Thermodynamics of Gases at Low Pressures

Authored by: Igor Bello

Vacuum and Ultravacuum

Print publication date:  November  2017
Online publication date:  November  2017

Print ISBN: 9781498782043
eBook ISBN: 9781315155364
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Thermodynamics assists in elucidating important physical quantities and behavior of gases at low pressures. Among them, internal energy of gas, enthalpy, work performed by gases, gas heat capacities, and their relationships at low pressures are discussed in this chapter. The laws of thermodynamics can be applied to phase transformation, inherent saturated vapor pressures of solid and liquids, and evaporation processes at vacuum conditions in thin-film technology.

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Thermodynamics of Gases at Low Pressures

Thermodynamics assists in elucidating important physical quantities and behavior of gases at low pressures. Among them, internal energy of gas, enthalpy, work performed by gases, gas heat capacities, and their relationships at low pressures are discussed in this chapter. The laws of thermodynamics can be applied to phase transformation, inherent saturated vapor pressures of solid and liquids, and evaporation processes at vacuum conditions in thin-film technology.

3.1  First Law of Thermodynamics and Enthalpy Applied to Ideal Gases

If gas molecules are invariable, the internal energy is equal to the sum of kinetic energies of individual molecules and their own energies. The increase of the internal gas energy by a differential value dU is equal to the supplied thermal energy dQ subtracted in an energy equivalent to the differential portion of mechanic work dW performed by the gas system, that is,

which can be rewritten to the equation
which is the first law of thermodynamics. 71 Accordingly, the supplied heat can partially be used for increasing the internal gas energy and performing the mechanical work. Equations 3.1 and 3.2 represent the law of conservation of energy, including mechanical energy. Both the energy and work have equivalent effects. The experiments show that the energy change does not depend on the way of gas changes from one state to another, but it only depends on the initial and final gas states. Therefore, the energy is a state function.

By adding and subtracting Vdp, the first law of thermodynamics can further be recast to

For the practical reasons, we can introduce a new state quantity
called enthalpy. The quantity of enthalpy is introduced because the energy of the system does contain not only the internal energy of gases that is stored in molecules but also additional energy equivalent to the mechanical work pV that is stored in the environment. Enthalpy is the measurable quantity that describes the internal energy of extended gas systems. When we consider chemical reactions inside a gas, we can express the change in enthalpy by full differential of enthalpy, that is, dH = dU + d(pV) = dU + Vdp + pdV, which at constant pressure is dH = dU + pdV. Accordingly, at constant pressure, the increase of volume by the differential portion dV implies that the gas performs the work dW = pdV. Hence, the change in enthalpy can also be expressed by equation dH = dU + dW, being equivalent to the heat absorbed by the gas. Then, introducing enthalpy into Equation 3.3 gives the first thermodynamic law in the form

Since the way of passing the gas system from one state to another does not play any role, the internal energy is the state function, and it is therefore a total differential. However, the work performed at a transition from one state to another depends on the transition path, and the work dW is neither the total differential nor the state quantity.

3.2  Definition of Gas Heat Capacities

Two gas systems with different temperatures have a tendency to equalize their temperatures via heat transfer phenomena. The measure of the heat transfer depends on the initial and final temperatures of these gas systems. In general, heat can be transferred by radiation, convection, and conduction.

Heat transfer by radiation depends only on the temperature of objects between which the heat transfers exist since all objects can emit and absorb radiation energy. Transfer of heat between two objects by radiation is very effective at large temperature differences. In vacuum conditions and even at a lower temperature differences, transfer of heat by radiation may overwhelm other mechanisms of the heat transfer even at lower temperatures.

In heat transfer by convection, the mass density of a gas alters upon the variation of the local gas temperature, which violates the mechanic balance of the gas system and drives the gas into motion. In nature, the convection phenomenon is known as wind. In practice, the transfer of heat by the convection process can technically be enforced. However, heat transfer by convention becomes notable only in the rough vacuum region when pressures are greater than 100 Pa.

The third form of the heat transfer by conduction depends on gas conductivity, which is further discussed in Chapter 5 on transfer phenomena.

In heat measurement, it is possible to choose a heat unit based on equality of heat quantities that raise the temperature of defined amounts of a substance by equal temperatures. Using this approach, the unit of calorie (cal) was introduced. For the first time, French physicist and chemist Nicolas Clément-Desormes introduced this heat unit 72 as kilogram-calorie in 1824. One of the several introduced calories is defined as the heat quantity needed to raise the temperature of 1 g chemically pure water at standard atmosphere from 14.5 to 15.5 °C. This is known as 15 °C calorie. Calories were also defined at other temperatures, including 4 °C, 17 °C, and 20 °C calories. A multiple unit of kilocalorie (kcal) has also been used frequently. In the British Imperial Unit System, a British thermal unit (Btu) was introduced as a heat amount. The unit Btu is the heat amount that raises the temperature of a pound of water from 63 to 64 °F. These definitions come from the understanding of heat as the ability to increase temperature of the defined substance amounts. The understanding of heat as a form of energy led to the same units for both heat and energy. In consistence with the SI unit system, the heat unit is Joule (J), and the 15 °C calorie is equal to 4.1855 J. Despite the intension of the definition of the heat in Joules without any reference to water, recalculation factor from calories to joules may deviate because of different definitions of calories. For example, heat needed to increase temperature of 1 g water at standard atmospheric pressure from 0 °C to 1 °C is also called calorie but with the recalculation of 1 cal = 4.184 J. The amount of the heat of 4.184 J stands for the thermochemical calorie. Herein, we use 15 °C calorie for which the conversion is


The needed heat to increase the temperature of a gas with a mass of M by a degree is called the heat capacity, which in the mathematical form is

Since two gas systems of the same kind have capacities proportional to their masses, for reference, the heat capacity related to the gas mass of unity is defined:
which is called specific heat capacity. The specific heat capacity is unique for each gas. In many calculations, it is also useful to refer the heat capacity to a gas amount of kmol (mol), which is called molar heat capacity. Thus, molar heat capacity is defined by equation
  • n m = M/M a is the gas quantity in kmol
  • M a is the molar mass in kg/kmol
  • M is the mass of gas in kg
The magnitude of the heat capacity applied to a quantity of 1 kmol (mol) is unique and refers to the chemical nature of a gas.

The heat capacity can be determined under the conditions at which either volume or pressure of a gas system is maintained constant. Thus, the heat capacity has to refer to the heating conditions of gas systems. Accordingly, at constant volume, the heat capacity is

and at constant pressure, the heat capacity is
respectively, following from the measurement conditions and the definition of internal gas energy U and enthalpy H.

Unless heat is used for performing work or transforming a substance to another phase, the definition of heat capacities may imply that the heat supplied into the gas system increases its temperature. The assumption of constant heat capacity in a considered range of temperatures allows us to separate variables and integrate equations for heat capacities. For instance, the integration of Equation 3.8 within the boundaries of heat from 0 to Q and temperatures from T 0 to T shows that heat supplied into a gas system is

  • c is the specific heat capacity
  • M is the mass of substance
  • T 0 is the initial temperature
  • T is the final temperature of a substance

Now, in terms of thermodynamics, we may analyze the simplest reversible processes. These processes relate to ideal balanced systems, where even tiny changes of the system conditions can overturn the direction of the process. For example, an extremely small decrease in environmental temperature results in a heat transfer from a gas system to the environment and vice versa.

3.3  Isochoric Processes: Molar Heat of Ideal Gases at Constant Volume

In general, ideal gas systems are described by three macroscopic parameters, that is, pressure p, volume V, and temperature T. In the isochoric process, the gas system is either heated or cooled while the gas volume V is maintained constant, as illustrated in Figure 3.1. Thus, volume change is invariable (dV = 0), and consequently the work performed by such an ideal gas has to be zero as well (dW = pdV = 0). As a result, the heat supplied into the gas only raises its temperature, and accordingly increases the gas pressure and thus the internal energy of the gas. Then, in the terms of thermodynamics, the isochoric process is described by equation

from where the molar heat at constant volume is
If the internal energy of 1 kmol of the ideal gas is U = iR 0 T/2, then its derivative is
is the molar heat capacity at constant volume. If the degrees of freedom for monoatomic, diatomic, triatomic, and polyatomic molecules are i = 3, 5, 6, and 8 (see Chapter 2.11), respectively, then the corresponding molar heat capacities are
These heats per 1 kmol at constant volumes refer to the gas systems that do not perform mechanical work. If temperature of 1 kmol of a monatomic gas is increased by 1 K, the heat supplied into the gas is equal to 12.7 kJ/(K kmol). Accordingly, the molar heat is the same for all monatomic ideal gases (12.47 kJ/(K kmol)). Similarly, the molar heat of diatomic gases at constant volume is equal to 20.786 kJ/(K kmol), though there are two additional quadratic vibration components (potential and kinetic energies). These vibrations can make differences at higher temperatures. In the case of triatomic (molar heat is C V = 24.786 kJ/(K kmol)) and polyatomic molecules (molar heat is 33.258 kJ/(K kmol)), vibrational components of energy cause more obvious discrepancies in molar heats because the number of vibrational states depends on the molecular structures. Since these states are progressively activated with increasing temperature, the inconsistency in heat capacity C V is more noticeable, particularly for polyatomic molecules.

Isochoric process when the heat supplied only increases the gas temperature.

Figure 3.1   Isochoric process when the heat supplied only increases the gas temperature.

Following the treatment of the discussed case, it can be suggested that at constant volume, the molar heat capacity depends on the degrees of freedom of molecules, but it is independent of the temperature.

In assumption of the constant molar heat capacity, Equation 3.15 can be integrated over considered temperatures, which gives the change in gas internal energy:

where the ideal gas is heated from temperature T 0 to temperature T.

Nevertheless, comparing the theoretical and empirical molar heat capacities indicates some inconsistency. Larger deviations of the empirical values from those predicted are observed in the case of polyatomic molecules. In particular, increasing the temperature parameter amplifies the discrepancies between experimental and theoretical data of heat capacities. The plots of heat capacities at constant volume constructed from selected numerical data, 73 in Figure 3.2, could serve as an example. At temperature of a few 100 °C above room temperature, the heat capacities C V significantly exceed the theoretical values, except for helium. Thus, the assumption of the equal distribution of energy based on the degree of freedom, defined in the equipartition theorem, is reasonably consistent with experimental data only for rigid gas molecules and at the temperatures close to the room conditions. Otherwise, the heat capacities determined based on the equipartition theorem may fairly deviate from the empirical heat capacities especially obtained at higher temperatures and for molecules that are more complex.

Heat capacities

Figure 3.2   Heat capacities C V of some gases at constant volume.

(Numerical data: From Lide, D.R., CRC Handbook of Chemistry and Physics, 72nd edn., CRC Press, Boca Raton, FL, 1991–1992, pp. 6-18–6-27.)

3.4  Isobaric Processes of Ideal Gases and Mayer’s Formula

Heating or cooling a gas while gas pressure is maintained constant (p = constant) is called the isobaric process. In the isobaric process, the pressure change is zero (dp = 0). Thus, heating of ideal gas raises the temperature and internal energy of the gas, which consequently causes a volumetric expansion of the gas. Since a part of the supplied heat is used for the gas expansion, the gas also performs a work. Accordingly, the first law of thermodynamics applied to the isobaric process leads to the equation

where C p is the molar heat capacity at constant pressure. The substitution of dU = C V dT and the ideal gas law applied to 1 kmol: pdV = R 0 dT into Equation 3.21 gives the well-known Mayer’s formula:
named after German physician and physicist, Julius Robert von Mayer. The Mayer’s formula gives the relationship between the heat capacities at constant pressure and constant volume. Accordingly, the molar heat capacity at constant volume of an arbitrary gas is smaller than the molar heat capacity at constant pressure by the value of the universal gas constant.

The division of the Mayer’s formula by the molar mass M a of the ideal gas yields

The expression comprises specific heat capacities at constant pressure and constant volume instead of molar heat capacities. Using the Mayer’s formula (3.22) and Equation 3.16, the molar heat capacity at constant pressure is transformed to
In view of that, the theoretical molar heats of ideal gases at constant pressure only depend on the freedom number of molecules, and they are temperature-independent. However, comparison of experimental and theoretical data of heat capacities at constant pressure indicates considerable discrepancies. The larger deviations exhibit gases with more complex molecular structures and gases with temperatures far from the room temperature. Therefore, the empirical data are approximated by an expansion series:
  • T is the absolute temperature in kelvins
  • C p is the molar heat capacity at constant pressure in kJ/(K kmol) units

The constants a, b, c, d can be tabulated. Approximations with three constants (C p  = a + bT + cT 2) are given in Table 3.1 for some gases. This polynomial approximation can be fitted to the empirical molar heat capacity at constant pressure (101,325 Pa) with accuracy of about 5% when temperature is in the range of 273–1500 K. The table also compares the data obtained by polynomial approximation and those acquired theoretically using Equation 3.24. Another polynomial approximation C p  = a + bT + cT 2 + dT 3 + eT 4 for many gases can be found in Reference 74.

Table 3.1   Constants for Polynomial Approximation 75–78 (Equation 3.25) of Molar Heat Capacities C p


C p  = a + bT + cT 2

Heat Capacity C p (kJ/kmol K)




273.15 K

500 K

1000 K




4.075 × 10−3

−14.9 × 10−7







6.97 × 10−3

−8.2 × 10−7







43.50 × 10−3

148.3 × 10−7







10.14 × 10−3

−2.72 × 10−7







−0.836 × 10−3

20.1 × 10−7



30. 244




1.82 × 10−3

15.5 × 10−7







75.50 × 10−3

−180 × 10−7







5.23 × 10−3

−0.04 × 10−7







12.98 × 10−3

−38.6 × 10−7







9.61 × 10−3

11.8 × 10−7




Considering the heat capacity at constant pressure to be temperature-independent, the change in gas internal energy of the extended system, enthalpy, due to the temperature increase from T 0 to T may be calculated at constant pressure using the following integration:


The isobaric process is presented by an abscissa called isobar, which is parallel to V axis in the pV diagram (Figure 3.3). The shaded area

represents the performed work by the gas at an isobaric process.

Work performed at isobaric process.

Figure 3.3   Work performed at isobaric process.

It can easily be found that the ratio of heat capacities

describing the properties of gases is a very useful quantity.

In Equation 3.28, i is the freedom number of molecules, R 0 is the universal gas constant, c p is the specific heat capacity at constant pressure, C p is the molar heat capacity at constant pressure, c V is the specific heat capacity at constant volume, and C V is the molar heat capacity at constant volume. Since the ratio of heat capacities is the function of the freedom number of molecules, this quantity is related to the molecular structure. When for monatomic gases the freedom number of molecules is i = 3, then the ratio of heat capacities yields

In accordance with the equipartition theorem (see Chapter 2.10), diatomic linear molecules have the number of degrees of freedom of i = 5, assuming the condition at which two degrees of freedom arising from stretch vibration (kinetic and potential energies) contribute to the total gas energy insignificantly. Hence, the ratio of heat capacities is
In the same condition, triatomic nonlinear molecules have the number of degrees of freedom of i = 6, which gives the ratio of heat capacities
We have already found that the experimental heat capacities differ from the theoretical values calculated using the principle of the energy partition. Now, we can compare the heat capacity ratio based on the freedom number and that determined from the specific heat capacities summarized in Table 3.2.

