latent heat of evaporation, is the gas constant.
As it follows from equation (9), at constant pressure, the difference of Gibbs energy is linearly proportional to the overheating/ overcooling
In the future, assuming the difference of energy to be equal to the rate of the phase transformation, one can obtain, that in the thermodynamic approach, the rate of conversion at constant pressure for small deviations from equilibrium is linearly proportional to the overheating / overcooling ΔT
where is the constant of proportionality between the normal speed limits and its overcooling. The constant does not have any clear physical sense and is chosen experimentally for each material. The dependence (14) by its form coincides with the well known relation for the determination of the linear crystal growth rate obtained on the basis of classical molecular-kinetic models in which the constant of proportionality is called the kinetic coefficient . The main application of the relation (14) and its various modification [22, 23] is found in the description of different processes of melting - solidification. Comparison with experiment showed that the equation (14) gives good agreement mainly at small overcoolings [24, 25]. The kinetic coefficient is the main parameter, characterizing the mobility of the boundary crystal -melt. Despite the great importance of this characteristic, there are only a few experiments to successfully measure the kinetic coefficient in metals and alloys . The main difficulties of experimental determination are associated with the great complexity of measuring the overcooling at the solidification front. Currently, the main approach to the determination of the quantitative evaluation and qualitative understanding of the physics of the processes behind the coefficient are the methods of molecular dynamics  - .
The structure particles of matter are in continuous motion and appear in the main provisions of the molecular-kinetic theory in which all processes are considered at the atomic or molecular level, the particles obey the Boltzmann statistics, and the speed of processes is given by
where is the activation energy, is the Boltzmann\'s constant, is the average thermal energy for one atom, is the pre-exponential factor that affects the process rate. The exponential term is known as Boltzmann\'s factor, that determines the part of atoms or molecules in the system that have the energy above at the temperature.
For the first time, the conditions of crystal growth from liquids were formulated by Wilson , who suggested that the atoms have to overcome the diffusion barrier in order to make the transition from liquid to solid phase. The rate of accession of atoms to the crystal lattice is expressed by the relation analogous to (15)
where is the activation energy for overcoming the diffusion barrier, is the frequency of attempts of transitions, is the atom diameter.
The rate of accession of atoms to the crystal was estimated in  using the diffusion coefficient of liquid, that allowed to estimate the velocity of the crystallization front as
where, is the temperature of the interphase boundary.
Later, Frenkel , using the Stokes-Einstein relation between the coefficients of diffusion and viscosity expressed the crystal growth rate in terms of viscosity
The expressions (17), (18) allowed establishing a linear relationship between the rates of crystal growth and overcooling. Wilson-Frenkel theory [31-33] contributed to a better understanding of the microscopic processes associated with the growth of crystals from the melt. A generalization of the obtained results allowed to obtain that, except for very large deviations from equilibrium, where the homogeneous nucleation mechanism can dominate, the process of melting-solidification proceeds heterogeneously. The heterogeneous nucleation mechanism involves the inclusion of the motion of the liquid–solid interface into the consideration. The velocity of this interface, as a function of the deviation from the equilibrium melting temperature is called the response function of the interface and is the main value characterizing the processes of crystallization and melting. As with the similar equation (14) obtained in the thermodynamic approach, the equations of Wilson-Frenkel showed a good agreement with experiments for very small overcooling of the boundary, i.e. for small deviations from equilibrium.
Theoretical studies of the velocity of the interface are based on some modifications and generalizations of Wilson-Frenkel theory. Their meaning is reduced to taking into account several factors, such as the latent heat of melting, interatomic distance, efficiency coefficient, showing the proportion of atoms that remain in the solid phase at the border crossing. In the modifications [24,34,35], it was considered that the crystal has always lower enthalpy than the melt. This is the amount of energy needed for atoms of the crystal to make the transition from crystal to melt. Escape rate of atoms of the crystal followed by the addition to the active points of the liquid, contains this energy difference in the form of the Boltzmann factor
The rate of the reverse flow of atoms into the crystal from the melt depends only on the diffusion process in liquid
where, are some constants that should be determined.
In the equilibrium point, the rates and are equal, which gives, and.
The velocity of the interphase boundary is equal to the difference of the rates и
The constant is associated with other physical constants using the following relation:. The studies have shown that the resulting equation (21) well predicts the velocity of the melting-solidification front of silicon  in a fairly wide temperature range.
