Importance of First-Order Reactions in Materials 57

Chapter 2 Introduction to Kinetic Processes in Materials 41 Chapter 3 Second-Order and Multistep Reactions 61 Chapter 4 Temperature Dependence of the Reaction Rate Constant 95

2.7 Importance of First-Order Reactions in Materials 57

2.8 Chapter Summary 58

Exercises 58

References 60

process in MSE of Si deposition illustrates that diffusion in the gas phase and reaction kinetics need to be part of MSE education today.

The process variables under a materials engineer’s control that determine the rate of Si deposition are the various gas pressures, gas flow rates, and temperatures. Reactant diffusion to the surface, followed by a surface reaction, finally followed by product diffusion away from the surface are three kinetic steps in series. As will be demonstrated later, the slowest step in the sequence determines the overall rate of a series sequential step process. Usually, at low temperatures, the surface reaction rate is the slowest step and limits the rate of deposition. In contrast, at high temperatures, diffusion is slower and limits the rate of deposition. Therefore, in order to develop a rational model of this process, it is essential to understand both the kinetics of diffusion in the gas and the kinetics of surface and other reactions. The first part of this book focuses on what could be called “classical kinetics” that might be taught in a physical chemistry course. But most MSE students do not take physical chemistry as a separate course today, so there is a need to expose the student to the terminology and concepts of classical reaction kinetics. This chapter is the first of several chapters that attempts to do this.

2.1.2 E

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: D

^issolution

r

^atEsof

n

^a

c

^lanD

a

^l₂

o

₃

, c

^ontrast

D

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The kinetics of dissolution of solids in aqueous or other solutions, an area certainly important in the pro- cessing and/or performance of all materials—metals, ceramics, polymers, and biomaterials— illustrates the effect of the relative speed of these two series processes: reaction and transport. For example, sodium chloride (NaCl) is thermodynamically quite soluble in water and dissolves by the reaction:

NaCl(s)Na (solution) Cl (solution);⁺ + ⁻ ∆ ° ° = −G (0C) 7 84. kJ mole/ which gives for the equilibrium constant (Roine 2002):

Na Cl

a K

NaCl

e

+ − −∆ °

    = ( C)° = = .

G

0 e RT 31 53

(where [Na⁺] and [Cl⁻] are concentrations in mol/kg-soln, that is, molal concentrations). If the solid is pure NaCl, its thermodynamic activity is aNaCl = 1, so the equilibrium concentrations of Na⁺ and Cl⁻are [Na+]e=[Cl−]e= 31 53. = 5.62 mol/kg-soln at 0°C. The NaCl-H2O phase diagram with data taken from a variety of sources is sketched in Figure 2.2, showing the solubility of NaCl in H2O as a function of temperature, and the depression of the freezing point of water down to

Substrate

Gas 3 HCl

Diffusion Diffusion

Surface reaction SiHCl₃ + H₂

Si film

FIGURE 2.1 Schematic showing the deposition of a silicon film on a substrate by chemical vapor deposition from trichlorosilane (SiHCl₃) in a hydrogen carrier gas. The three series steps are as follows: (1) transport of the reactant gases to the surface, (2) the actual reaction at the surface, and (3) transport or diffusion of the prod- uct gases away from the surface. The slowest of the three steps controls the overall kinetics of the deposition.

the eutectic temperature of −21.1°C at 26.6 w/o (w/o = weight percent), or 10.05 m/o (m/o = mole percent), or 4.5 mol/kg-soln. Figure 2.3 schematically shows the sequential reaction and diffusion steps. In this case, the Na⁺ and Cl⁻ ions give up their nearest neighbor anion and cations in the solid surface for the polar H2O molecules in solution. The enthalpy for the reaction at 0°C is actu- ally positive, 8.132 kJ/mol, because the solvation energy of the ions is less than the energy of the ions in the crystal. But it is the high increase in entropy, 58.462 J/mol-K, going from the ordered solid to the disordered liquid,

∆G=∆H T S− ∆ =8 132 10. × ³−273 15 58 462. × . = −7 837. kJ mol

that produces the high solubility (Roine 2002). This reaction step is followed by the diffusion of the solvated ions away from the surface. For the dissolution of NaCl in water, the rate of dissolu- tion is controlled by diffusion in the liquid, which is easily demonstrated by stirring the solution—

essentially decreasing the diffusion distance—which increases the rate of dissolution. Experience shows that salt (NaCl) dissolves quite rapidly in water, at least at room temperature and above.