Table 3.2   Theoretical and Experimental Values of the Heat Capacity Ratio κ = c p /c V for Some Gases from Table 5.3

Monoatomic Molecules

Diatomic Molecules

Polyatomic Molecules

Theoretical κ = 1.666

Theoretical κ = 1.4

Theoretical κ = 1.333



T (K)



T (K)



T (K)
















































































Despite inconsistency in theoretical and experimental heat capacities (C V , C p ), the deviations of theoretical heat capacity ratios from experimental values are not as significant as indicated by heat capacities, which is one of the most surprising outcomes of the molecular kinetic theory of gases. Aside a large temperature variation, the heat capacity ratio of gases does not change significantly. For example, the heat capacity ratio of air changes from κ = 1.401 at –40 °C to κ = 1.321 at 1000 °C. 79 At 50 °C, κ = 1.4 and at 200 °C, κ = 1.399. The ratio of heat capacities weakly depends on the temperature. The specific heat capacities with weak dependences can still be approximated by polynomial fitting. 80 , 81

3.5  Isothermal Processes of Ideal Gases

In pumped vacuum systems, a gas volume expands, and as a result, the gas temperature may reduce. The variation of temperature should be taken into consideration at the calculation of state quantities. At low pressure, smaller than 10 Pa, the process of expanding gas can be considered as isothermal because the process is slow and gas molecules can exchange heat with surrounding walls and maintain practically the temperature of chamber walls constant. This process can be described on the thermodynamic principles at which both the temperature and internal energy of gas are constant, while the gas performs a mechanical work. Assuming the temperature and internal energy of a gas system to be constant, both the temperature differential and the differential of internal energy have to be zero (dT = 0; dU = C V dT = 0). This implies that at constant temperature the supplied heat to the gas from an external source entirely converts to mechanical work. In thermodynamics, this energy conversion describes the equality

However, at a constant temperature, when the gas is compressed, the entire work used for compression changes to heat, which is then transferred to the environment. In the isothermal process, an ideal gas accepts the heat, but it neither heats up nor alters its internal energy. Then, the question is, what is the heat capacity of the gas at these particular conditions? If in the isothermal process, the accepted heat is dQ, which is greater than zero, and temperature is constant, then from the definition of the heat capacity
Thus, the gas exhibits infinite heat capacity. In isothermal expansion or compression, the ideal gas law should be identical with the Boyle’s law pV = R 0 T = constant. As illustrated previously, the pV plot of the Boyle’s law in Figure 2.3 shows a hyperbolic isotherm. The isotherms for higher temperatures are above those for lower temperatures.

Since the heat supplied into the gas entirely changes to mechanical work, the work in the pV diagram is the area constrained by the volumetric change of the gas and the isotherm of p = R 0 T/V (see Figure 3.4). Then, the differential work carried out by the gas is

Since at the initial volume V 0, the work is equal to zero, at the gas expansion from the volume V 0 to the volume V, the work performed by the gas is given by the integral
Introduction of V = R 0 T/p and V 0 = R 0 T/p 0 into Equation 3.36 yields an alternative equation:
which is the work performed by an ideal gas with a gas quantity of 1 kmol (mol) in isothermal expansion. Then, for n m = M/M a kmol of the ideal gas, the reversible isothermal work is
The derived equation represents the maximal work, which is proportional to the volumetric ratio of an ideal gas in its final and initial states, while its temperature is maintained constant. This isothermal work can alternatively be determined from the relative change of initial and final pressures. In a reversible process, both the alternative equations show that the isothermal work is the functionality of relative volumetric or pressure changes, and the absolute values of volumetric and pressure parameters do not matter because the same ratio may yield from different initial and final volumetric and pressure variables. Example 3.5 illustrates that this work has to be distinguished from the work at an irreversible process.

Work performed at an isothermal process.

Figure 3.4   Work performed at an isothermal process.

3.6  Adiabatic Processes of Ideal Gases

At higher pressures in vacuum systems, the expansion of a gas can be so fast that molecules and wall cannot equalize their temperatures. The process at which the transfer of energy between the walls and molecules is negligible is the adiabatic process. Strictly speaking, the adiabatic process is a thermally isolated process, at which a gas neither accepts heat from environment nor gives to it. Accordingly, the heat change of the gas is


Since the system is thermally isolated, work performed by the gas system is to the detriment of internal energy. If the gas system performs the positive work, then the gas expands and the change in internal energy is negative, indicating reduction of the gas temperature. Conversely, if the gas is compressed, the gas performs negative work, while the change of the gas internal energy is positive and the gas is heated.

It should be noted that like ideal gas, the adiabatic process is ideal. It can be imitated using a very good thermal insulation. The best thermal insulator is high vacuum, but vacuum does not restrict heat transfer by radiation. The transport of energy by radiation is reduced by special preparation of the wall surfaces between which the heat transfer is undesirable. The surfaces of the walls are polished and coated by radiation-reflective films.

The phenomenon of adiabatic expansion is used in cryogenic pumps with closed helium circuits, where temperatures as low as 15–6 K are induced upon helium gas expansion. Another example of the adiabatic expansion is a phase liquefaction of gases (see Chapter 4.7). However, the heat developed in the adiabatic compression ignites fuel in diesel engines.

Based on the definition of heat capacities, in the adiabatic process, the gas behaves as a substance with zero heat capacity (dQ = 0 at dT ≠ 0).

Now, let us examine the p = f(V) function at adiabatic process, called the adiabat. Figure 3.5 shows the courses of both calculated adiabat and calculated isotherm. The adiabat, seen in Figure 3.5, differs from the isotherm with its steeper course. For a single kmol of an ideal gas undergoing an isothermal process, it is valid that the gas pressure is p = R 0 T/V and the product of temperature T and the universal gas constant R 0 is invariable.

Comparison of the adiabat and isotherm for 1

Figure 3.5   Comparison of the adiabat and isotherm for 1 mol of an ideal gas at 300 K.

However, different behavior is observed at adiabatic processes. The value of R 0 T decreases as the volume increases from V 0 to V, and it increases with a reduction in the gas volume. This implies that for V > V 0, the adiabat has to be below the isotherm and for V < V 0, the adiabat has to be above the isotherm. This analysis indicates that in adiabatic processes, three variables, pressure p, volume V, and temperature T, have to be taken into account, contrasting isothermal processes at which only pressure p and volume V are variable, while temperature T is maintained constant.

The analytical form of the adiabat can be derived from the first thermodynamic law applied to the adiabatic process at which the heat change is equal to zero. Assuming dQ = 0, the first thermodynamic law is

For the derivation of analytical expression of the adiabat, we exclude the variable T from Equation 3.40 using the ideal gas law applied to a gas quantity of 1 kmol (pV = R 0 T) in the differential form
Then, the substitution of Equation 3.41 into Equation 3.40 gives
The performed substitution is only permitted for the description of gas systems with controlled expansion satisfying the conditions of reversible processes. This means that at any point of expansion, the external and internal pressures differ by an infinitesimal value.

Further recasting the last equation leads to

Since C V + R 0 = C p , the equation can be transformed to
The integration of Equation 3.44 and substitution of the heat ratio κ = C p /C V gives
from where the final analytical form of the adiabat affiliated with reversible process is
because V 0 = M0 and V = M/ρ, where the symbols with zero indices (p 0—pressure, V 0—volume, ρ0—mass density) denote the quantities in an initial state and those without indices correspond to parameters in the final state. We may write ideal equations for the final and initial gas states as and , then their ratio gives . Since for adiabatic equation it is valid , for temperature ratio due to adiabatic process, we may write

Similarly, from the ideal gas equation for the final and initial gas state we can write

and , and for their ratio . Substitution of this pressure ratio into gives and then by a couple of algebraic operations, we may obtain . Accordingly, in the adiabatic process, the relative temperature change is expressed by the equations
The adiabatic equation supplements the ideal gas law and it can be applied only at the adiabatic conditions. However, the ideal gas law is valid at any condition under which the ideal gas is defined.

Since the ratio of heat capacities κ = C p /C V = (i + 2)/i > 1, the courses of adiabats have to be steeper than the courses of isotherms. (C p and C V are molar capacities at constant pressure and volume and i is the number of freedom degrees.)

Using the ideal gas law for 1 kmol (p = R 0 T/V), the equation of the adiabat may also be written in other forms.

At adiabatic reversible process, the work in the differential form is given by Equation 3.39, that is, dW = –C V dT, from where the integral value of the work performed by the gas system is

due to the temperature changes from initial temperature T 0 to T. The integral value of the work performed at adiabatic process is then
which is equal to lowering the content of gas energy. It represents the maximum work carried out at the reversible adiabatic expansion. Since for a gas quantity of 1 kmol, it is valid
the temperatures T and T 0 may be excluded from Equation 3.49 as follows:
the gas work at an adiabatic process can be rewritten to
The final modification of the analytical expression for the work at adiabatic process may be done by excluding pressure via introduction of the equation of the adiabat
as follows:
Hence, the final equation for the work performed by an ideal gas at an adiabatic reversible process is
  • p 0 is the initial pressure
  • V 0 is the initial volume
  • V is the final volume
  • κ is the heat capacity ratio at constant pressure and volume, respectively

The last equation represents the maximum work performed by a gas system. The maximum work of the gas system can be obtained when any change in the system is fully reversible.

3.7  Polytropic Processes of Ideal Gases

In practice, the heat exchange between a gas and the walls of a vacuum system may occur. Both the isothermal process, given by Equation 2.31, and the adiabatic process, given by Equation 3.46, refer to two limiting ideal cases that are characteristic of the behavior of ideal gases. The general process, however, takes place between these boundary cases whenever the heat exchange between the gas and the container enclosure is incomplete. The processes between the two boundary cases are known as the polytropic processes, and they are described by the equation of polytrops

where the exponent of the polytrop is
The exponent of the polytrop can attain values from 1, corresponding to isothermal process, to κ referring to adiabatic process (1 < χ < κ). In the exponent χ, the capital C denotes the molar heat at a polytropic process.

Up to now, all the processes discussed above are polytropic. The polytropic process is a general process. It can be shown that, in certain ideal cases, the polytropic process can be coincident with the isochoric, isobaric, isothermal, and adiabatic processes. Simple analyses of polytropic process can demonstrate that the isochoric, isobaric, isothermal, and adiabatic processes are boundary cases of the polytrophic process.

  1. Polytropic—isochoric process: At this analysis, first we recast equation of polytrops (3.57) to
    This equation can further be transformed to an equation describing isochoric process, if the molar capacity of polytrop C is equal to molar capacity C V at constant volume (C = C V ). At this condition, the exponent χ → ∞. Thus, for C = C V , the polytrop is pV 0 = const, which is equal to the isochor V = const.
  2. Polytropic—isobaric process: The equation of polytrop p V χ = const can be transformed to isobar p = const, when the exponent of polytrop χ = 0, that is, when the polytrop pV 0 = const. The exponent of the polytrop (Equation 3.58) is zero when the molar capacity of polytrop C is equal to the molar capacity C p at constant pressure (C = C p ).
  3. Polytropic—isothermal process: Isothermal process is the limiting case of the polytropic process, which can be illustrated when the exponent of polytrop (Equation 3.58) is transformed to
    Hence, when the heat capacity C → ∞, the ratios of heat capacities C p /C and C V /C tend to zero and thus the exponent of polytrop χ = 1. Then, the equation of the polytrop is identical with the equation of the isotherm pV = const.
  4. Polytropic—adiabatic process: The adiabatic process is the limiting case of the polytropic process. This statement can easily be proved becuase at the adiabatic process, the heat capacity is C = 0. Accordingly, the exponent of the polytrop is equal to the heat capacity ratio (χ = κ), so that the equation of the polytrop turns to the equation of the adiabat pV κ = const.

3.8  Measurements of Gas Heat Capacities

Heat capacities are measured in calorimeters. Calorimetric measurements enable us to determine reaction heat, enthalpy, and entropy. The calorimeter consists of a thermally isolated container filled with water in which there is a processing chamber. If temperature increases in the processing chamber, then the heat is transferred to the water. The temperature increase can then be recorded. Knowledge of the mass of the water and its specific heat capacity and the recorded increase of temperature allow us to determine the heat transferred to the processing chamber.

The specific heat capacity of gases is conveniently measured at constant pressure 82 , 83 (c p ), while the specific heat capacity (c V ) of solids and liquids is determined at constant volume. A measurement principle of the specific heat capacity at constant pressure is illustrated in Figure 3.6. At a preset pressure, a gas can flow via the measurement system. First, the gas is thermally stabilized at temperature T 0 in a gas container 10. When the valve 2 is opened, the flow meter 9 measures a constant gas flow Q′. The assembly comprises a calibrated capillary 4 with two absolute capacitance gauges (3 and 5) placed at each end of the capillary to serve as a flow meter. The measured gas throughput Q′ passes via a copper tube coil located in a water thermostat 6, which provides a constant gas temperature T. Then, the gas enters the second copper coil immersed in a known amount of thermally insulated water in calorimetric container 8. The heat transferred from the flowing gas into water increases the water temperature, which is measured. The gas cools down to the temperature at which it exits at constant pressure, for instance, to the standard atmosphere. If the temperature change of water in the calorimetric container 8, gas mass M that passed via the calorimeter for the measured time, mass of water, and individual masses and the specific heat capacities of all components, from which calorimeter is composed, are known, then the following equation

enables us to calculate the heat capacity of a gas at constant pressure. In Equation 3.61, total mass
includes the mass of water M W and masses M i of all components from which the calorimeter is composed. Thus, specific heat capacities c W of water and specific heat capacities c i of the corresponding material components of the calorimeter have to be known because all of them absorb heat. In summary, when the gas mass M passed via the calorimeter, temperature T of the flowing gas, initial temperature T 1, and final temperature T 2 in the calorimeter are measured at constant pressure, then the specific heat capacity of the gas at constant pressure c p can be determined from Equation 3.61.

Measurement of heat capacities of gases at constant pressures.

Figure 3.6   Measurement of heat capacities of gases at constant pressures.

3.9  Measurement of the Heat Capacity Ratio

A number of articles illustrate measurements of heat capacity ratios. 84 , 85 The principle of the measurement of the heat capacity ratio can be explained with a simple gas system. A gas is initially confined in a container with higher pressure than atmospheric pressure. This gas system, with absolutely determined pressures, is in a thermal equilibrium with the surrounding atmosphere. When a gate valve that is mounted on the gas container is opened suddenly, the gas flows from the confined volume to surroundings with atmospheric pressure. The pressure equilibrium occurs, but the temperature of the expanding gas alters. When pressures equalize, the gate valve is closed. Thus, the internal and external pressures are equal, but temperature of the confined gas differs from that of surroundings. The confined gas starts to exchange heat with surroundings, which causes the pressure drop in the container. So, we observe two characteristic steps. Firstly, the confined gas, in the volume V 1, suddenly expands from the initial pressure p 1 to the atmospheric pressure p a and it changes its temperature when the gate valve is opened. This is consistent with the adiabatic expansion described by the equation of the adiabat,

. Secondly, the separation step of the residual gas is followed by the heat exchange between the confined residual gas and surroundings. When the gas temperature stabilizes at temperature of surroundings, the gas pressure also attains a constant value p 2. The pressure p 2 can also be reached when the gas expansion from the initial pressure p 1 is carried out very slowly without the temperature change. In this case, Boyle’s law (p 1 V 1 = p 2 V) can be applied. Excluding volume V from these two equations leads to the equation (p 1/p 2)κ = p 1/p a from where the heat capacity ratio can be calculated on a base of simple absolute pressure measurements.