The problems of melting-crystallization of monatomic metals in the modes of rapid heating/ cooling, typical of ultra-short laser irradiation, use a different expression for the velocity. It is based on the assumption [37, 38], that crystallization of single-atom metals, (that are characterized by high velocities m/c [35, 39]) is not diffusion-limited, but is limited only by the collision frequency during transition from liquid to crystal surface. The modification of the equation (21) consists of replacement of the diffusion term with the thermal velocity. The velocity of interphase boundary is written as:
where, is the mean free path, is the atomic mass.
Thus, the kinetic approach allows us to obtain an expression for the response function without fitting coefficients and suitable for a wide range of overheating/overcooling. The kinetic dependence (22) is asymmetric for the processes of melting and solidification. However, large deviations from equilibrium require additional modification of the kinetic dependences (21) (22), because they do not take into account the dynamic effects associated with the occurrence of high pressures generated by the high velocity of propagation of phase fronts.
One of the key processes in the zone of laser irradiation is the transition of condensed matter to the gaseous state. Evaporation process is characterized by high power consumption and large increase in the specific volume of the substance.
Investigation of evaporation process began in the 19th century [40, 41] and continues to this day [42 -]. This fact is defined by practical importance and not fully clarified features of non-equilibrium behavior of matter when it evaporates.
184.108.40.206. The simplest model of kinetics of evaporation in vacuum
The thermodynamic relation (9) gives that at a constant temperature the difference between the Gibbs energy between the two phases, one of which is an ideal gas, is linearly proportional to the pressure difference:
On the basis of the formula (23), it can be assumed that the rate of phase transformation at constant temperature should be linearly proportional to the pressure difference:
The simplest model for the growth kinetics of vapor phase was developed by Hertz  and Knudsen  about 100 years ago. Its formulation is qualitatively the same as the thermodynamic model (23), (24)
where is non-equilibrium flow of atoms at the surface of evaporation, is the flux of atoms that collide with the surface under the assumption that the adhesion coefficient is 1. Formally, this flow can be determined using the relation connecting the vapor pressure with the equilibrium particle flux directed to the condensed surface:, where is the average velocity in one direction, , what gives. Since the nature of the flux remained undetermined and the values of and correspondingly unknown, then during the formulation of the boundary conditions in the problem of evaporation into vacuum it is supposed [20, p.281] to use a single-term version of the Hertz-Knudsen formula to determine the evaporation rate:
that takes into account the connection between the pressure of saturated vapor and flux of evaporated atoms at the surface temperature. The equilibrium vapor pressure depends on the temperature, as it is shown in the equation (12).
However, the representation of the process of surface evaporation in the form of a simple model, which does not take into account the reverse influence of evaporated atoms, does not remove the internal contradictions inherent in the model of Hertz - Knudsen. Let us write the expressions for the fluxes of momentum and and energy of the particles, moving away from the evaporation surface
It is easy to see that these fluxes, which completely describe this one-dimensional flow, are impossible to be characterized by any temperature. If we equate these fluxes to the corresponding thermodynamic expressions containing the velocity, temperature and density and find these values, we will see that the system of equations has two distinct complex solutions, which have no physical meaning.
The ambiguity of the solution for real values of the thermodynamic parameters is associated with the possibility of discontinuous solutions of the type of shock wave. The complexity of the solution in this case is due to the thermodynamic non-equilibrium of the evaporation flow of Hertz-Knudsen, which can not be described in terms of thermodynamic concepts. When taking into account collisions in the non-equilibrium layer, the evaporative flux is thermalized, but the temperature on the outer side of this layer no longer coincides with the surface temperature.
220.127.116.11. Approximation of the Knudsen layer
Intense surface evaporation is essentially non-equilibrium process. In addition to the thermodynamic equilibrium, this process also has a gas-kinetic non-equilibrium in a thin (Knudsen) layer of vapor, directly adjacent to the interface. Gas-kinetic non-equilibrium is due to the flow of material through the phase boundary. The mass flow increases with the growth of the evaporation rate and, consequently, the degree of non-equilibrium of the process increases. From the physical considerations, the maximum velocity of material flow on the outside of the Knudsen layer is limited to the local speed of sound, or, where is the gas-dynamic velocity, is the Mach number. The maximum deviation from equilibrium is determined by the maximum value of mass flow, which is known to be achieved at.