In contrast, the dissolution or corrosion of Al2O3 in acidic solutions occurs by the following reaction:

Al O s2 3 H solution Al solution3 H O solution G

6 2 3 2

( )+ ⁺( )→ ⁺( )+ ( ); ∆ °°( ° )30 C = −101 5. kJ mol with an equilibrium constant of (Roine 2002)

Al a

H a ^{H O} K C

Al O e +

+

 

  = = ×

3 2 3 6

2 17 2 3

30 3 09 10

( ° ) . .

The thermodynamic activities of water and alumina are 1.0 (not exactly true for the water but close enough for estimation purposes). If the solution is acidic, so that [H⁺] ≅ 1 mol/kg-soln (pH ≅ 0), this

Liquid Liquid + solid NaCl

Ice+ liquid

Ice + solid NaCl

Temperature (°C)

w/o NaCl

H2O NaCl

0

−21

26.6 w/o 10.05 m/o 4.5 mol/kg-soln

Tmp = 801°C

FIGURE 2.2 Sodium chloride–water phase diagram based on data from a number of sources.

+ + +

+ +

+ +

+ +

+ + +

+

+

+ +

+ +

NaCl crystal Surface Water molecules Cl⁻

Na⁺

Solution

Solvated Na⁺and Cl⁻ ions

−

−

−

−

−

− −

−

−

−

−

−

−

−

−

− −

−

FIGURE 2.3 The dissolution of a sodium chloride crystal in water with the solvation energy of the water molecules with the Na⁺ and Cl⁻ ions providing the necessary free energy for dissolution. The rate of dissolution of NaCl is generally controlled by liquid diffusion of the solvated ions away from the surface.

would make the equilibrium concentration of [Al⁺³] (or Al2O3) on the order of 10⁸ molal, which is clearly ridiculous because the molar concentration of pure water is only

H O2 1000M 1000 mol kg 18 55 6

[ ]

^{= = = . /}

and that of pure Al2O3,

Al O2 3 1000 mol kg 102 9 9

[ ]

^{= = . .}

Nevertheless, it is obvious that, thermodynamically, Al2O3 is quite soluble in acidic solutions.

However, the dissolution of Al2O3 in acids occurs so exceedingly slowly that, for all practical purposes, Al2O3 does not dissolve in acids! Why not? Diffusion of the ions in the solution to and from the surface of the solid should occur at about the same rate as the diffusion of the ions that con- trols the rate of dissolution of NaCl because diffusion coefficients in most liquids are about the same (DL ≈ 10⁻⁵ cm²/s)—as will be shown later. However, for Al2O3, the rate of dissolution is controlled by the rate of the surface chemical reaction between the H⁺ ions in the solution and the aluminum and oxygen ions at the surface of the alumina. This reaction rate is vanishingly small in spite of the fact that the thermodynamics for the dissolution are quite favorable. Here, stirring does not help because the reaction at the alumina surface controls the dissolution rate under all conditions and decreasing the distance that the ions diffuse by stirring does not affect the dissolution rate.

2.1.3 c

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E

^xamplEs

These examples of two identical processes for two solids—dissolution in an aqueous solution— illustrate the importance of the combined effects of the relative rates of the surface chemical reaction and diffusion in the liquid. They demonstrate the necessity of studying both chemical reaction kinetics and diffusion processes to understand more broadly the “phase transformation kinetics” of solids. These examples also illustrate two other important points: (1) the thermodynamics of the process must be known before the kinetics can be studied and (2) for reactions in series, sequential steps—in these examples, a sur- face reaction followed by liquid diffusion—it is the slower of the two steps that controls the overall rate of the dissolution reaction. Therefore, it is essential to investigate both chemical reaction kinetics—

particularly at solid surfaces and interfaces—and the macroscopic and atomistic aspects of diffusion in solids, liquids, and gases in order to develop a firm grasp of kinetic processes in materials.