The other method to determine the heat capacity ratio is the method of the speed of sound in a gas environment. 86 In a homogeneous environment, the speed of sound is given by elastic properties of a material, namely bulk modulus and mass density, which are specifically described by equation

where mass density is
and elastic bulk modulus is
In an environment of an ideal gas, the propagating sound waves cause very fast compressions and expansions at which heat can barely be exchanged. Therefore, the volume and pressure changes are considered to be adiabatic, and the adiabatic process is described by equation V κ p = const, where κ is the ratio of thermal heat capacities, which is about 1.4 for air. Because of fast changes, the equation of the adiabat can be recast to p = const ⋅ V −κ. Hence,
The substitution of mass density given by Equation 3.64 and elastic bulk modulus given by Equation 3.66 into Equation 3.63 yields
from where speed of sound is
where we substitute R 0 T = pV/n m from the ideal gas equation.

Since speed of sound, u s , is a measurable parameter, the ratio of thermal heat capacity can be determined from equation

  • M a is the molar mass of the gas of interest
  • R 0 is the universal gas constant
  • T is the absolute gas temperature
  • M a /R 0 T = ρ/p is the mass density (see Equation 2.24) at pressure of unity
It should be noted that the last equation refers to ideal gases.

In early years, the measurement of the heat capacity ratio led to conclusions that noble gases do not combine with any other elements. In measurements of the heat capacity ratio of mercury vapors it was discovered that the mercury vapors consist only of monatomic molecules. Thus, the measurement of the heat capacity ratio points at the molecular structure.

The simplest kinetic theory predicts the heat capacity ratios being remarkably consistent with empirical data of many gases at near room temperatures and some of them even at higher temperature. Deviations from this simple mechanistic theory are observed for gases with more complex molecular structures and gases with molecules formed of atoms with less rigid bonding. For illustration, the speed of sound in some gases at standard conditions is given in Table 3.3.

Table 3.3   Speed of Sound in Some Gases at the STP Conditions (273.15 K, 101,325 Pa)



















u s (m/s)









u s (m/s)









3.10  The Second Law of Thermodynamics Applied to Ideal Gases

Prior to the discussion of the second law of thermodynamics applied to ideal gases, it is useful to highlight the uniqueness of reversible and irreversible processes that can be carried out in ideal gases. Irreversible processes are those in which gas systems pass from one state to other, but the gas system cannot be returned to its initial state unless mechanical energy is supplied to the gas system. For irreversible process at low pressure, a good example is two vacuum systems with different pressures isolated by a gate valve. Opening the gate valve leads to pressure equalization. This process is irreversible since there is no spontaneous process, without supplying mechanical work, which can return the systems to their initial states. However, reversible processes are those that allow transition of physical systems from their initial states to other states and back, either directly or via other interstate transitions.

We may presume a thermodynamic gas system that performs a reversible process, passing through its initial state in cycles. A gas system may perform positive work to the detriment of the external energy, which is supplied into the system. However, in consistence with the second law of thermodynamics and reversibility of the process, there has to be a method at which transition in the reverse direction is induced due to the heat transfer from the gas system to an environment or a cooler. Accordingly, we could imagine a reversible cycle process in a system comprising a heater and a cooler with defined temperatures. Both the heat transfer at heater and cooler sides are two individual isothermal processes between which there is a transition that can be bridged by adiabatic processes. Thus, two isothermal and adiabatic processes provide a loop transition, which is graphically presented by isotherms and adiabats. A cyclic process with two isothermal and two adiabatic processes, originally investigated by French physicist and military engineer Sadi Nicolas Leonard Carnot, is called the Carnot cycle. The Carnot cycle, in the pV diagram (Figure 3.7), indicates the following elementary processes:

  1. The isothermal expansion from a volume of V A to a volume of V B at the heater temperature T H , which is associated with performing positive work W AB by an ideal gas. This work is given by the area ABB′A′.
  2. The adiabatic expansion from the volume V B to the volume V C , inducing a decrease of temperature from a value of the heater temperature T H to a value of the cooler temperature T C . In this process, the gas performs the work W BC = C V ′(T H – T C ), which is given by the area BCC′B′.
  3. The subsequent isothermal compression of the gas from the volume V C to the volume V D at the cooler temperature T C at which gas performs negative work W CD is given by the area CDD′C′.
  4. The final adiabatic compression from the volume V D to the volume V A at which temperature increases from the value T C to the value T A and the gas performs the negative work W DA = (T C – T H ) corresponds to the area DAA′D′.

Carnot cycle of an ideal gas.

Figure 3.7   Carnot cycle of an ideal gas.

The efficiency of the heat system is then the ratio

of the work W performed and total heat Q H supplied to the gas from the heater. A part of the heat is supplied to perform the mechanical work by gas, while the rest heat is delivered to the cooler. Since
the efficiency of the heat system is the ratio of the performed work and supplied heat

The efficiency of the Carnot cycle is given by the Equation 3.70, where W is the total work performed by the gas at the Carnot cycle and Q H is the heat accepted from the heater at temperature T H , at the isothermal expansion. The heat accepted from the heater is equal to the performed work by the gas system (Q H = W AB ).

Then, the total work is

If for two adiabatic processes it is valid
and the negative work W CD is rewritten to W CD  =  − W DC , then the total work is
Introducing Equation 3.75 and Q H = W AB into Equation 3.70 gives the Carnot efficiency
It can be proved that
Since the ratio of the heats Q H /Q C is directly proportional to the ratio of temperatures T H /T C , the efficiency given by Equation 3.72 can be transformed to

Accordingly, the Carnot heat efficiency is only a function of the heater and cooler temperatures. Efficiency of the Carnot cycle has important implication for the theory of equilibrium of gases and vapors with other phases and derivation of the Clausius–Clapeyron equation for determination of saturated vapor pressures and evaporation rates of materials.

It should be noted that the empirical efficiency of real heat cycles is lower than that of the theoretical Carnot cycle because the efficiency of the Carnot cycle is derived for ideal gas systems and ideal adiabatic and isothermal processes. More details on the reversible Carnot cycles can be found in the books devoted to thermodynamics. 87–89

3.11  Entropy of Gas Systems

If the heat accepted by a gas is positive and the heat delivered by a gas to the environment is negative, then on the base of the Carnot cycle, the heat ratio is equal to the ratio of corresponding temperatures, that is,

from where
After the cycle process, the sum of the arbitrary quantity is equal to zero, implying that the final and initial values are equal. The quantities with such properties are state functions. This simple analysis indicates the existence of a state function quantity whose change is given by the expression dQ/T. This quantity is called entropy, denoted by the symbol S in thermodynamics and herein. Entropy is the quantity unavailable for mechanical work. The term “entropy” comes from a Greek word that means “to give a direction.” It was first introduced by German physicist Rudolf Julius Emanuel Clausius in 1850. The entropy increase is given by equation
When the system passes from the state A to the state B, the total change in entropy is
In the case of the Carnot cycle, the total change in entropy is
which means that in reversible processes, the energies of the isolated systems do not change.

However, entropy increases in irreversible processes. Entropy of an isolated system generally only increases or remains the same, S B  ≥ S A , which is the common statement of the second law of thermodynamics. Obviously, the change in entropy is equal to the change in heat per unit of temperature and hence its unit is joule per kelvin (J/K).

The mathematical expression for entropy enables us to find only the entropy changes in different processes, but it does not allow us to determine the absolute values of entropy. The substitution of a state change of the ideal gas formulated by the first principle of thermodynamics dQ = dU + pdV = C V dT + n m R 0 TdV/V into Equation 3.81 yields the following change of entropy:

Assuming heat capacity C V independent of the temperature, integration of the last equation gives the change of entropy in the form
Thus, for the isothermal expansion, when T = constant, the entropy change is
and for the isochoric process when V = constant, the simplified expression of the change in entropy is
which follows from Equation 3.85.

Based on the study of heat capacities at low temperatures, Walter Nernst (1906) revealed the third law (also called zero law) of thermodynamics. However, a satisfactory formulation of the third law of thermodynamics is given in a later book Thermodynamics and Energy of Substances (1923), 90 where the statement on entropy is as follows: If entropy of a single crystal element is taken to be zero at 0 K, then entropy of each substance will have a final positive value. This statement, on entropy equal to zero at 0 K, is met for perfect crystalline substances.

This way the formulated third law of thermodynamics allows us to find the integration constant in the equation for entropy on the base of boundary conditions at which entropy of a homogeneous substance approaches zero if temperature goes to the absolute zero (0 K).

3.12  Thermodynamic Free Energy

Intrinsic processes in material systems drive these systems to their equilibrium states. One of these equilibrium states may attain the minimal internal energy and the other may reach the maximal entropy. The maximal entropy is reached when the internal energy is constant. However, the minimal internal energy can be obtained when the entropy is constant.

In practice, the systems are usually studied at constant temperature or pressure. For all cases where processes are investigated at state variables of volume, temperature, and other quantities needed for determination of composition, it is very appropriate to use a state function called the Helmholtz free energy defined by formula

  • U is the internal energy, which could be taken as the energy required to form the considered system
  • T is the absolute temperature
  • S is the final entropy
The formula indicates that for the system formation there is not necessarily needed energy equal to the internal energy U because a part of energy TS can be transferred from environment with temperature T in a spontaneous process to form a system. Helmholtz free energy is named after German physicist and physician, Hermann Ludwig Ferdinand von Helmholtz.

The introduction of the first law of thermodynamics, expressing the change of the internal energy (dU = dQ + dW) into the total differential of the Helmholtz free energy, yields

If the system can only exchange the volumetric work with surroundings, then the Helmholtz free energy is
At constant temperature and volume, Helmholtz free energy is dA h = dQSdT, which means that for any change of the independent state variables it is valid that dA h ≤ 0, where dQTdS considering both reversible and irreversible processes. (The sign of equality corresponds to reversible processes.) At equilibrium, constant temperature and constant volume and at the zero exchange of work, the Helmholtz free energy reaches minimum.

For studying processes in a system defined by temperature, pressure, and other variables corresponding to composition, we use the state function

called Gibbs free energy, originally introduced by American mathematical-engineer, theoretical physicist and chemist, Josiah Willard Gibbs. Again, U is the internal energy needed for the system formation if there are no temperature and volumetric changes, S is the final entropy, H is the enthalpy, p is the pressure, V is the final volume, and T is the absolute temperature. However, whenever the system is formed from a small volume, there is needed additional energy pV to perform work in consistence with the content of enthalpy.

If the first thermodynamic law is substituted into the total differential of the Gibbs free energy

at constant pressure and temperature, then the Gibbs free energy can be written in the form
Since for the reversible process dQ = TdS and for irreversible process dQ < TdS, then for any change of an independent variable of the system, the change in Gibbs energy has to be smaller or equal to zero (dG ≤ 0), specifically at constant temperature and pressure and in a case when only the volumetric work is permitted to exchange.

At equilibrium, the Gibbs free energy is at its minimum and system is characterized by constant temperature and pressure and conditions at which only volumetric work is permitted to exchange. The Gibbs free energy is a vital quantity that explains the transformation of one substance phase to another occurring at constant pressure and temperature. Wherefore, the change of a thermodynamic function of the Gibbs free energy is

The substitution of the change of the free energy dA h = –pdVSdT into Equation 3.94 gives the change of the Gibbs free energy as follows:
At phase transition of matter, the differentials of pressure p and temperature T are dp = 0, dT = 0, and thus differential of Gibb free energy dG = 0, which means that the thermodynamic potentials of the initial and final states are equal at the phase transition.

The Helmholtz and Gibbs free energies are in fact thermodynamic potentials if they are given as functions of independent state variables A h (V,T) and G(p,T). Then, the gradient of these functions is the driving force of physical and chemical processes.

3.13  Thermodynamic Equilibrium of Gases with Their Other Phases

Each physically, chemically, or mechanically resolved part of the system represents an independent phase. Matters can thus exist in different phases, that is, solid, liquid, or gas, depending on the system variables. Gas mixture is, however, single phase because gases can be mixed in any ratios. The system with more phases is called a heterogeneous system. For instance, water that contains ice pieces is a two-phase system.

For a description of a phase system in its thermodynamic equilibrium with other phases, we need some numerical values of chosen variables such as pressure, temperature, volume, and the density of individual phase components. The important feature of equilibrium between phases is that all the phases are independent of the substance quantities of components in any heterogeneous system. For example, pressure of water vapor above the surface of water does not depend on the volume vessel where water is placed.

Behaviors of gases and vapors in thermodynamic equilibrium with their other phases describe phase diagrams. For description of thermodynamic equilibrium between different phases, water is a suitable example because of our extensive experience with this substance.

The pT plot in Figure 3.8 is the phase diagram of water. Note, the diagram displays behavior of water and its other phases at vacuum conditions, that is, below 105 Pa. The curve (a) is the line of fusion representing the thermodynamic equilibrium between the solid (ice) and liquid phases (water). The line (a) is plotted at pressure below the standard pressure (<101,325 Pa). Taking into account that the freezing point of water at atmospheric pressure is at 273.15 K, the line (a) between the freezing point (273.15 K) at atmospheric pressure and the denoted triple point at 273.16 K and 611.73 Pa, in Figure 3.8, is nearly parallel with the pressure abscissa up to the atmospheric pressure. If the system is in states corresponding to the line (a), any change in temperature or pressure leads to transition of either the ice to water or water to ice. The curve (b) is the boiling line comprising pairs temperature/pressure points at which water and water vapor are in thermodynamic equilibrium. The water vapor, existing at saturated pressure p (see Figure 3.8) in thermodynamic equilibrium with its liquid phase, passes to gas phase (vapor) when the pressure decreases below the value p and vice versa. The last line (c) is called sublimation line, where both solid and gas coexist in equilibrium. Any change in pressure leads to the transition from the solid to the gas phase when pressure is reduced or vice versa. Obviously, ice may rapidly sublimate in vacuum. This phenomenon is widely used in food and pharmaceutical industries. First food is frozen and then water sublimates under vacuum conditions to last longer and preserve its nutrition values. The lines (a), (b), and (c) meet in a single point called triple point. The triple point of water is at 611.73 Pa and 273.16 K. In the triple point, all three phases are in a state of thermodynamic equilibrium. The state of the system is stable and time-independent, which means that none of the three phases increases to the detriment of others. At given temperature, the transition of substance from one phase to another takes place at certain pressure called saturated vapor pressure. For instance, if a liquid is located in an enclosed container, it evaporates, but its pressure does not increase above the value of saturated vapor pressure at an invariable temperature. The liquid is in a thermodynamic equilibrium with its vapor. The evaporation and condensation rates are equal.

Phase diagram of water at vacuum conditions.

Figure 3.8   Phase diagram of water at vacuum conditions.