Under the conditions of phase equilibrium, when the saturated vapor pressure is equal to the external pressure, the flow of vaporized material is balanced by the return flow of particles and the total mass flux through the boundary is zero. The distribution of particle velocities in vapor is in equilibrium and can be described by the Maxwell function with zero average velocity. In cases where the vapor pressure above the surface is less than the saturated vapor pressure, in the system condensed matter-vapor, the directed movement is formed with and is characterized by non-zero material flow through the phase boundary. The decreasing reverse flow leads to a deviation from the equilibrium in the distribution of the particles. The magnitude of the flux of returning particles decreases with the increase of the rate of evaporation, and the distribution function at the evaporation surface becomes increasingly different from Maxwellian one.
In general, the non-equilibrium distribution function is found by solving the Boltzmann equation in a region with a characteristic size of a few mean free paths. This area is adjacent to the evaporation surface, where the kinetic boundary conditions are set, taking into account the interaction of individual particles with the interface. Similar problem was solved by various methods in many studies, taking into account, in particular, the differences from unity and variability of the coefficient of condensation, which determines the probability of attachment of the particle in its collision with the evaporation surface. (See, for example. [42 - 46]).
Methods of non-equilibrium thermodynamics are used to describe the evaporation process together with other approximate phenomenological approaches [47 -52]. A more general and fundamental approach is to use molecular dynamics method, which was used in [53, 54] to analyze the evaporation process. A recent review on the issue of non-equilibrium boundary conditions at the liquid-vapor boundary is given in .
For the equations of continuum mechanics, thin Knudsen layer is a gas-dynamic discontinuity. The knowledge of the relations at this break, connecting the parameters of the condensed medium and the evaporated material, is needed to deal with the full gas-hydrodynamic problem that arises, for example, during the description of laser ablation, taking into account the variability of the Mach number and instability of the evaporation front . The use of kinetic approaches, which explicitly consider the structure of the Knudsen layer, in such cases is difficult because of the emerging problem of significant difference of space-time scales. The solution of these problems is associated with the additional computational difficulties and is not always possible. Therefore, usually another approach is used that allows to determine the matching conditions with certain assumptions about the form of the non-equilibrium distribution function inside the break [57 - 60] without solving the kinetic problem. Approximation of the distribution function in the Knudsen layer was carried out in different models. But to obtain physically reasonable boundary conditions it is necessary to formulate criteria which these conditions must meet. In addition, attention is paid to the peculiarities of the behavior of the fluxes of mass, momentum and energy as the Mach number tends to unity that allows to use the requirement of extremum of total fluxes of mass, momentum and energy as one of the criteria for- model. In – model [61, 62], a compound distribution function is used to describe one-dimensional non-equilibrium flow of particles on the inner side of the planar Knudsen layer:. Here, the distribution for particles, flying out of the surface is given as a Maxwell-shaped function with density of saturated vapor for the surface temperature. The distribution characterizes the flow of particles returning to the surface and is supposed to be proportional to a “shifted” Maxwell function with density, temperature and mean velocity steady flow of vapor on the outside of the Knudsen layer:
The flows of mass, momentum and energy calculated using must be equal to their gas-dynamic values that are determined by function:
where is the vapor heat capacity per particle at constant pressure, for single-atom gas. From the solution of equations (27), it is possible to obtain gas-dynamic conditions on the break, allowing to determine the magnitude of, и in terms of, and. The calculations showed that total fluxes have extrema depending on at correspondingly.
All 3 flows.. will have extrema at, if the function is set to be equal to, where the values of are written in terms using additional fitting parameters and
For example, at and all three flows will have extrema at with values. It is clear, that this version of selection of fittings coefficients is not the only one.- model. It is possible to suggest another phenomenological model , where strict localization of extrema is achieved without usage of fitting coefficients for such distribution function, that do not depend on gas-dynamic values. An example of such function is function
It takes into account the decrease of temperature of the reverse flow of particles as compared to the surface temperature. Due to this change of, the ratio of the normalized fluxes ceases to be a constant and takes the form, which ensures that the correct limiting value is equal to 1.25 in the equilibrium case for. The equation for, that is obtained from the equity of fluxes (27) has a relatively simple form:
The right-hand side of the equation (28) has a maximum at, that determines localization of the extrema of and. The values of are equal correspondingly to.
Modified Crout model. The property of localization of the flows also present in the model suggested by D. Crout . It uses non-equilibrium function of distribution of particles, written in analytical form with temperature that is anisotropic by directions: and.
where is the lateral temperature along x axis, is the transversal temperature along y,z axis; are the components of the velocity vector along corresponding axes, u0 is the drift velocity.
The modification of the Crout model  consists of explicit introduction of the Mach number into the main relations that allows to obtain:
The value of m is determined from non-linear equation
For the numerical solution of the nonlinear equation (30), one can use the Newton\'s iterative procedure. All fluxes have extrema at M=1. Calculations using an anisotropic non-equilibrium particle distribution function give the corresponding extreme values of the fluxes:.
The calculations show that specific choice of the model has relatively little effect on the magnitude of the momentum flux j2, but significantly affects the flux of mass and energy.\n\t\t\t\t\t\tFig.1 shows the dependencies of the ratio of the normalized fluxes on for all discussed models. From the comparison of the extreme values of the fluxes and behavior of the curves it follows, that due to bad choice of f(-), the less favorable for the description of evaporation kinetics is – model.
Table 1 for all models shows numerical values of and (normalized by and correspondingly) depending on the Mach number that changes from zero to unity. Comparative analysis of tabular data, as well as the behavior of the curves, show a marked difference of the values of and, obtained using model, from the values, obtained using other models. The values of turned out to be underestimated, and overestimated as compared to their real values at the outer side of the Knuden layer.
|“β”-model (Knight)||“ε-δ”-model||“α”-model||Crout model|
The modified Crout model,– model,– model fulfill the requirement of the extremum of the flows at =1. The difference of the results that were obtained using these models does not exceed 1.5%. Any of these models can be used to describe the kinetics of the process of non-equilibrium surface evaporation.
The performed brief analysis of the kinetics of phase transitions is an introduction to the construction of the models that combine mathematical description of kinetics of high-speed phase transformations with dynamics of the macro-processes (heat and mass transfer) under conditions of a strong deviation from local thermodynamic equilibrium that are typical for ultra-short super-power laser action on metals.
Determination of physical characteristics of a medium, including equations of state under conditions of local thermodynamic equilibrium can be carried out either experimentally or by means of calculation using distribution functions - a Maxwell-Boltzmann function for ideal gas and ideal plasma and Fermi one for degenerate electron gas, and for the phonon gas – Bose function. In case of violation of the conditions of local thermodynamic equilibrium distribution functions are determined by solving the classical kinetic Boltzmann equation or quantum-kinetic equations. The presence of distribution function is just the required minimum of information that can be used to describe nonequilibrium processes with reasonable accuracy.
The influence of ultrashort high-energy laser on a strongly absorbing media (metals, semiconductors) is in a very short temporal and spatial scales and leads to disturbance of their general local-thermodynamic equilibrium. Irradiated targets in these conditions are presented in the form of two subsystems - electron and phonon each of which is in local thermodynamic equilibrium and are characterized by their temperatures and equations of state. As a consequence, all processes are described in the two-temperature approximation , . The target at pico- and femtosecond influence may be heated to very high temperatures and pressures at which the thermal and mechanical properties of matter are not known in general. One of the most important problems for the mathematical modeling is the necessity to determine thermophysical, optical and thermodynamic properties in a wide (tens and hundreds of electronvolts) temperature and frequency ranges for each of the subsystems.
The most important thermophysical and thermodynamic characteristics of the electron Fermi gas within the scope of heat-conducting mechanism of energy transfer are: heat capacity, thermal diffusivity and thermal conductivity. For its determination using fundamental physical quantities, which include the electron mean free paths, and the characteristic times (frequency) of interaction for two scattering mechanisms: the electron-electron and electron-phonon.
For a quantitative description of electrical, thermodynamic and thermophysical properties of degenerate electron gas with distribution function
widely used Fermi-Dirac functions expressed in terms of integrals of the form
where, - temperature and energy of electron, - chemical potential, - Fermi energy.
In the future, the Fermi integrals will be represented as a function of dimensionless energy of the chemical potential:
where, - dimensionless energy and chemical potential of electrons.
The integral of form is used to determine the electron density, where is a density of states, is distribution function of the free electron energy,. Then
From the known distribution function of particle energy, using the ratio the average value of any physical quantity depends on energy can be found. Since the average energy of the electron gas is defined as the ratio of Fermi integrals
Similarly, we can determine the other physical quantities of the electron gas.