We are familiar with heating and cooling water when it is exposed to vacuum. When water in an opened container is loaded into a vacuum chamber and the chamber is pumped down from atmospheric pressure, water starts to release absorbed gases while it intensively evaporates, as observed at the boiling point given at atmospheric pressure. This process leads first to an increase of the water temperature. The container is indeed hot. When pumping continues, water “boiling” stops and water cools down due to the heat loss carried away by rapidly evaporated water molecules, while the heat flow back via vacuum is poor. The intense cooling effect finally causes water to turn into ice, from which water molecules continue to escape by sublimation. When rapid pumping starts from atmospheric pressure and the chamber is well illuminated, an “explosive” evolution of a white fume from the internal surface can be observed. The fume quickly diminishes as the major amount of adsorbed water is removed by the pumping action.

The saturated vapor pressure can be determined on thermodynamic principles derived in Chapter 3.10. Using the second law of thermodynamics, the relation between the saturated vapor pressure and temperature of a substance can be found from a Carnot cycle, vaporization heat, and volumetric change of the substance at vaporization. For simplicity, consider water with an amount of 1 kmol in the thermodynamic state (1) in Figure 3.9 to derive phase transition and saturated vapor pressure of water. However, the derivation and its outcome may describe many substances including sublimating materials. Vaporization of 1 kmol of water at temperature T, from the initial state (1) to the state (2), requires vaporization heat Q H equal to the enthalpy change ΔH of 1 kmol water. At this isothermal phase transition from liquid to vapor phase, the water volume increase is ΔV = V G V L , where V G is the volume of water vapor and V L is the volume of water in liquid phase.

Carnot cycle for the derivation of Clapeyron equation.

Figure 3.9   Carnot cycle for the derivation of Clapeyron equation.

The subsequent adiabatic expansion causes dropping of both pressure and temperature by dp and dT, respectively. In the Carnot diagram (Figure 3.9), the adiabatic expansion represents the thermodynamic transition of water from the state (2) to the state (3).

Then, we proceed with liquefaction of water vapor by an isothermal compression, which is transition from the state (3) to the state (4), at temperature TdT in such a way that the subsequent adiabatic compression returns the water to its initial state (1).

Now, using previously derived Equations 3.70 and 3.78, we can analytically express the efficiency of the Carnot cycle:

where W is the work performed by the system while Q H is the heat supplied to the system, which is equal to the change of enthalpy ΔH, at the phase transition. The work performed during a single Carnot cycle is given by the area of the Carnot loop, which is dW = (V G – V L )dp = ΔVdp. Since in this case T H = T and T C = T – dT, the efficiency of Carnot cycle is
which can be recast to the well-known Clausius–Clapeyron equation, 91 also called Clapeyron equation
The Clausius–Clapeyron equation is named after German mathematician and physicist Rudolf Julius Emanuel Clausius and French engineer, Benoit Paul-Emile Clapeyron. At temperature considerably higher than the critical temperature, the volume of the gas phase is much greater than that in its liquid phase. For example, at a temperature of 100 °C, the volume per kmol of water vapor V G is 1606.73 times greater than that for liquid water phase V L . Thus, the liquid volume may be neglected. Hence, in the Clausius–Clapeyron equation, the volume difference is simplified to ΔV = V G – V L V G ; (V G V L ). Assuming that vaporized water of 1 kmol behaves as an ideal gas, the volumetric increase is ΔVV G = R 0 T/p. Then, substitution for the volumetric change ΔV in the Clausius–Clapeyron equation gives
  • p is the saturated vapor pressure
  • T is the vapor temperature
  • R 0 is the universal gas constant
  • ΔH is the change in enthalpy, vaporization heat, which is assumed to be temperature-​ independent
Then, integration of Equation 3.99 yields
If we denote A = ln (const) = ln(D′) and B = ΔH/R 0, then the natural logarithm of saturated vapor is
where A and B are constants that are characteristic of particular materials. The constants A and B have been found for many materials empirically. Obviously, the measurement of pressures p of a substance at two different temperatures T enables us to determine constants A and B. Since the constant B = ΔH/R 0, the change in enthalpy, that is, evaporation heat, can be determined too.

The Clausius–Clapeyron equation is derived from thermodynamic principles for reversible processes and a relatively narrow temperature range. Theoretically, this equation permits us to determine the change in enthalpy referring to a substance amount of kmol (mol) based on the measurements of the temperatures and corresponding vapor pressures in assumption that the enthalpy is temperature-independent 92 in the given range of temperatures. Below ~100 Pa, vapor pressures can be calculated using the derived Equation 3.101.

The constants A and B are in Table 3.4 for single chemical elements, while the saturated vapor pressures plotted against temperature are in Figure 3.10 for some chemical elements. The figure shows that the saturation vapor pressures of zinc and cadmium are on the orders of 10–3 and 10–1 Pa at 500 K, respectively. Thus, using zinc- and cadmium-plated components such as screws and washers are not recommended in vacuum application. In addition to vacuum contamination, these materials can be evaporated by a relatively small increase in temperature, and deposited in undesirable areas in vacuum systems.

Table 3.4   The Constants A and B for Calculation of Saturated Vapor Pressures of Single Element Materials with Aid of Equation 3.101


Constants in Equation 3.101


Constants in Equation 3.101












































































































































Saturated vapor pressures of some pure chemical elements in dependence on temperature.

Figure 3.10   Saturated vapor pressures of some pure chemical elements in dependence on temperature.

Equation 3.101 is valid for the conditions as assumed at derivation of the Clausius–Clapeyron equation. Thus, the illustrated conclusions can be applied to various substances and defined temperature ranges in which the enthalpy change (vaporization heat) is temperature-invariable. However, in many cases, the vaporization heat is temperature-dependent. The simplest case is when the vaporization heat is a linear function of temperature ΔH = ΔH 0 + cTH 0 is the evaporation heat at T = 0 and c is a constant characteristic of a vaporized material). Then, substitution of this linear function of temperature for the vaporization heat in Equation 3.99 gives

from where by integration, natural logarithm of saturated vapor pressure is
or the saturated vapor pressure is
Thus, three constants A = ln D′, B = ΔH 0/R 0, and C = c/R 0 are determined from experimental measurements of saturated pressures at three different temperatures at least. The measurement at larger numbers of temperatures and averaging leads to more accurate data.

The saturated vapor pressures are also discussed in articles by Flatau et al. 94 and Rasmussen. 95

An equation for calculation of vapor pressure of metallic materials in the pressure range from 10–10 to 100 Pa is also given by Alcock et al. 96 However, the equation contains five fitting constants.

3.14  Equilibrium of Gaseous Phases from Kinetic Theory of Gases

The equation for saturated vapor pressure above is derived based on the concept of thermodynamic equilibrium of vapors with their other phases using macroscopic thermodynamic parameters. Similar equation can be obtained using Maxwell–Boltzmann distribution of molecules according to velocities or energies.

Particles in liquids or solids are held together by bonding forces. On average, individual particles have energy proportional to the material temperature T. Unlike the forces inside the material, the bounding forces on the surface interface are unbalanced. The surface particles form fields of surface forces with a range of a couple of monolayers.

When the energy distribution of particles at the solid–vapor interface describes Maxwell–Boltzmann distribution, some particles may have velocity component v y in the direction of the surface normal and energies exceeding a threshold velocity u and energy w to escape from the field of the attractive surface forces. Then, the dn s (v y ) fraction from the total number of particles n s of the solid may have velocities (energies) from the range of v y to v y + dv y , which is expressed by a one-dimensional Maxwell–Boltzmann distribution function

; (see Equation 2.103). The number of surface particles that may escape from a unit area per second with velocities from the interval of v x to v x + dv x is
Hence, the flux density of particles with velocities greater than the escape velocity u is
By a couple of arithmetic operations and substitution of average thermal velocity v s  = (8kTm)1/2 we obtain the particle flux density leaving the surface in the form
  • R 0 is the universal gas constant
  • The symbol w denotes the escape energy (vaporization energy) for a single particle
  • W = w N a is the vaporization energy referring to the particles with an amount of kmol

However, the vapors above the surface may also condense on the surfaces of liquids or solids. If the volume density of vapor molecules is n v and the average thermal velocity is v v , then the flux density impinging on a unit surface of solid is

and in a dynamic equilibrium, the two flux densities should be equal (Φ s  = Φ v ), that is,
However, a thermodynamic equilibrium state is also characterized by momentum equilibrium and energy equilibrium of evaporating and condensing particles. The equilibria for momentum and energy are respectively expressed by equations
where masses m of individual molecules and constants 1/4 and 1/2 are canceled in the last two equations. These equations can simultaneously be satisfied only when velocities v s and v v are equal (v s = v v ), which means that at the interface, molecules vaporize and condense with the same velocity. Accordingly, the vapor density is
Introduction of n v = p v kT yields vapor pressure
which is indeed similar to Equation 3.100, being exponentially dependent on vaporization heat (energy), or change in enthalpy, ΔH, following from the thermodynamic principles and the Clausius–Clapeyron equation.

The derivation leading to Equation 3.112 has a greater impact, since it can be applied elsewhere, for example, to plasma processes.

3.15  Saturated Vapor Pressure of Some Materials Used in Vacuum Technology

All solid and liquid materials have vapor pressures above their surfaces and they can be converted entirely to vapors by increasing their temperature. Matters that are in gas phases at standard conditions can be liquefied or solidified by reducing temperature and/or increasing their pressure. Particularly, unstable gases and vapors can be converted into liquid or solid forms at low temperatures; for example, carbon dioxide, CO2, can be in a solid form at –78.5 °C. At production of low pressures using low temperatures, the saturated vapor pressures of common gases are limiting factors of the ultimate pressure in vacuum systems. For reference, the saturated pressures of common gases are plotted against the temperature in Figure 3.11, using numerical data from Honig and Hook. 97 The boiling and melting points of these gases compiled from individual gas properties are in Table 3.5, and can be compared with those by Lide (1991–1992). 98

Saturated vapor pressure of common gases at low temperatures, plotted from the numerical data by Honing and Hock (1960).

Figure 3.11   Saturated vapor pressure of common gases at low temperatures, plotted from the numerical data by Honing and Hock (1960).

Table 3.5   Melting Points (m.p.) and Boiling Points (b.p.) of Common Gases Abstracted from Individual Properties of Chemical Elements and Gases













m.p. (K)












b.p. (K)












Organic solvents are often used for cleaning material surfaces in vacuum applications, and some of them are employed in leak testing on vacuum systems. Most of these solvents have saturated vapor pressures from units to tens of kPa at 20 °C. For example, at 20 °C, vapor pressures are as follows: isopropyl alcohol—4.1 kPa, ethanol—5.85 kPa, methanol—12.93 kPa, trichloroethylene—14 kPa, carbon tetrachloride—20.7 kPa, and acetone—24.59 kPa.

Mercury is a liquid metal at normal (NTP) conditions. It has well-defined physical and chemical properties, and can be purified by evaporation, and then it can be used as a manometric liquid for low-pressure measurement. Owing to the mass uniformity and chemical stability at higher temperatures, mercury was also used as a pumping medium in diffusion pumps, and sealing substance in vacuum applications. The saturation vapor pressures of mercury at 20 °C, 0 °C, and –78.5 °C are 1.87 × 10–1 Pa, 2.92 × 10–2 Pa, and 1.33 × 10–6 Pa, respectively. Obviously, its vapor pressure rapidly decreases when the temperature reduces, as illustrated in Figure 3.12. The saturated vapor pressure of mercury is presumed to be barely measurable at temperature lower than liquid nitrogen (77 K). Since at temperature of 92.15 K, the theoretical estimate of mercury pressure is 4.63 × 10–30 Pa, diffusion pumps operating with mercury as a pumping medium and equipped with additional accessories (low-temperature traps at their inlets) were able to obtain the ultimate pressure in the order of 10–11 Pa. However, the high saturated vapor pressure of mercury at the standard conditions has always been of concern particularly because of health and safety issues. Therefore, different special fluids are employed in ejector and diffusion pumps. For example, the dimethylpolysiloxane fluids are based on (CH3)3SiO–[(CH3)2SiO] n –Si(CH3)3 molecular structures. They are commercially available under trademarks DC702, DC703, D704, and DC 705 produced by the Dow Corning or Midland Silicons under trademark MS702 – MS705. These silicon polymers with n ≥ 5 have heavy molecules. For instance, the average molecular number of DC702 is 530 for n = 5. At room temperature, they have low saturated vapor pressures, while at the boiler temperature of 175 °C, they provide sufficiently high pressure to form suitable molecular jet beams in diffusion pumps.

Saturated vapor pressure of mercury as a function of temperature.

Figure 3.12   Saturated vapor pressure of mercury as a function of temperature.

The saturated vapor of DC dimethylpolysiloxane fluids can be estimated using semiempirical Equation 3.101, as deduced from the Clausius–Clapeyron equation:

where “ln” is the natural logarithm, pressure is in Pa, and constants A and B are related to material properties and they are determined from the pressure and temperature measurements. Using this equation and A and B constants, in Leybold’s Taschenbuch, 99 we recalculated the constants to SI units, computed the saturation vapor pressures, and made plots in Figure 3.13.

Saturated vapor pressure of dimethylpolysiloxane fluids (DC702, DC703, DC704, and DC705) for diffusion pumps, plotted following

Figure 3.13   Saturated vapor pressure of dimethylpolysiloxane fluids (DC702, DC703, DC704, and DC705) for diffusion pumps, plotted following Equation 3.114.

3.16  Vacuum Thermal Evaporation

Thermodynamic equilibrium of a gaseous phase with its other phases takes place at the saturation vapor pressure, which can be attained in any sufficiently small and enclosed volume. At equilibrium conditions and invariable temperature, the saturation vapor pressure does not change, becuase the flux densities of evaporating/sublimating molecules and the condensing molecules are equal. However, if the saturation pressure p above the surface of an evaporant is greater than the background pressure p b of that substance, the evaporant mass reduces because the molecular flux from the background is smaller than the flux leaving the evaporant surface. Then, the evaporated molecular flux density can be calculated from the difference of two molecular flux densities (see Equation 2.111):

  • A E is the evaporant surface area
  • N is the number of molecules evaporated form area A E for time t
  • p is the saturated vapor pressure above the evaporant surface at temperature T
  • p b is the background pressure of the evaporated substance
  • m is the mass of single evaporated molecule
  • k is the Boltzmann constant
For further insight into the physical phenomenon and relation between pressure and molecular flux density, see the derivation of this quantity given by Equation 2.111.

When the background pressure p b of an evaporant is lower that its saturation pressure p under high vacuum, evaporated molecules travel along straight paths and may impinge on a colder substrate. A fraction α of the flux density can be adsorbed on the colder surface area of the substrate, while the remaining fraction (1 − α) leaves the substrate surface and contributes to the background pressure, p b , of the evaporated substance. In the sorption theory, the coefficient α is termed the sticking coefficient. Hence, the deposited flux density can be written in the form

introduced as the Hertz–Knudsen equation for deposition, previously.

At vacuum evaporation of metals, it is reasonable to assume that virtually all metal atoms impinging on a substrate surface with temperature close to the room temperature are adsorbed. The sticking coefficient is then practically equal to one. 100 Similar conjecture can be made at evaporation of the organic liquids with high boiling points. 101 In assumption that all molecules are adsorbed at their impact on the substrate, that is, α = 1, the deposition flux density is equal to evaporation density (see Equation 3.116). Then, the evaporated mass from a unit area of the evaporant per second is

However, at vacuum evaporation of materials, the pressure of evaporant p b is usually much smaller than the saturated vapor pressure p above the evaporant surface because the vacuum background pressure p B is very low. Thus, at such conditions and dynamic evaporation/condensation process at which all molecules are adsorbed at their impact on the substrate, that is, α = 1, the last equation can be transformed to
Hence, the total evaporated mass from the source with surface area A E for time t is
  • M is the mass evaporated from the source area A E for time t
  • M E is the evaporated mass from an area unity per second
  • N is the number of molecules corresponding to the mass M
  • m is the mass of a single evaporant atom/molecule
  • M a is the molar mass of evaporated substance in kg/kmol
  • k is the Boltzmann constant
  • R 0 is the universal gas constant
  • T is the absolute temperature of evaporated material and thus also its vapors
Then, the saturated vapor pressure can be determined using equation
The illustrated determination of vapor pressures was used by Langmuir and several other authors later. The data on vapor pressures of various materials were compiled by Dushman 102 and Honig. 103

The evaporated mass M is proportional to the saturated vapor pressure p of the evaporant, and it is affected by the background pressure of evaporant. However, the background pressure of evaporant is usually negligible because the vapors of solid evaporants are rapidly adsorbed at high vacuum conditions and low temperatures. Hence, the derived Clausius–Clapeyron equation provides us not only with the base for determination of the saturated vapor pressures of materials, but it also allows us to determine the mass evaporation rate given in kg m –2 s –1 if the saturation pressure is substituted into Equation 3.117.

Now, let us consider a point source (Figure 3.14a) and total evaporated mass M, as given by Equation 3.119. The evaporated mass M can also be determined by a gravimetric method, that is, by weighting the evaporant before and after deposition. The point sources are isotropic, which means that at high vacuum conditions, the vapor molecules propagate radially along straight paths via a spherical volume without scattering. Obviously, at a radius r, their mass surface densities are equal in any point of the sphere with an area of A = 4πr 2. So, at r, the mass flux density is

, and at conditions of low background pressure p b of the evaporant under high-vacuum conditions, sticking coefficient α =1. At such circumstances, the evaporated and condensed mass flux densities are equal. Since at a distance (radius) r, mass flow density is equal around the sphere, and M is the total mass deposited for time t, we can determine the deposited film thickness from equation
  • ρ is the mass density of the deposited film over the spherical area, A = 4πr 2, at distance r
  • d is the film thickness at r deposited for time t
Thus, with the assistance of Equation 3.121, we can find the mass flux density, that is
As said, this deposited mass flux density is equal to the mass flux density deposited from the gas phase on an area of unity per second. If we denote deposition rate, r d = d/t, which refers to the mass flow density mΦ, and molecular flow density Φ at the radius r, at the given conditions, the molecular mass flow can be rewritten as
and hence,
The substitution for pressure p from Equation 3.100 gives the deposition rate
where we substitute for molecular flux density and saturated vapor pressure p = e A − B/T as given by Equations 3.100 and 3.101. It should be noted that D′ is not diffusion coefficient, but a material constant (A = ln D′), as follows from A and B being empirical material constants fitted to Equation 3.101 for a particular evaporated material based on pressure and temperature measurements. The constants A and B are listed in Table 3.5 for pure chemical elements.

Evaporation sources: (a) point source and (b) surface source.

Figure 3.14   Evaporation sources: (a) point source and (b) surface source.

In the case of a point source, as seen in Figure 3.14, the evaporated mass is distributed evenly over any concentric sphere with a radius of r. Thus, the evaporated mass referring to a unit area of the sphere with radius r is

and the mass evaporated on an elementary area of a sphere with radius r is . At conditions when the background pressure of the evaporant is significantly smaller than its saturation pressure (p b  ≪ p), and when all the evaporated molecules are adsorbed (α = 1), we may consider that evaporated mass is equal to the deposited mass, for which we may write
  • ρ is the mass density of deposited film
  • dV r is the elementary volume
  • dA r is the elementary area
  • d is the film thickness
Now, if the surface area is tilted by an angle of θ with respect to the radial direction, the tilted area dA T is deposited with the same mass of evaporant as the area dA r (being normal to the radius r) because it is valid that dA r  = dA T  cos θ. Hence,
Then, the division of the last equation by the mass density ρ and the tilted area dA T gives the film thickness
In contrast to spherical surface, the thickness deposited from a point source on a planar surface is not uniform. In this case, the film uniformity on a planar substrate can be determined trigonometrically as illustrated in Figure 3.15. Since r 2 = h 2 +  2, r = (h 2 +  2)1/2, and cos θ = h/r, the last equation can be rewritten to
Apparently, the maximal film thickness is at  = 0, which is
Then, the relative change in film thickness with the  -displacement is
As an example, we can calculate the relative change of the thickness, for instance, for a wafer with a diameter of 80 mm, when the distance between the point source and wafer center is 0.3 m. Employing the last equation, the relative change of the film thickness is
Thus, if the film thickness is 100 nm in the center of the wafer, then the film thickness is 94.7 nm at a radius of 40 mm, when evaporated from a point source that is 30 cm apart from the substrate.

Schematic for thickness calculation of film deposited on a planar flat surface from a point evaporation source.

Figure 3.15   Schematic for thickness calculation of film deposited on a planar flat surface from a point evaporation source.

Based on similar considerations, we may deduce the mass and thickness of films deposited on a planar surface substrate from a surface evaporation source as seen in Figure 3.14b.

By recasting Equation 3.133 and its division by mass density, we receive
which can be transformed to
Obviously, the maximal thickness is at  = 0, that is,
Then, the relative change in thickness with -displacement is
The relative change in thickness or nonuniformity of thickness can be estimated using the last equation when the film is deposited from a surface source.

3.17  Thermal Evaporation from Multiple Sources

The theoretical section above indicates that uniformity in film thickness depends largely on the geometrical configuration of a source and substrate. In the case of very thin films, an ununiformed film thickness is visualized as color differences over opaque surfaces due to the light interference, and variations in depth of color on transparent substrates. In electronic structures, the thickness nonuniformity may affect the transient time of charge carriers, which may be related to the device performance. Therefore, the thickness uniformity is a vital parameter in thin-film deposition. The source-to-substrate distance and size of the substrate as well as its shape are geometrical parameters that should be taken into account. For example, organic electroluminescence devices (OELDs) are designed with multilayer structures and film thickness as thin as 60 nm. Two and more individual sources loaded with different materials cannot be placed to the same location unless they are placed on a rotatable carousel and used in sequence. Figure 3.16 schematically depicts two geometrical configurations of evaporation sources and a substrate. The first configuration, in Figure 3.16a, is unsuitable because the deposited film especially from source 2 is highly ununiformed (indicated by Figure 3.16c) unless the substrate rotates. Upon the displacement distance, the deposition time is prolonged, which may increase the incorporation of impurities from vacuum background. In addition, depending on the size of the substrate and source-to-substrate distance, source 1 may not provide satisfactory uniformity even with the substrate rotation. A more favorable arrangement is the confocal configuration in Figure 3.16b, where the normal of each source intersects the substrate center and substrates rotates.

Source—substrate configuration: (a) nonrecommended; (b) recommended confocal configuration with a substrate rotation; (c) thin-film profile without substrate rotation; 1,2, two sources; 3, substrate; 4,5, films.

Figure 3.16   Source—substrate configuration: (a) nonrecommended; (b) recommended confocal configuration with a substrate rotation; (c) thin-film profile without substrate rotation; 1,2, two sources; 3, substrate; 4,5, films.

When using masks to deposit patterned structures, we should pay attention to the angle at which molecular flux arrives to the substrate and mask thickness due to shadowing effect. The impact angle of molecules on substrate may also affect adsorption.

Some chemical elements evaporate in molecular rather than atomic forms. Examples are arsenic, which evaporates as As4, and sulfurs, which evaporate as S2, S4, S6, and S8 molecules. However, it can be more beneficial in some cases to evaporate material in atomic form rather than in large molecules. Therefore, there are available special sources, two zone effusion cracker cells, to break molecular clusters to atoms or smaller evaporant entities that can easier migrate over surfaces when adsorbed.

Alloy deposition: Thin-film alloys and various structures consisting of different chemical elements have wide technical applications, but there is no method, which enables us simple deposition of alloys. The composition of alloy source is usually unpreserved by a standard single source deposition. With a few exemptions, the alloy constituents evaporate at different mass rates, since they segregate upon their melts, and the segregated elementary constituents exhibit dissimilar saturation vapor pressures. Thus, preferential evaporation of one chemical element over another commonly takes place. The resulting film composition thus differs from that of the source material. For the same reason, the film composition varies upon the evaporation time. We can deposit films with desired composition if the content of the component with the lower saturation pressure in the source alloy is correspondingly increased. However, the more volatile component of the alloy gradually depletes in the source. Therefore, the source alloy has to be replenished after some operation time.

The composition of film alloys can be controlled well using evaporation from two or few individual sources in confocal configurations, as previously described. They can also be evaporated employing flashed techniques from a hot and dry source. This technique requires alloy materials in the form of well degassed wires fed to the heated surface with a sufficiently high temperature where the materials melt and spontaneously evaporate.

3.18  Conditions at Vacuum Thermal Evaporation

Thermal evaporation of solid materials to form thin films takes place upon heating of solids to their evaporation temperatures at high-vacuum conditions. The exception is the synthesis of nanomaterials, which can be carried out at subatmospheric pressures of different gases and elevated temperatures. At such conditions, specific gases and catalysts are supplied to mediate desired reactions. However, this process differs from the conventional evaporation because it involves mediated chemical reactions. In this chapter, only thin-film deposition by conventional evaporation methods, that is, at high-vacuum conditions (~10–4 Pa), is briefly discussed.

For better elucidation, let us consider the specific evaporation case of aluminum. Aluminum rapidly turns to vapor at 2519 °C, which is its boiling point (Table 3.8) defined at atmospheric pressure.

In thin-film deposition by conventional evaporation, the saturated vapor pressure over evaporant surface is in the order of 10–100 Pa (~0.1 to ~1 Torr). The saturated pressure is related to the evaporation rate at the point where the phase transformation occurs. For example, although aluminum boils at 2519 °C under atmospheric conditions, at high-vacuum conditions, it intensively evaporates at 1547 °C when its saturated vapor pressure is ~100 Pa. So, at vacuum conditions, aluminum is deposited at much lower temperatures than its boiling point. At 1082 °C, when saturated vapor pressure is 1.33 × 10–1 Pa, the aluminum deposition rate is 8.23 × 10–5 kg m –2 s –1. For evaporation, we need to supply the energy for fusion (W Fusion ), and vaporization (W Vapor ), as well as the energy (W KE ) that is imparted on evaporated molecules. Thus the total supplied heat into evaporant is given by

and parazitic heat loss in radiation and heat loss by conduction via electrical leads.

The heat imparted on the evaporated atomic molecules is proportional to the average kinetic energy of molecules at evaporation temperature. So, if evaporation is at temperature of 1082 °C = 1355.15 K, the average energy carried by an evaporated molecule is

We can say that the average energy carried by a single evaporated molecule is approximately w KE ≈ 0.2 eV = 3.2 × 10−20 J at evaporation temperature of ~1000 °C, and w KE  ≈ 0.4 eV = 6.41 × 10−20 J at evaporation temperature of ~2000 °C. These values are temperature-limited, and they are the measure of the velocity of molecules. Since the aluminum mass density is 2700 kg/m 3 and the molar mass is 26.98 kg/kmol, the atomic mass density is
Hence, the estimate of specific energy imparted on evaporated molecules in reference to cm 3 of solid aluminum in the form of kinetic energy is
For comparison of the heat needed for fusion and vaporization with the heat transfer by molecules to the substrate, we can calculate the heat transferred by evaporated monoatomic molecules with molar quantity of a mole, which gives the value
As seen in Table 3.6, this value is comparable to the fusion energy needed for melting aluminum.

Table 3.6   Some Aluminum Properties at Evaporation


Transformation to Vapor

Molar heat

24.30 kJ/(kmol K)

Fusion heat

10.71 kJ/mol

396 kJ/kg

Molar mass

26.98 kg/kmol

Vaporization heat

284 kJ/mol 10,526 kJ/kg

Melting point

660.32 °C

Kinetic energy

16.89 kJ/mol @ 1082 °C

Boiling point

2519 °C

The kinetic energy is given up to the deposited substrate in the form of heat. Thus, the heating effect from deposited molecules is proportional to the number of molecules deposited on a unit area for the time of unity. This heat can be considerable in some cases. Since the source and substrate are in the line of sight, transfer heat by radiation has to be taken into account, too.

Although we define vacuum in Chapter 1.6, we need to highlight specific vacuum characteristics at evaporation processes. We need to consider vacuum as a complex environment, which is neither free of gas molecules, nor chemically inactive. Therefore, any thin-film deposition should be done with attentiveness on the vacuum environment and its effect on the evaporant vapor flux and the growing film structures.

In conventional evaporation processes, there are several parameters that deserve considerable attention. At deposition of compact solid films and the first approximation, we compare the mean free path of molecules with the distance between the evaporant source and the substrate. The mean free path is taken as a criterion for the assessment of vacuum quality as well as transport processes that may occur under different vacuum conditions.

In vacuum deposition systems, evaporated molecules should have straight paths from the source to the substrate. They should not be scattered, and they should not react with background molecules. For the estimate of molecular mean free path L, it is sufficient to take into consideration just molecules in the residual vacuum environment. Say, that we need vacuum conditions that provide the mean free path L in the molecular background in vacuum equal to the source-to-substrate distance of, for example, d = 662 mm. The mean free path can be estimated using Equation 2.135 (see also Example 2.23). Thus, for L = d = 662 mm, we calculate the corresponding pressure using the referenced equation

This estimate is acceptable though the last equation is applied to air molecules at 20 °C and though the residual molecules are mostly water (60%–90%) interacting with atoms/molecules of an evaporant at high-vacuum conditions. More suitable calculation can be performed with specific equations for mean free path in gas mixtures, which are derived in this publication (see Chapter 2.21). However, this will not make a tangible difference.

Since the mean free path is the average value of the free molecular paths, at pressure p = 10–2 Pa, there is a considerable number of molecules that make collisions on shorter distances than the distance d = 662 mm. The above calculations do not directly show how many molecules pass the given distance without collisions. The answer to the question how many molecules n of the total number n 0 actually travel the distance longer than x can be found out from the distribution law of molecules in accordance with their free paths, that is, n = n 0 e x/L or Φ = Φ0 e x/L (see Equations 2.153 and 2.154), where Φ0 and Φ correspondingly stand for the molecular flux densities. In our case, x is the source-to-substrate distance (x = d = 662 mm). Because of the formulation of task, d is equal to the mean free path L, the fraction molecules that travel a distance greater than d is n/n 0 = e x/L  = e d/L  = e L/L  = e −1 ≈ 0.37. Obviously, at the pressure of 10–2 Pa, the majority of molecules (63%) make collisions on the shorter distances than the source-to-substrate distance, d = 622 mm. Although at pressure of 10–2 Pa, the molecular density (2.47 × 1018 m –3) is reduced by 7 orders of magnitudes with respect to that of 2.47 × 1025 m –3 at 105 Pa (~atmospheric pressure—101,325 Pa), the molecular density is still high (see Table 3.7) in terms of molecular collisions and scattering. When the background pressure is further reduced by an order of pressure magnitude to 10–3 Pa, then by the same calculation it can be found that more than 90.5% of molecules travel the distance d = 662 mm without collisions, while the rest (9.5%) are still scattered.

Table 3.7   Gas Pressure p, Molecular Density n, Mean Free Pass of Molecules and Their Relation with Molecular Flux Density, and the Ratio, Φ E /Φ B , of Evaporant-to-Background Molecular Flux Densities at 20 °C

p (Pa)











n (m –3) 2.47×











L (m) 6.62×






















If it is desired that 99% of molecules travel the distance longer than the source-to-substrate distance, d = 0.662 mm, without collisions, then it can be written

Hence, the pressure is
At pressure of 10–4 Pa (7.5 × 10–7 Torr), mean free path is already , which is 100 times longer than the geometrical source-to-substrate distance. Out of curiosity, we may find the mean free distance of molecules at 10–4 Pa, when the mean fee path of molecules is 66.2 m. First, we determine the molecular density, using equation n = p/kT. Substitution for the pressure p = 10–4 Pa, temperature T = 293.15 K, and Boltzmann constant k = 1.38 × 10 23 J/K gives the molecular density n = 2.47 × 1016 m –3 (see Table 3.7), and then we calculate the mean distance of molecules, which is
Comparison of the mean molecular distance (3.43 μm) and mean free path (66.2 m) of molecules gives the impression that the mean free path of molecules is too long when compared to the mean free distance among neighboring molecules. However, taking into consideration the size of molecules (in the order of 10–10 m) virtually clarifies the problem.

So, the transit and the scattering of molecules in a vacuum environment and following prerequisites for vacuum evaporation are understood in the frame of the discussion above. Nevertheless, this is still an unsatisfactory understanding because the background pressure (molecular density) is the source of molecules that unceasingly impinge on the internal surface area of any vacuum system and, thus, the substrate where the deposition of an evaporant takes place.

The molecular flux density from vacuum background includes all kinds of molecular species found in vacuum environment. However, chemically reactive species are of particular concern at thin film deposition. In high-vacuum systems, the major gas source is in desorption of gases from the internal surface of vacuum systems making up the residual gas environment with 70%–90% of water vapor. These desorbing gases and vapors behave as permanent gas sources. The knowledge on the presence of reactive gases is vital especially at low deposition rates, when the reactive gases can incorporate into the growing films in large quantities and affect their compositions, microstructures, and deteriorate the desired film properties. A good illustrative example is the evaporation of aluminum in high-vacuum environment at a very low deposition rate, which leads to the formation of aluminum oxide instead of pure aluminum metal. For such reasons, the knowledge of potential chemical activity of a high vacuum environment should be kept in mind at any technological and analytical processes carried out under vacuum conditions.

At a flash evaporation, the pressure is usually in the order of 10–4 Pa (~10–6 Torr) and the evaporation takes a second or a fraction of a second. The incorporation of background impurities proportionally reduces by shortening the evaporation time. However, we still need to be concerned if the detrimental quantities of chemically reactive molecules exist in the residual vacuum atmosphere. A higher concentration of reactive species in vacuum, even easily evaporable materials like aluminum form islands rather than uniform films due to adsorbed gases and vapors on substrate interfaces. Whenever the evaporation takes longer under the discussed conditions, the material identity of films becomes problematic. Low evaporation rate is needed for the deposition of high-quality films and films with single crystal structures. At low evaporation rate, the adsorbed atoms/molecules have a time-space to migrate over the surface to be assemble in thermodynamically favorable configurations instant of fast piling molecules and quick formation of immobile molecular clusters. However, the drawback also is the flux of impurities that can incorporate into the growing film structures from the vacuum background in detrimental quantities. This problem can be solved by lowering the pressure to much lower pressure (UHV) but at a very high price. The deposition in UHV is only economical for relatively small areas in research and high-value products/devices.

Other important parameters are substrate purity, homogeneity in adsorption sites, as well as material properties such as surface energies with respect to deposited films. These considerably affect the film structures and their properties.

Let us be more specific. When functional films made of organic materials with small molecules are deposited, the evaporation takes a longer time. It typically takes around 3.0–3.5 min to grow a functional layer with a thickness of 60 nm. Obviously, these films grow from the evaporant flux Φ E , but there are also molecules arriving to the growing films from the vacuum background, as indicated in Figure 3.17. The background molecules can be deposited in detrimental quantities. Their flux density Φ B can be estimated using the Hertz–Knudsen equation. Taking into account 70% of water vapor and the gas temperature of 293.15 K, the deposited molecular flux density is

By substitution for the constants, and gas parameters at background temperature of T = 293.15 K (20 °C), we obtain
  • p B is the total pressure, 0.7 refers to 70% of water molecules
  • m is the mass of a single water molecule in the background
  • α = 1.0 is the sticking coefficient
  • k is the Boltzmann constant
  • T is the absolute temperature of gas in vacuum background
  • m = M a /N a is the mass of a single water molecule
  • M a = 18 kg/kmol is the molar mass of water
  • N a is the Avogadro constant
If we take the sticking coefficient α = 0.3, which is already low value, we receive Φ B  ≈ 7.62 × 1017m −2s −1. Below, we use even a lower value for the sticking coefficient “to please ourselves” when we deal with harsh reality, referring to the unfavorable arrival ratio of the molecular flux densities.

Elucidation of mixing the molecular flux density Φ

Figure 3.17   Elucidation of mixing the molecular flux density Φ E from the evaporation source and molecular flux density Φ B from vacuum background.

The deposited molecular flux density can also be found out from the deposited thickness over an area of unity per second. Assume that a film made of small organic molecules is deposited with thickness of 60 nm for 3 min (180 s), which gives the deposition rate of

The product of the uniformly deposited area A and deposition rate r is the volumetric growth rate
by multiplication with mass density, we obtain the mass growth rate
which can be rewritten as
Since the molar amount is , the last equation can be recast to
where N/At is the deposited evaporant flux density. Thus Equation 3.153 takes the form
Assuming that the molar mass of organic molecule is hypothetically 100 kg/kmol, while the mass density is estimated to be 2.5 g/cm 3 = 2500 kg/m 3, we can calculate the deposited evaporant flux density
Thus, the ratio of evaporant-to-background molecules flux densities is
which means that for each two deposited organic molecules, there is one water molecule that has arrived to the growing organic film.

The amount of incorporated impurities into the organic film structure depends on the adsorption coefficient of water molecules on a particular surface, and this coefficient certainly is not small. In a very favorable case, consider the adsorption coefficient to be 0.1 for water, and 100% adsorption of organic molecules at their impact on the surface of the growing film. Then, the incorporated ratio of the two fluxes is

Accordingly, one molecule of water incorporates per 20 deposited organic molecules. We may compare this value with the practical arrival rate ratio (being >104) used in molecular beam epitaxy (MBE). In MBE such a high value is needed for growing high-quality films with crystal lattice structures.

At the deposition where a water molecule incorporates per 2–20 organic molecules, the properties of organic films may measurably be affected even at small changes in outgassing of any internal parts of the deposition system. Consequently, poorer film properties may result in unsatisfactory performance of fabricated devices, shortening their lifetime and their ultimate failure.

Usually the devices produced in industrial plants have higher performance than those fabricated in many laboratories at “similar deposition conditions.” The major difference is in degassing the deposition system. In the industrial plants, the evaporant sources operate continuously and therefore the sources and systems are degassed well. However, in laboratories, the deposition systems operate for a short time, and they are periodically vented and therefore often poorly degassed, which makes a difference in incorporation of impurities into the functional layers. However, in industrial operations, for the sake of the film and device quality, small organic molecules have to be replenished continuously not only because of their evaporation, but also because of the degradation of small organic molecules due to the long heating process.

The reactive species (water and oxygen) may also incorporate into the device structure by diffusion from substrates and by permeation via passivation and capsulation barriers. Therefore, the substrates (glass or plastic web) have to be degassed under vacuum conditions and then they should be passivated by a uniform barrier layer, for example, by SiO2 or better by an SiO2–Al2O3 possibly using an atomic layer deposition (ALD). The ALD films grow two-dimensionally, layer-by-layer. Molecules do not pile one on another and provide uniformly distributed adsorption sites for further deposition. This way the formation of surface pinholes and spikes, which are the reasons for short contacts upon metallization and localized burning organic films due to strong electric fields and large current densities, is obstructed. Other source of reactive gases is surrounding atmosphere from where gases permeate or leak. These processes can be suppressed using effective capsulation and getter trapping. Again, the first several capsulation layers should be deposited by ALD for the film uniformity and minimization of defects.

The incorporation of molecules from the vacuum background depends largely on the partial pressure of reactive vacuum constituents and mutual chemical reactivity of interacting counterparts. For convenience, we present Table 3.7, which lists relationships of pressure p, molecular densities n, mean free paths L, and arrival rate ratios Φ E B of two impinging molecular fluxes (Φ E –evaporant, and Φ B –background). Say, at pressure of 10–4 Pa, the arrival rate ratio is Φ E B = 10, which means that for each 10 evaporant molecules there is a molecule from the vacuum background. Obviously, the higher film purity can be prepared when the arrival rate ratio is greater. The evaporated films are usually deposited for a very short time (flash evaporation) or by continuous evaporation of films on fast moving webs. If the increase of evaporation rate is impossible, or it is inherently limited, the only way to increase the ratio Φ E B is to reduce the background pressure. Table 3.7 indicates that reducing the pressure by an order of magnitude means the increase of the arrival rate ratio by the same factor. Assuming the evaporation rate constant as that at 10–4 Pa, then only one molecule per 10,000 evaporant molecules arrives to the growing film from the vacuum background at 10–7 Pa. Accordingly, at evaporation rate as low as 0.1 Å/s, used in MBE, we needed pressure of ~10–8 Pa to provide high film purity.

The formation of film structures also depends on the deposition rate. The low deposition rate allows the adsorbed molecules and particles (ad-atoms/ad-molecules) to migrate over the solid surfaces to form thermodynamically favorable structures. Therefore, deposition can be carried out in short pulses (interrupted deposition) to improve crystallographic structures.

3.19  Scaling of Evaporation Systems

Deeper insight into evaporation and evaporation systems shows that the system scaling is not straightforward. The total molecular flux (denoted just here as Φ in s –1) leaving the surface of a point source propagates radially via a high vacuum, as seen in Figure 3.18. At the distance of r and R, the flux densities are

with units of m –2 s–1. Accordingly, at the longer source-to-substrate distance, the evaporant flux density is smaller due to the larger spherical surface. As a result, the arrival rate ratio of Φ E B is smaller too, when the saturation pressure and evaporation temperature are invariable. Consequently, the incorporation of impurities from the background is higher. The possible solution is to provide the same flux densities at both radii R and r. Hence, the total flux density Φ has to be increased to the value of Φ′, at which the flux density is the same as that at r. This allow us to write
from where
Since the mass flux density is the product of molecular flux density and mass of molecule, the corresponding mass flux densities are in the ratio of M′/M = R 2/r 2. Thus, the total flux density has to increase by the square of radii R 2/r 2 to keep the same density as that at radius r. This also means that the temperature has to increase correspondingly. However, the higher temperature may affect many other parameters. The evaporation source with higher temperature also increases the temperature of surrounding components via radiation heat transfer, which subsequently induces the higher degassing rate of materials. The increase in the degassing rate raises the background pressure of reactive molecules, which consequently leads to the higher incorporation of molecules into the growing film structure. In addition, due to the increase of temperature, the higher kinetic energy is imparted on the evaporated particles that heat the growing films and sensor of thickness monitor.

Scaling of evaporation systems.

Figure 3.18   Scaling of evaporation systems.

An alternative would be reducing the pressure to the level of p/(R 2/r 2) instead of increasing the evaporation rate. However the evaporation is typically carried out at nearly ultimate pressure of the deposition system that usually makes difficulties to facilitate this approach.

For illustration, increasing the distance from 50 to 70.5 cm, approximately, requires either raising the deposition rate twice or reducing the pressure by a factor of 2 to maintain the same molecular arrival rate ratio of fluxes from the evaporation source and vacuum background.

The reduction of partial pressures of condensable species, particularly water vapors, is possible using a Mixner’s trap operating at temperature of liquid nitrogen (–196 °C) or cryosorption pump connected to the chamber. The effective capturing of reactive gases can also be facilitated by different types of getters and additional gas activation in a deposition environment.

The last two methods can reduce the pressure of reactive gases by nearly an order of magnitude, and thus proportionally lower the incorporation of impurities into the growing films. In this aspect, it is also suitable to use cryogenic pumps in which nominal pumping speeds for water far exceed those for ordinary gases.

Reducing the geometrical dimension may not have a positive effect with regard to impurity incorporation, because at shorten distances, radiation heat can cause heating of surrounding components and their outgassing. The desorbing reactive gases then may incorporate into the growing films.

Because of the radiation heat, the evaporation sources are shielded using water-cooled jackets, especially in the cases of longer deposition. The cooled shielding also suppresses the undesirable heating of the shield and consequential gas desorption. The shield also prevents the deposition of an evaporant on the undesirable areas.

Now, let us consider that the source-target distance R is reduced to the distance r. If the total evaporated flux from a point source is Φ Tot , then the flux densities in the distances of r and R are

and , respectively, while the corresponding deposited masses for the deposition time t are and , where m is the mass of a single deposited molecule. Since R > r, the mass and thickness of the film deposited on a unit area for time t is smaller when the film is deposited on the surface sphere with a larger radius. For the deposition of the same mass on an area of unity at the same evaporation temperature, the deposition time t r has to be shorter for the sphere with a smaller radius. Thus, for equality of two masses we can write
from where the deposition time at the shorter distance r is
This calculation is useful when the thickness monitor is calibrated for a particular configuration, and then used in the other geometry where the radiation heat and the heat transferred by evaporant are different. These differences in heat transfer cause an electronic drift in the crystal of the thickness monitor, which consequently gives incorrect reading of the film thickness. For example, if the film thickness is greater than in the optimized structure of OELDs, then the driving voltage of the device can be high or eventually the device is nonfunctional.

3.20  Different Thermal Evaporation Techniques

Thermal evaporation can be carried out using different evaporation techniques in high vacuum. The techniques are chosen depending on the type of coated workpieces, evaporant properties, expected film properties, coating flow process, and economical motivations. The thermal evaporation techniques can be classified into (a) resistance evaporation (RE), (b) electron-assisted evaporation (EAE), (c) electron beam evaporation (EBE), and (d) evaporation by induction heating. MBE could also be classified in this group, since it is a refined form of the evaporation technique that uses evaporation of materials under ultrahigh vacuum condition (10–8 Pa) and highly controlled deposition rate including pulse-controlled deposition.

3.20.1  Resistance Evaporation

In resistance evaporation, we can use coils, boats, and special sources heated by passage of electric current. The coils are made in many shapes, from a simple V-shape via helical spiral shape to conical baskets. Figure 3.19 depicts a multistrand helical coil source (1) having rather high load capacity. The weight of evaporant should be 10% of the coil mass. Multistrand coils can be used in large drum coaters as isotropic sources. Conical basket coils can hold a small amount of evaporant while they minimize dripping of the evaporant upon their melting. Basket coils can hold various crucibles with an evaporant and heat them. The crucible can be made from different materials: pyrex, quartz, alumina, pyrolytic boron-nitride (PBN), or even electrically conductive materials. Hexagonal boron-nitride sources are highly inert, but they are hydroscopic, and they are difficult to degas. Similar properties have also graphite sources. Crucible can be of sizes from 1 ml to as large as 1 l. The large volume crucibles are used in continuous roll-to-roll aluminum coating of plastics for many applications including packing materials and capacitors.

Sources for resistance evaporation: 1, multistrand wire coil; 2, rod overwound by an evaporant wire; 3, folded boat; 4, dimple boat; 5, buffer source.

Figure 3.19   Sources for resistance evaporation: 1, multistrand wire coil; 2, rod overwound by an evaporant wire; 3, folded boat; 4, dimple boat; 5, buffer source.

Boats are other alternatives of resistance sources that can be prepared in many shapes, as seen in Figure 3.19 denoted by the positions 3 and 4. Obviously, boats are constructed with folds and dimples, which prevent dripping of the melted evaporants. Evaporation from a boat with a surface point source has a typical angular cosine distribution of the flux density (Figure 3.20). The flux density around the illustrated surface sphere is practically equal. At specific conditions, an evaporation cell (Figure 3.21) used in MBE can be treated similarly, when a material is evaporated from a molecular cell, where molecular velocities follow Maxwell–Boltzmann distribution and where molecular motion is unaffected by the cell walls. A small aperture of the cell (Figure 3.21) separates the ultrahigh vacuum and cell environment and maintains the constant pressure in the cell. The molecular flux density Φ0 passing the aperture is calculated from the saturated vapor pressure in the cell. The total molecular flux passing through the aperture is proportional to the aperture area (Φ T  = Φ0 A a ). Then, the molecular flux is evenly distributed over the surface of a sphere, as illustrated in Figure 3.21, when the aperture is small and works as a point source of molecules. Hence, molecular flux follows cosine angular distribution. The molecular flux density in any point of the sphere is

The flux density Φ0 is determined using the equation , where p is saturated vapor pressure in the cell interior. However, practical evaporation cells have conical shapes. They are very well thermally shielded and externally cooled to prevent undesired degassing during the evaporation process.

Evaporation flux density from a surface point source.

Figure 3.20   Evaporation flux density from a surface point source.

Molecular flux density from an evaporation cell; thin wall aperture.

Figure 3.21   Molecular flux density from an evaporation cell; thin wall aperture.

Special resistance evaporation sources are heated rods that are overwinded by several turns of an evaporant wire around a refractory metal rod, seen in Figure 3.19 denoted by the position 2. Unique sources are also rods made of refractory metals plated with evaporant materials. Due to the good thermal contact, such sources are used for the evaporation of materials with higher melting points like chromium (M.P. = 1907 °C). For evaporation of powder materials, for example, silicon monoxide (SiO), and similar materials (spitting materials) that intake microparticles into evaporated vapor flux, there are available baffle sources, illustrated in Figure 3.19 denoted by the position 5. The baffle sources comprise a heated zone loaded by an evaporant powder and zone with baffle plates/fins that permit entrance of vapors to exit to a chimney zone, but filter out microparticles from the deposition vapor flux.

For larger area of uniform coating and moving objects, linear evaporation sources are used, as illustrated in Figure 3.22. The linear evaporation sources can be constructed with multiple orifices with spacing between neighboring openings, usually designed by computer simulation. The source can be designed with a bottom and top resistance heaters, which are individually controlled. The top heater provides evaporation of material from its surface while it maintains the temperature of the shields with orifices just a little above of the surface temperature of the evaporant load. This arrangement maintains the orifices free of evaporant condensation and thus prevents the blocking of the orifices.

Example of a linear evaporation source.

Figure 3.22   Example of a linear evaporation source.

The heated components are made of refractory metals (tungsten, tantalum, molybdenum). These materials often alloy with evaporant. The alloyed materials may have lower evaporation temperatures than the pure evaporant and pure refractory materials, which may result in cross-contamination of the deposited film by refractory metals. In addition, alloying is mostly the reason for failure of resistance evaporation sources. Alloying makes these sources brittle, which leads often to their cracking upon thermal expansion and contraction. For prevention/minimization of these effects, materials for evaporation sources should be carefully selected. The selections are based on experience of an operator and practical tables that are available with recommended source materials and techniques for many evaporants. A source coated by inert materials (e.g., alumina) is often a good choice to prevent alloying and thus the premature failure and film cross-contamination. However, these sources require higher deposition temperatures.

For lower temperature evaporation from crucibles, kanthal (FeCrAl alloy) is recommended. Kanthal is inherently passivated by oxidation of an aluminum constituent forming Al2O3 on its surface. Kanthal has a melting point of about 1500 °C. It is durable, chemically stable, and suitable for heating of crucibles and evaporation materials with low melting points, particularly such as small organic molecules.

In laboratory metal evaporation, electric current is often from tens to hundreds of amperes at relatively low voltage. At evaporation of small organic molecules, currents are much smaller (units of amperes) because of lower evaporation temperatures. Using high-current sources for the low-current evaporation of small organic molecules causes difficulties in the precision control of evaporation temperatures.

3.20.2  Electron-Assisted Evaporation

Electron-assisted evaporation sources are alternatives to the resistance thermal sources. These evaporation sources are heated by electron emission current (Figure 3.23). The filament 4 is heated to an incandescent temperature (Figure 3.23a) by an electric current passing between terminals 1 and 2. The filament floats on a high potential with respect to the heated TiB2–BN intermetallic bar that is at the ground potential. Then, for evaporation, the electric power is the product of high voltage placed on the heated bar and emission current. The evaporant wire is automatically supplied to the heated bar as the wire melts and evaporates. The TiB2–BN intermetallic bar sources can evaporate aluminum wire with a mass flow rate of several g/min on a plastic web moving with a speed of ~3.4 m/s. Electron-assisted heating can also be applied to heat a crucible with a loaded evaporant, as seen in Figure 3.23b.

Electron sources: (a) electron heated rod source; (b) electron heated crucible source; (c) induction heated crucible: 1,2, contacts for heated filament 4; 2,3, high-voltage contacts; 5, evaporant wire; 6, TiB

Figure 3.23   Electron sources: (a) electron heated rod source; (b) electron heated crucible source; (c) induction heated crucible: 1,2, contacts for heated filament 4; 2,3, high-voltage contacts; 5, evaporant wire; 6, TiB2–BN intermetallic bar; 7,8, heated crucible with evaporant load, 9, high-frequency induction coil.

3.20.3  Electron Beam Evaporation

In electron beam evaporation (EBE), thermal energy for evaporation is supplied to an evaporant by an electron beam. The energy of evaporated atoms/molecules ranges from 0.2 to 0.4 eV, which is similar to that at the resistance evaporation. For a brief clarification, see Figure 3.24. Unlike in resistance evaporation, an electron beam 4 supplies thermal energy only to the top surface of evaporant 8, which is placed in a water-cooled hearth 3. Electron beam is supplied by an electron gun located in housing 7. Electrons emitted from a hot filament floating at a high potential (10 kV) against the hearth anode is prevented from reaching the grounded body by shielding electrode at a higher negative potential than that of the filament. The accelerated thermionic electrons thus enter between the magnetic steering poles (5, 6) prior to their impact on the evaporant 8 in the hearth 3. The magnetic field between the poles is induced by an electric current passing via a coil 9. The electron beam is bended by 270° and finally its energy is mostly absorbed in a rather small evaporant volume with a depth of a fraction of mm. At evaporation, the saturation pressure reaches ~10 Pa. At high evaporation rate, the vapor pressure can be as high as 1000 Pa. The evaporant area of 0.25–2 cm 2 is exposed to the electron beam current of 1.5 A with energy of 10 keV. These parameters then represent the deposited power density of 60 kW/cm 2, when the area exposed to the electron beam is 0.25 cm 2.

Schematic of a system for electron beam evaporation: 1,2, water cooling; 3, hearth for evaporant; 4, electron beam; 5,6, poles of an electromagnet; 7, housing of an electron source; 8, evaporant load; 9, hidden coil of electromagnet.

Figure 3.24   Schematic of a system for electron beam evaporation: 1,2, water cooling; 3, hearth for evaporant; 4, electron beam; 5,6, poles of an electromagnet; 7, housing of an electron source; 8, evaporant load; 9, hidden coil of electromagnet.

The deposited power induces localized melt and vaporization of evaporant. So, a part of the energy is used for fusion and evaporation and some portion is imparted on the evaporated molecules. The other part of thermal energy induced by electron beam is dissipated by conduction through a solid material to the water-cooled hearth and thermal radiation. However, electrons also make elastic collisions, and their energy is not converted to the heat. The elastically recoiled electrons carry their energy away to the points of their secondary impacts. Stable evaporation/deposition takes place only at stable energy losses.

Since only a small volume of evaporant near the surface region melts, the alloying and the potential cross-contamination of growing films are minimized.

3.20.4  Thermal Induction Evaporation

An electrically conductive crucible is heated by a contactless induction method (Figure 3.23c). The crucible 8 is placed in the center of an induction coil 9. The coil with a few turns of thick copper conductors is a part of a high-power and high-frequency oscillator (200–500 kHz) and is inductively coupled with the crucible. Via this coupling, alternative electromagnetic field induces eddy currents, also known as Foucault currents, in the crucible, causing thermal heating of the crucible loaded by an evaporant. Thus, the evaporant deposits on a substrate. Alternatively, evaporant continuously supplied in the form of a wire is evaporated from the surface of an electrically conductive bar heated by electromagnetic induction.

The induction heating has been used for degassing of electron tubes and evaporation of getters extensively in the final steps of the production process, as well as for crystal growth from the melts.

3.21  Cross-Contamination at Thermal Evaporation

Although for construction of resistance sources we use refractory materials such as tungsten, tantalum, and molybdenum, they can readily be deposited as cross-contaminants in measurable quantities when heated to medium temperatures. For example, molybdenum heated to 1897 °C has a saturated vapor pressure of 1.33 × 10–3 Pa, and it is deposited with a mass deposition rate of 1.20 × 10–6 kg m –2 s –1 (1.20 × 10–7 g cm –2 s –1). However, serious cross-contamination can be caused by alloying of an evaporant with a construction material of the source. Some alloys can exhibit extremely low melting points. For example, eutectic gold/germanium alloy (Au/12% Ge) has a melting point at just 356 °C, while the melting points of pure gold and pure germanium are 1064.18 °C and 938.23 °C, respectively. Similarly, some alloys, oxides, and sulfides can have much lower melting points than those of pure refractory metals. Examples of refractory metals whose oxides can be formed at vacuum condition are listed in Table 3.8.

Table 3.8   Melting Points of Selected Refractory Metals and Their Composites


Melting Point (°C)


Melting Point (°C)


Melting Point (°C)































Molybdenum trioxide has melting point of only 795 °C, which contrasts very much with the melting point of metallic Mo (2623 °C). As a very good example can serve etching of diamond in hydrogen, hydrogen/argon, and hydrogen/argon/3% O2 using a hot filament plasma reactor. Etching in an H2/Ar gas mixture activated by tungsten filaments with temperature of 2100 °C and a substrate bias clearly indicates etching process at which the flat faceted surfaces of diamond polycrystallites turn to the surface with evident etching pits. When oxygen is added in an amount of 3% to a hydrogen/argon gas mixture, the surface morphology is drastically changed.

Although oxygen is in a trace amount in the deposition environment, the etching process may convert to deposition. The tungsten filaments oxidize and form tungsten trioxide (WO3) which rapidly evaporates at the filament temperature of 2100 °C and then it condenses at temperature of ~850 °C on diamond surfaces, where it is partially reduced in a strong reducing hydrogen atmosphere. A portion of reduced tungsten reacts with carbon atoms of diamond to form tungsten carbide. As a result, tungsten, tungsten carbide, and WO3 are deposited on the diamond substrate. The fairly low melting point of 1473°C and boiling point of 1700 °C for WO3 with respect to the melting point of tungsten (3422 °C) are explanatory. Accordingly, the transport of the tungsten mass is mediated by small oxygen additives. 104

3.22  Degassing of Evaporation Sources

Degassing is performed in a two-step process. Each new evaporation source is degassed without loaded evaporant first. Then, in the second step, it is degassed with the loaded evaporant. The two-step process is essential because, in the first step, the degassing temperature is well above the evaporation temperature of evaporant to minimize the content of gases in the heated source and obtain a sufficiently clean evaporation source for the intended evaporation. In the first step (Figure 3.25), gases are removed from both the surface and bulk of an evaporation source. After cooling, the cells are exposed to nitrogen with atmospheric pressure for loading an evaporant material. When exposed to atmospheric pressure, the environmental gases are adsorbed on the source surfaces instantly (nanoseconds), but these gases cannot enter the material bulk in detrimental quantities, since at the room temperature, the diffusion of gases in solid is very low. Thus, after loading evaporant into degassed sources, in the second degassing process, only gases from the source surface and bulk of evaporant are removed in quantities as high as possible (Figure 3.25b). This is carried out by increasing the temperature in steps (while monitoring pressure) up to value just a little below the evaporation with the closed shutter. In each step, we maintain the temperature until pressure stabilize at the value close to that measured before starting the degassing. Finally, we open the shutter and rapidly increase the evaporant temperature to deposit a film on the substrate. At slow deposition, the evaporation temperature is set up and held for better degassing before opening the shutter.

Two-step degassing process: (a) degassing of source at temperature above that for evaporation; (b) degassing the source with loaded evaporant followed by evaporation.

Figure 3.25   Two-step degassing process: (a) degassing of source at temperature above that for evaporation; (b) degassing the source with loaded evaporant followed by evaporation.

During a slow increase in temperature, the pressure rises, but it is not allowed to increase above 5 × 10–3–10–2 Pa. The degassing is considered to be completed when, at the intended temperature, the pressure drops back close to the value at which the degassing started. Failure of degassing leads to extensive direct incorporation of gases released from the evaporation source and the other heated components in the chamber. Thus, compositional identity of the deposited films becomes a considerable problem. In the case of EBE, the degassing process is termed “conditioning.” Unless the evaporant is conditioned, the evaporation is unstable and intense sparks can be induced due to the huge gas evolution. At conditioning, the electron beam scans over a defined surface area of evaporant while power slowly increases in small increments. The temperature increase of evaporant is accompanied by consequential gas desorption. Larger gas desorption may be observed with instabilities and electric sparks. When these effects are observed, the supplied electric power is reduced down just below the first instability. The incremental increase in the power continues until the gun operates at maximum power with no instabilities observed. The conditioning is performed with closed shutter.

In the case of so-called ideal evaporant, which are low melting point materials (such as Al, Cu, Ag, Au), notable evaporation starts at the vapor pressure of 10–2 Pa. Evaporation rate increases until the evaporant pressure over the surface of the electron beam impact reaches a viscous flow. This state occurs at ~10 Pa, when mean free path is a fraction of a millimeter. Since the change in temperature is small from the point of the measurable evaporation to steady evaporation, the thermal losses by conduction to the hearth and radiation do not increase considerably, while the evaporation rate increases rapidly until it is limited by the vapor density above the evaporant surface. Beyond this point, the excess electron beam energy is absorbed in ionization of vapor above the evaporant surface. Therefore, further increase in evaporation rate is only possible by increase in evaporation area. However, this means that deposited power density is reduced. For demonstration, we may use the example of aluminum for which the measurable evaporation starts at saturated pressure of 10–2 Pa occurring at 1245 K. The relatively small temperature increase of aluminum to 1640 K then causes increase of the saturation pressure by several orders of pressure magnitude, to ~10 Pa.

Conditioning of semimelting and sublimating materials is not easy. Conditioning of these materials requires some skill and patience of the operator because despite the methodical increasing power in steps, considerable instabilities may appear.

3.23  Examples Applied to Thermodynamics of Gases and Thin-Film Deposition


Example 3.1

Gas undergoing an isobaric expansion at 1 atm increases its volume from 20 to 80 l. What is the internal energy of the gas, when the heat of 30 kJ is supplied into the gas during the expansion process?


When the gas increases its volume from 20 l = 0.02 m 3 to 80 l = 0.08 m 3 at 1 atm = 101,325 Pa, it performs the work as given by Equation 3.27:

If during this isobaric expansion, the heat of 30,000 J is supplied to the gas, according to Equation 3.2, the change of the gas internal energy is
The internal energy of the gas increased by 23.92 kJ.


Example 3.2

Helium with mass of 20 g underwent a reversible isothermal expansion at 0 °C. During this process, the helium pressure dropped from 101,325 Pa to 10,132.5 Pa. What is the helium volume after its expansion and what is the work performed by helium at this process?


Using the ideal gas law, the initial volume is

At isothermal reversible expansion, calculated from the Boyle’s law, the helium volume increases to the value of
and the work performed at this isothermal expansion is maximal and equal to the heat accepted from the environment given by Equation 3.38:
where molar gas quantity is n m = 0.02 kg/(4 kg/kmol)= 0.005 kmol. The work performed by helium at this process is about 26.147 kJ.


Example 3.3

What is the work needed to compress 30 g of nitric oxide (NO) from 80 kPa to 0.8 MPa, when a reversible and isothermal process is carried out at 300 K?


The molar mass of NO found from the periodic table of chemical elements is approximately

Thus, 30 g = 0.03 kg of NO corresponds to gas molar amount
The reversible isothermal work on compression is given by Equation 3.38:
Hence, the required work to compress NO is about W = 5743.11 J.


Example 3.4

Argon with the amount of 20 g is in a container at standard conditions (273.15 K, 101,325 Pa). At a reversible process, argon expands adiabatically to 33.6 l. What is the temperature of the argon and the work performed by it?


Since both initial argon volume V 0 and standard molar volume V a are given at the same conditions, Equations 1.4 and 1.5 can be used to calculate the initial argon volume V 0.

The gas molar amount is

  • M is the total mass of argon
  • M a is the argon molar mass
Thus, the argon volume before the expansion is
For reversible process, we may write the equation of adiabat . Hence, pressure after the reversible adiabatic expansion is
where κ is the heat capacity ratio. For monatomic gas argon, the theoretical value of the heat capacity ratio is (see Equation 3.29). For Ar, the experimental value of κ is practically the same (see Table 3.2).

After the gas expansion, the gas temperature can be calculated from the ideal gas law

The work performed by the gas, on the adiabatic reversible expansion, can be calculated using Equation 3.53, which is .

Thus, the work performed by the gas is

The calculated values of temperature and work performed by the gas at the adiabatic reversible expansion can be compared with the values calculated for irreversible adiabatic process in the following example.


Example 3.5

Argon with the amount of 5 × 10–4 kmol is in a container at 273.15 K and 101,325 Pa. When the external pressure suddenly drops to the value of 16,237 Pa, argon adiabatically expands against this external pressure. What is the argon temperature and performed work by argon at this adiabatic expansion process?


At sudden change of the external pressure, argon undergoes irreversible adiabatic expansion. Therefore, the equation of adiabat (3.46) cannot be used in this case. The first law of thermodynamics applied to the adiabatic process is given by Equation 3.40, which is dQ = 0 = C V dT + pdV (where C' V = n m C V is the heat capacity of the given gas amount at constant volume; C V is the molar heat capacity at constant volume; n m is the argon molar amount; and p is the external pressure against which gas expands).

The change of the internal gas energy and work depends only on the initial and final states of the gas. Thus, at constant heat capacity: –p(V – V 0) = n m C V (T – T 0). The substitution for V and V 0 following from the ideal gas law (V = n m R 0 T/p and V 0 = n m R 0 T/p 0) and subsequent algebraic operations give the final temperature

  • T 0 is the initial temperature
  • p 0 is the initial pressure
  • R 0 is the universal gas constant
  • C V  = 3R 0/2 = 3 × 8,314 J/(K kmol)/2 = 12, 471 J/(K kmol) is the molar heat capacity at constant volume for monatomic molecules
Then, the gas temperature after the expansion is
The change of the gas internal energy is equal to the work performed by the gas (see Equation 3.39). Thus, −W =  − ΔU = n m C V (T 0 − T)

Hence, −W = 5 × 10−4 kmol × 12, 471 J kmol −1K −1(273.15 K − 181.17 K) = 573.54 J

Although in Examples 3.4 and 3.5, representing both reversible and irreversible adiabatic processes, the argon states were set up equally, disparate results are obtained. The gas cools less and performs less work at the adiabatic irreversible process than it does at the reversible process.


Example 3.6

The helium pressure drops from atmospheric pressure to 30 kPa when helium undergoes a reversible isothermal expansion at 300 K. What is the change in entropy when the expansion is performed with 4 g of helium?


4 g of He corresponds to He molar amount n m = M/M a = 4 × 10−3 kg/(4 kg/kmol) = 10−3 kmol.

Then, using Equation 3.86, the change of entropy is

The change of entropy is 10.12 J/K at this isothermal and reversible process.


Example 3.7

Saturated vapor pressures of 1 μ and 100 μ were measured at 1142 °C and 1427 °C for aluminum, respectively. What is the vaporization heat of aluminum?


At T 1 = 1142 + 273.15 = 1415.15 K vapor pressure is p 1= 3.4 μ = 3.4 × 10−3 Torr = 0.452 Pa.

At T 2 = 1427 + 273.15 = 1700.15 K vapor pressure is p 2 = 260 μ = 0.26 Torr = 34.58 Pa.

where the vaporization heat is equal to the enthalpy change: W = ΔH and R 0 = 8341 J/(kmol K) is the universal gas constant. Hence,

By subtracting (E.3.7.2) from (E.3.7.1), we obtain

from where
Then, the vaporization heat is approximately ΔH = 3.05 × 108 J/kmol = 305 kJ/mol.

This value can be calculated from B constant in Table 3.4 because B = ΔH/R 0 where R 0 is the universal gas constant. ΔH = BR 0 = 36,966.75 K × 8,314 = J kmol –1 K –1 = 3.07 × 108 J/kmol = 307 kJ/mol.


Example 3.8

Since saturation vapor pressures of materials are often expressed by semiempirical equations containing different logarithms and non-SI units, it is useful to illustrate the conversion of these equations to yield SI units as well as the conversion from one logarithm to the other.


The equation with natural logarithm

can be recast to the decimal logarithm. For the conversion to the decimal logarithm, the base is adapted to be 10. Then, the last equation can be transformed to p = 100.43429  (A − B/T), because
from where
Accordingly, the constants A and B given in Equation E.3.8.1 with the natural logarithm have to be multiplied by the factor of 0.4342945 to be tabulated as A′ and B′, for the equation with the decimal logarithm.

Alternatively, the transformation of the equation from the decimal to natural logarithm, we may use similar algebraic operations:

from where
Since ln10 = 2.302585, the constants A′ and B′ in Equation E.3.8.4 with the decimal logarithm have to be multiplied by the factor 2.302585 to obtain A and B constants, which can then be used in the equation with the natural logarithm.

If the equation gives the value of pressure p in the units of microns, it can be transformed to the SI pressure units (Pa). Since 1 μ = 10−3 Torr = 10−3 × 133.3 Pa = 0.1333 Pa, we may write

and hence, , where A″ = A′ − 0.875169.


Example 3.9

What is the saturated vapor pressure of cadmium at 273.15 and 500 K?


For the estimation of saturated vapor pressure of cadmium, we may use Equation 3.91 and Table 3.4:

Practically, the same value can also be read from the graphs for Cd in Figure 3.10. So, if there is a screw that is cadmium-plated close to a thermal source in vacuum, we may find that Cd is deposited on a sample. Cadmium also has relatively high saturation pressure (1.8 × 10–9 Pa) at room temperature. Therefore, Cd-plated surfaces are not recommended for vacuum applications. Very high saturation pressure also exhibits zinc (the constants A = 24.776 and B = 15,056.05), which is often used for plating metal surfaces. Zinc-plated surfaces are therefore not recommended for vacuum applications, either. For the same reasons, brass, being an alloy of copper and zinc, is unsuitable for ultrahigh vacuum applications.


Example 3.10

What is the saturated vapor pressure of red phosphorus at temperature of 455 K?

For calculation of vapor pressure of phosphorus, use equation

where the fitting constants A′ = 10.12, B′ = 5600, C = a′ + bT, a′ = 0.64, b′ = 5.27 × 10–4 are empirical constants, and pressure p is in pascals.


From which, the saturated vapor pressure of red phosphorus is about 1.02 Pa at 450 K.


Example 3.11

A boron-nitride (BN) film was deposited by a reactive radio frequency magnetron sputtering. The deposition rate was 200 nm/h. The XPS (X-ray photoelectron spectroscopy) compositional analysis shows that boron-to-nitrogen atomic ratio is 1/1, while the FTIR (Fourier transform infrared) analysis indicates that the ratio of cubic phase (cBN) to hexagonal (hBN) phase is 80%/20%. What was the ion-to-deposition flux ratio at the substrate bias of 60 V when the ion current density was 4 mA/cm 2?


For the atomic ratio B/N = 1/1, average molar mass is

Since the mass density of hBN is ρ H  = 2.50 g/cm 3 and that for cBN is ρ H  = 3.480 g/cm 3, the boron-nitride film composed of cBN 80%/hBN 20% has the mass density ρ H  = 0.8 × 3.48 g/cm 3 + 0.2 × 2.50 g/cm 3 = 3.224 g/cm 3.

At the deposition rate of 200 nm/h = 2 × 10−5 cm/h, the film with thickness of 200 nm is deposited for an hour. Thus, the volume grown over an area of 1 cm 2 is: V = 1 cm 2 × 2 × 10−5 cm = 2 × 10−5 cm 3.

The number of deposited atoms found from molar amount is

If current density j = 4 mA/cm 2, the current over the area of 1 cm 2 is I = (4 mA/cm 2) × 1 cm 2 = 4 × 10−3 A, then the number of incident ions on an area of 1 cm 2 for 1 h is
Hence, the ion to deposited flux ratio is

To form the boron-nitride film with a phase composition of cBN/hBN = 80%/20%, approximately 29 energetic ions per deposited atom are needed.


Example 3.12

Compare the mass deposition rates of cubic boron-nitride (cBN) films prepared by a DC Jet CVD (direct current jet chemical vapor deposition) and an ECR MW CVD (electron cyclotron resonance microwave chemical vapor deposition). The cBN films were deposited with the growth rates of 18 μm/h over 1 cm 2 using the DC Jet CVD and 0.8 μm/h over an area of 3 in. in diameter employing the ECR MW CVD.


Mass density of cBN: ρ = 3.48 g/cm 3; 1 Carat = 200 mg

  • DC Jet CVD: growth rate r = 18 μm/h = 1.8 × 10−5 m/h = 1.8 × 10−3 cm/h; A = 1 cm 2 Volume grown per unit time: V′ = Ar = 1 cm 2 × 1.8 × 10−3 cm/h = 1.8 × 10−3 cm 3/h
  • ECR CVD: growth rate r = 0.8 μm/h = 0.8 × 10−6 m/h = 8 × 10−7 m/h = 8 × 10−5 cm/h Diameter of area: d = 3.0 in. = 3.0 in. × 2.54 (cm/in.) = 7.62 cm;
    At the given deposition rate and areas, the mass deposited by ECR MW CVD is greater than that by DC Jet CVD, and it seems to be more effective because the deposition area is larger. In addition, DC Jet CVD is carried at temperatures above 1000 °C and high gas consumption. The cBN by DC jet is also contaminated with electrode materials. In contrast, ECR MW CVD allows to growth cBN with low gas consumption. The deposition is at temperatures below 1000 °C, and the cBN films do not show measurable traces of metallic contamination.

3.24  Sputtering, Etching, and Deposition

Sputtering is one of several processes that can take place at the interaction of energetic ions with the surfaces of solids (see also Appendix A.4). Sputtering is the process used for the deposition of thin films made of many materials including conductors, semiconductors, and insulators. The deposited films and source materials can be pure elements, oxides, nitrides, or a vast number of composite materials. We also use sputtering for cleaning surface in surface analytical methods as well as in sputtering ion pumps. We discuss the sputtering pump in Chapter 15.8. Therefore, we elucidate sputtering process very briefly herein.

Generally, in sputtering process, we use direct current (d.c.), radio frequency (r.f.), and pulse electric discharges. Their combinations are also employed. For sputtering, we can also use energetic ion and molecular beams. Energetic molecular beams are produced using ion neutralization by electrons, when ion beams pass via electron clouds emitted from hot filaments.

Sputtering devices can be designed in diode, triode, and electrode systems equipped with magnetic fields. The latter devices are known as magnetrons. Magnetrons can be in planar or cylindrical configurations. The planar configuration can be circular or rectangular, which may have very large sputtered areas. All these devices operate on the principle of sputtering, which is the process at which energetic ions impinge on a cathode (target) where they transfer their momenta to the surface atoms. The impact ions induce collisions, collision cascades, and displacement of surface particles. Due to the displacement, the target structure is stressed and compressed to such extent that at the structure relaxation, the surface particles are ejected to vacuum and then deposited. The number of particles sputtered per incident is called sputtering yield, that is,

The sputter particles include electrically neutral particles and ions (positive and negative). Although, the neutral particles absolutely dominate, ions may play very important role at the deposition and formation of the film microstructures. In material analyses, we are interested in ion sputtering yield and its enhancement. Therefore, we also define ion sputtering yields (positive or negative ions), which in contrast to sputtering yield is much more complex because it is strongly affected by the so-called matrix effect, that is, by surface chemical environment of the target. Depending on the electron affinity and ionization energy of the surface particle and those sputtered, some electrons can be transferred from the surface particles to the escaping neutral particles and vice versa. Thus, neutral particles can be converted to negative or positive ions to a considerable extent. Such enhancement of ion sputtering yields via the electron transfer is employed at secondary ion mass spectrometry (SIMS). However, the sputtering rate of neutral particles is far higher than the ion sputtering yield.

The number of sputtered particles N can be expressed by their mass M using equation for molar quantity:

while the number of ions N + can be determined from the total electric charge eN + impinging on an area A of the target for time t and electric current I in assumption that each ion carries elementary charge e. Thus,
The substitution for N and N + into the sputtering yield gives
  • h is the depth of the sputtered volume V
  • A is the sputtered area
  • ρ is the mass density of the sputtered material
  • r = h/t is the sputtering rate
The volume of sputtered material can be measured by a profilometer. Hence, the sputtering rate is
This indicates that the sputtering (etching) rate is dependent on the current density, nature of material, and sputtering yield that is dependent on the energy of ions and their masses. Sputtering yields are available for different ions and their energies and target materials in the forms of tables, graphical dependences, as well as there is software available for calculations.

The mass and deposition rate are proportional to the etching rate and can be theoretically estimated and experimentally measured, too.

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