The chemical potential depends on the temperature, so the Fermi integrals can be conveniently represented as a function of dimensionless temperature. In  for the integrals of the form (33) has been proposed convenient approximation, which allows to express the integrals through the transcendental gamma-functions and the dimensionless temperature:
where, coefficients expressed in terms of gamma-functions. Equation (36) has correct asymptotics at and:
The integral of order can easily be determined from the expression (34):.
The most frequently used Fermi integrals that are expressed through (36) have the form:, ,
The maximum error compared with the exact solution  for integrals at and does not exceed 8%, but increases slightly with increasing of k.
Approximation (36) allows to obtain simple analytical expressions for the physical quantities of the electron gas at arbitrary temperatures.
Using the approximating expressions (36) and (37) equations of state for degenerate electron gas can be written as simple analytical expressions at arbitrary temperatures. Since the average electron energy and its pressure can be represented as
The expression for the heat capacity of the electron gas can be obtained from the relations, ,
Using the approximating expressions (37), heat capacity of electron gas can be represented with an error not exceeding 5% as the following function
where, z - the number of valence electrons, Na - the concentration of atoms (ions) of lattice.
The resulting expression gives the classical linear dependence of heat capacity of a degenerate electron gas vs. temperature  in low temperature region, and constant value at equal to heat capacity of gas with Maxwell distribution. Dependences for copper and aluminum are shown in Fig. 2.
The thermal diffusivity of electron gas is proportional to the product of the mean free path le and average velocity of the electron:
In metals electron mean free path due to several mechanisms: pair of electron-electron collisions, electron-phonon collisions and scattering by plasmons.
Electron-electron collisions dominate at temperatures comparable to the Fermi energy. Electron-phonon interaction is dominant at low temperatures. The interaction associated with the excitation of plasmons occurs at high temperatures, exceeding the plasma frequency energy (eV). Taking into account high temperature region of occurrence and limitation of experimental data about reducing of the mean free path of electrons due to plasmon excitation (it is known only for some metals), electron scattering by plasmons will not be considered.
18.104.22.168. The electron-electron thermal diffusivity
The mean free path of an electron in pair electron-electron collisions is determined from the known gas-dynamic formula
where - scattering cross section with energy transfer for electrons with energies. The cross section is expressed through the transport cross section of the collision of two isolated electrons and Fermi integrals. In turn, the transport cross section of collision of two isolated electrons in a field of screened Coulomb potential is expressed through the differential scattering cross section determined in the Born approximation [70, p.560]. The final cross sections will be written in form
where \n\t\t\t\t\t\t= 0.529 10-8cm – Bohr radius, - the average distance between atoms, e - the electron charge, d – field acting radius (Debye).
At low temperatures, the effective cross section is small and amounts to. Maximum of cross section is achieved at, Fig.3, when the degeneracy is passed and the electron-electron collisions with large energy transfer become possible. At very high temperatures cross section becomes the Coulomb one, Fig.3, and decreases logarithmically. The mean free path of electrons is determined by the formula (42):
Calculations indicate that the mean free path for Al and Cu change in a wide range (~10-2 ÷ 10-7) cm and have a minimum at.
The average thermal velocity of electron is expressed through its average energy:
Taking into account (44) and (45) electronic thermal diffusivity takes the form:
where is dimensionless function,.
Fig. 4 shows the temperature dependence of for Al and Cu. Temperature dependence of electron-electron thermal diffusivity of both metals has a deep minimum at. Its value reaches ~ 20÷30 [cm2/s], Fig.4. At removal of degeneracy, when the thermal diffusivity increases due to decreasing of the effective cross section. With further increase of temperature the thermal diffusivity continues to increase and its dependence coincides with temperature dependence of thermal diffusivity of Maxwell electron plasma. At low temperature dependence of thermal diffusivity is inversely proportional to square of temperature ~. Strong growth of with decreasing of leads to the fact that reach values which is 3-4 orders higher than the actual electron thermal diffusivity of metals at room temperatures. Thus, the resulting expression (46) is a good approximation only for high temperatures. Under normal conditions () it is necessary to consider the scattering of electrons of metals by phonons to determine thermal diffusivity, this interaction is dominant at low temperatures.
22.214.171.124. The electron-phonon thermal diffusivity
The mean free path, defined by electron-phonon interaction is described by the assumption of elastic scattering of conduction electrons of metal on lattice oscillations. To determine it is convenient to use the phenomenological approach , in which the crystal is considered as an elastic continuum. Lattice oscillations at the same time considered as a wave of elastic deformations. To simplify the density fluctuations are presented as deviations of each atom (ion) from the average, which square of amplitude is directly proportional to temperature. According to the macroscopic theory of elasticity we can obtain an expression for the mean free path, expressed in terms of macroscopic quantities  by expressing the force tending to return the atom (ion) to the equilibrium state through the Young\'s modulus E:
From the expression (47) it follows that is inversely proportional to the lattice temperature in electron-phonon interaction.
During melting of a metal the number of collectivized electrons remains practically unchanged. The modulus of elasticity is only one value (excluding the jump in specific volume, which usually does not exceed 10%) that changes. The melting of most metals is accompanied by decrease in elastic modulus by 2-3 times . This decrease causes a corresponding increase in density fluctuation and, consequently, an abrupt decrease in the mean free path:
where subscripts denoting membership in the solid and liquid phases, respectively, is a distance between the atoms. From below is limited by Bohr diameter value. The mean free path of electrons taking into account the scattering by phonons, calculated for aluminum and copper from the relation (48) showed that, compared with the mean free path of the electron-electron scattering values for both metals decreased by several orders of magnitude: for the high-temperature region of 1.5 ÷ 2 orders of magnitude, while at low temperatures for 4 ÷ 5 orders of magnitude.
The thermal diffusivity, defined by the mechanism in electron-phonon interaction is calculated by the formula:
Dependences for Al and Cu calculated from (49) at are shown in Fig.5. The calculations show that at room temperature values of the electron thermal diffusivity, for both materials are 3-4 orders of magnitude less than the thermal diffusivity and reach values of 102 cm2/s typical for metals under normal conditions. Under equilibrium conditions with temperature increasing, thermal diffusivity, undergoing break at the phase transition decreases rapidly to a value of ~ 1 cm2/s at a temperature of ~ 5 000K. Function increase by several orders of magnitude with electron temperature increase.
126.96.36.199. The resulting thermal diffusivity of electron Fermi gas
Calculations have shown that taking into account only a pair electron-electron collisions leads to a strong (by several orders of magnitude) overestimation of the thermal diffusivity of the electron gas at low temperatures. Accounting for electron-phonon collisions gives a more realistic values of at low temperatures.
By averaging mean free paths we obtain an expression for resulting thermal diffusivity, at arbitrary temperature
Fig.6 shows temperature dependences of total electron thermal diffusivity for Al and Cu in equilibrium Te\n\t\t\t\t\t\t= Tp.
According to elementary kinetic theory, the thermal conductivity of the gas is
Thus, the thermal conductivity can be determined through the heat capacity Сe and averaged thermal diffusivity of electron gas. The temperature dependences of for Al and Cu calculated for the equilibrium case when are shown in Fig.6. In accordance with the results, the equilibrium electron thermal conductivity at temperatures not exceeding the boiling temperature of the equilibrium is practically independent of temperature. In high temperature region >1eV thermal conductivity increases rapidly due to the dominance of electron-electron scattering. It is natural that in this region the thermal conductivity of the electron gas depends on electron density and increases with increasing of electron concentration. For this reason, electron thermal conductivity of aluminum is higher than the same one of copper at the high-temperature. In low temperature region where electron-phonon interaction is dominated, the ratio is inverse. Because of the greater mean free path, electron thermal conductivity of copper is higher than that of aluminum.
The main consequence of the existence of lattice oscillations is the possibility of its thermal excitation, which is appeared as a contribution to the heat capacity of solid.
Phonons are considered as a gas of particles. From elementary kinetic theory, thermal diffusivity of a gas is given by
where - phonon mean free path. It is assumed that the phonons move with velocity of sound.
The mean free path of phonons is determined from the description of thermal motion in a solid by means of notions of the phonon gas. The interaction between the phonons can be characterized by some effective cross section which is proportional to the mean square of thermal expansion of the body or the mean square of density fluctuation :, where is a coefficient of compressibility. Taking into account:
Assigning to phonons radius equal to thermal oscillations amplitude, we can taken into account scattering of sound waves and determine the mean free path
where M - the mass of the atom, - Griuneyzen constant.
Accounting for the expressions for the mean free path (54) and the velocity of sound  expressed through the Fermi velocity
thermal diffusivity of phonons can be written as: