CASTING The Science And Engineering Of Materials 6th Edition

Chapter

Principles of Solidification

Have You Ever Wondered?

• Whether water really does “freeze” at 0°C and “boil” at 100°C? • What is the process used to produce several million pounds of steels

and other alloys? • Is there a specific melting temperature for an alloy or a thermoplastic

material? • What factors determine the strength of a cast product?

• Why are most glasses and glass-ceramics processed by melting and casting?

O at some point during processing. Like water freezes to ice, molten materials solidify

f all the processing techniques used in the manufacturing of materials, solid- ification is probably the most important. All metallic materials, as well as many ceramics, inorganic glasses, and thermoplastic polymers, are liquid or molten

as they cool below their freezing temperature. In Chapter 3, we learned how materi- als are classified based on their atomic, ionic, or molecular order. During the solidifi- cation of materials that crystallize, the atomic arrangement changes from a short-range order (SRO) to a long-range order (LRO). The solidification of crystalline materials requires two steps. In the first step, ultra-fine crystallites, known as the nuclei of a solid phase, form from the liquid. In the second step, which can overlap with the first, the ultra-fine solid crystallites begin to grow as atoms from the liquid are attached to the nuclei until no liquid remains. Some materials, such as inorganic silicate glasses, will become solid without developing a long-range order (i.e., they remain amorphous). Many polymeric materials may develop partial crystallinity during solidification or processing.

The solidification of metallic, polymeric, and ceramic materials is an important process to study because of its effect on the properties of the materials involved. In this chapter, we will study the principles of solidification as they apply to pure metals. We will discuss solidification of alloys and more complex materials in subsequent chapters. We will first discuss the technological significance of solidification and then examine the mechanisms by which solidification occurs. This will be followed by an

330 CHAPTER 9

Principles of Solidification

examination of the microstructure of cast metallic materials and its effect on the material’s mechanical properties. We will also examine the role of casting as a materials shaping process. We will examine how techniques such as welding, brazing, and soldering are used for joining metals. Applications of the solidification process in single crystal growth and the solidification of glasses and polymers also will be discussed.

9-1 Technological Significance

The ability to use heat to produce, melt, and cast metals such as copper, bronze, and steel is regarded as an important hallmark in the development of mankind. The use of fire for reducing naturally occurring ores into metals and alloys led to the production of useful tools and other products. Today, thousands of years later, solidification is still considered one of the most important manufacturing processes. Several million pounds of steel, aluminum alloys, copper, and zinc are being produced through the casting process. The solidification process is also used to manufacture specific components (e.g., aluminum alloys for automotive wheels). Industry also uses the solidification process as a primary processing step to produce metallic slabs or ingots (a simple, and often large casting that later is processed into useful shapes). The ingots or slabs are then hot and cold worked through secondary processing steps into more useful shapes (i.e., sheets, wires, rods, plates, etc.). Solidification also is applied when joining metallic materials using techniques such as welding, brazing, and soldering.

We also use solidification for processing inorganic glasses; silicate glass, for example, is processed using the float-glass process. High-quality optical fibers and other materials, such as fiberglass, also are produced from the solidification of molten glasses. During the solidification of inorganic glasses, amorphous rather than crystalline materials are produced. In the manufacture of glass-ceramics, we first shape the materials by casting amorphous glasses and then crystallize them using a heat treatment to enhance their strength. Many thermoplastic materials such as polyethylene, polyvinyl chloride (PVC), polypropylene, and the like are processed into useful shapes (i.e., fibers, tubes, bottles, toys, utensils, etc.) using a process that involves melting and solidification. Therefore, solidification is an extremely important technology used to control the prop- erties of many melt-derived products as well as a tool for the manufacturing of modern engineered materials. In the sections that follow, we first discuss the nucleation and growth processes.

9-2 Nucleation

In the context of solidification, the term nucleation refers to the formation of the first nanocrystallites from molten material. For example, as water begins to freeze, nanocrys- tals, known as nuclei, form first. In a broader sense, the term nucleation refers to the ini- tial stage of formation of one phase from another phase. When a vapor condenses into liquid, the nanoscale sized drops of liquid that appear when the condensation begins are referred to as nuclei. Later, we will also see that there are many systems in which the nuclei of a solid (!) will form from a second solid material (") (i.e., "- to !-phase

9-2 Nucleation 331

transformation). What is interesting about these transformations is that, in most engi- neered materials, many of them occur while the material is in the solid state (i.e., there is no melting involved). Therefore, although we discuss nucleation from a solidification perspective, it is important to note that the phenomenon of nucleation is general and is associated with phase transformations.

We expect a material to solidify when the liquid cools to just below its freezing (or melting) temperature, because the energy associated with the crystalline structure of the solid is then less than the energy of the liquid. This energy difference between the liquid and the solid is the free energy per unit volume #G v and is the driving force for solidification.

When the solid forms, however, a solid-liquid interface is created (Figure 9-1(a)).

A surface free energy $ sl is associated with this interface. Thus, the total change in energy #G, shown in Figure 9-1(b), is

(9-1) where 4 3

4 ¢G = 2

3 p r 3 ¢G v + 4pr s sl

3 pr is the volume of a spherical solid of radius r, 4%r is the surface area of a spherical solid, $ sl is the surface free energy of the solid-liquid interface (in this case), and #G v is the free energy change per unit volume, which is negative since the phase transfor- mation is assumed to be thermodynamically feasible. Note that $ sl is not a strong func- tion of r and is assumed constant. It has units of energy per unit area. #G v also does not depend on r.

An embryo is a tiny particle of solid that forms from the liquid as atoms cluster together. The embryo is unstable and may either grow into a stable nucleus or redissolve. In Figure 9-1(b), the top curve shows the parabolic variation of the total surface energy (4%r 2

4 & $ sl ). The bottom most curve shows the total volume free energy change

term A pr 3 3 # ¢G v B . The curve in the middle shows the variation of #G. It represents the

Figure 9-1 (a) An interface is created when a solid forms from the liquid. (b) The total free energy of the solid-liquid system changes with the size of the solid. The solid is an embryo if its radius is less than the critical radius and is a nucleus if its radius is greater than the critical radius.

332 CHAPTER 9

Principles of Solidification

sum of the other two curves as given by Equation 9-1. At the temperature at which the solid and liquid phases are predicted to be in thermodynamic equilibrium (i.e., at the freezing temperature), the free energy of the solid phase and that of the liquid phase are equal (#G v = 0), so the total free energy change (#G) will be positive. When the solid is very small with a radius less than the critical radius for nucleation (r*) (Figure 9-1(b)), fur- ther growth causes the total free energy to increase. The critical radius (r*) is the minimum size of a crystal that must be formed by atoms clustering together in the liquid before the solid particle is stable and begins to grow.

The formation of embryos is a statistical process. Many embryos form and redis- solve. If by chance, an embryo forms with a radius that is larger than r*, further growth causes the total free energy to decrease. The new solid is then stable and sustainable since nucleation has occurred, and growth of the solid particle—which is now called a nucleus—begins. At the thermodynamic melting or freezing temperatures, the probabil- ity of forming stable, sustainable nuclei is extremely small. Therefore, solidification does not begin at the thermodynamic melting or freezing temperature. If the temperature con- tinues to decrease below the equilibrium freezing temperature, the liquid phase that should have transformed into a solid becomes thermodynamically increasingly unstable. Because the temperature of the liquid is below the equilibrium freezing temperature, the liquid is considered undercooled. The undercooling (#T) is the difference between the equilibrium freezing temperature and the actual temperature of the liquid. As the extent of undercooling increases, the thermodynamic driving force for the formation of a solid phase from the liquid overtakes the resistance to create a solid-liquid interface.

This phenomenon can be seen in many other phase transformations. When one solid phase (") transforms into another solid phase (!), the system has to be cooled to a temperature that is below the thermodynamic phase transformation temperature (at which the energies of the " and ! phases are equal). When a liquid is transformed into a vapor (i.e., boiling water), a bubble of vapor is created in the liquid. In order to create the trans- formation though, we need to superheat the liquid above its boiling temperature! Therefore, we can see that liquids do not really freeze at their freezing temperature and do not really boil at their boiling point! We need to undercool the liquid for it to solidify and superheat it for it to boil!

Homogeneous Nucleation As liquid cools to temperatures below the equilibrium freezing temperature, two factors combine to favor nucleation. First, since atoms are losing their thermal energy, the probability of forming clusters to form larger embryos increases. Second, the larger volume free energy difference between the liquid and the solid reduces the critical size (r*) of the nucleus. Homogeneous nucleation occurs when the undercooling becomes large enough to cause the formation of a stable nucleus.

The size of the critical radius r* for homogeneous nucleation is given by

where #H f is the latent heat of fusion per unit volume, T m is the equilibrium solidification temperature in kelvin, and #T = (T m - T) is the undercooling when the liquid temperature

is T. The latent heat of fusion represents the heat given up during the liquid-to-solid trans- formation. As the undercooling increases, the critical radius required for nucleation

decreases. Table 9-1 presents values for $ sl , #H f , and typical undercoolings observed exper- imentally for homogeneous nucleation. The following example shows how we can calculate the critical radius of the nucleus for the solidification of copper.

9-2 Nucleation 333

TABLE 9-1 ■ Values for freezing temperature, latent heat of fusion, surface energy, and maximum undercooling for selected materials

Freezing

Solid-Liquid

Typical Undercooling

Temperature

Heat of Fusion

Interfacial Energy

for Homogeneous

Nucleation (#T) Material

(J >>cm 3 )

(J >>cm 2 )

Example 9-1 Calculation of Critical Radius for the Solidification

of Copper

Calculate the size of the critical radius and the number of atoms in the critical nucleus when solid copper forms by homogeneous nucleation. Comment on the size of the nucleus and assumptions we made while deriving the equation for the radius of the nucleus.

SOLUTION

¢T = 236°C T m = 1085 + 273 = 1358 K

From Table 9-1 for Cu:

¢H 3

f = 1628 J>cm s sl = 177 * 10 -7 J

>cm 2

Thus, r* is given by

= 12.51 * 10 -8 cm

¢H f ¢T

Note that a temperature difference of 1°C is equal to a temperature change of 1 K, or #T = 236°C = 236 K.

The lattice parameter for FCC copper is a 0 = 0.3615 nm = 3.615 * 10 -8 cm.

Thus, the unit cell volume is given by

V ) 3 3 unit cell 3 = (a 0 = (3.615 * 10 -8 ) = 47.24 * 10 -24 cm

The volume of the critical radius is given by

V 4 3 r* 4 = 3 p r = Q 3 p

R(12.51 * 10 -8 ) 3 = 8200 * 10 - 24 cm 3

334 CHAPTER 9

Principles of Solidification

The number of unit cells in the critical nucleus is

V unit cell

8200 * 10 -24

= 174 unit cells

r*

47.24 * 10 -24

Since there are four atoms in each FCC unit cell, the number of atoms in the criti- cal nucleus must be

(4 atoms cell)(174 cells nucleus) > > = 696 atoms nucleus > In these types of calculations, we assume that a nucleus that is made from only a few

hundred atoms still exhibits properties similar to those of bulk materials. This is not strictly correct and as such is a weakness of the classical theory of nucleation.

Heterogeneous Nucleation From Table 9-1, we can see that water will not solidify into ice via homogeneous nucleation until we reach a temperature of

-40°C (undercooling of 40°C)! Except in controlled laboratory experiments, homoge- neous nucleation never occurs in liquids. Instead, impurities in contact with the liquid,

either suspended in the liquid or on the walls of the container that holds the liquid, provide a surface on which the solid can form (Figure 9-2). Now, a radius of curvature greater than the critical radius is achieved with very little total surface between the solid and liquid. Relatively few atoms must cluster together to produce a solid particle that has the required radius of curvature. Much less undercooling is required to achieve the criti- cal size, so nucleation occurs more readily. Nucleation on preexisting surfaces is known as heterogeneous nucleation. This process is dependent on the contact angle ( ') for the nucle- ating phase and the surface on which nucleation occurs. The same type of phenomenon occurs in solid-state transformations.

Rate of Nucleation The rate of nucleation (the number of nuclei formed per unit time) is a function of temperature. Prior to solidification, of course, there is no

nucleation and, at temperatures above the freezing point, the rate of nucleation is zero. As the temperature drops, the driving force for nucleation increases; however, as the temper- ature decreases, atomic diffusion becomes slower, hence slowing the nucleation process.

Figure 9-2

A solid forming on an impurity can assume the critical radius with a smaller increase in the surface energy. Thus, heterogeneous nucleation can occur with relatively low undercoolings.

9-3 Applications of Controlled Nucleation 335

Thus, a typical rate of nucleation reaches a maximum at some temperature below the transformation temperature. In heterogeneous nucleation, the rate of nucleation is dic- tated by the concentration of the nucleating agents. By considering the rates of nucleation and growth, we can predict the overall rate of a phase transformation.

9-3 Applications of Controlled Nucleation

Grain Size Strengthening When a metal casting freezes, impurities in the melt and the walls of the mold in which solidification occurs serve as heterogeneous

nucleation sites. Sometimes we intentionally introduce nucleating particles into the liquid. Such practices are called grain refinement or inoculation. Chemicals added to molten met- als to promote nucleation and, hence, a finer grain size, are known as grain refiners or inoc- ulants. For example, a combination of 0.03% titanium (Ti) and 0.01% boron (B) is added to many liquid-aluminum alloys. Tiny particles of an aluminum titanium compound

(Al 3 Ti) or titanium diboride (TiB 2 ) form and serve as sites for heterogeneous nucleation. Grain refinement or inoculation produces a large number of grains, each beginning to grow from one nucleus. The greater grain boundary area provides grain size strengthening in metallic materials. This was discussed using the Hall-Petch equation in Chapter 4.

Second-Phase Strengthening In Chapters 4 and 5, we learned that in metallic materials, dislocation motion can be resisted by grain boundaries or the forma-

tion of ultra-fine precipitates of a second phase. Strengthening materials using ultra-fine precipitates is known as dispersion strengthening or second-phase strengthening; it is used extensively in enhancing the mechanical properties of many alloys. This process involves solid-state phase transformations (i.e., one solid transforming into another). The grain boundaries as well as atomic level defects within the grains of the parent phase (") often serve as nucleation sites for heterogeneous nucleation of the new phase (!). This nucleation phenomenon plays a critical role in strengthening mechanisms. This will be discussed in Chapters 10 and 11.

Glasses For rapid cooling rates and or high viscosity melts, there may be insuf- >

ficient time for nuclei to form and grow. When this happens, the liquid structure is locked into place and an amorphous—or glassy—solid forms. The complex crystal structure of many ceramic and polymeric materials prevents nucleation of a solid crystalline structure even at slow cooling rates. Some alloys with special compositions have sufficiently com- plex crystal structures, so they may form amorphous materials if cooled rapidly from the melt. These materials are known as metallic glasses. Typically, good metallic glass formers are multi-component alloys, often with large differences in the atomic sizes of the elemen- tal constituents. This complexity limits the solid solubilities of the elements in the crys- talline phases, thus requiring large chemical fluctuations to form the critical-sized crystalline nuclei. Metallic glasses were initially produced via rapid solidification processing in which

cooling rates of 10 6 °C -1 were attained by forming continuous, thin metallic ribbons about 0.0015 in. thick. (Heat can be extracted quickly from ribbons with a large surface area to volume ratio.)

Bulk metallic glasses with diameters greater than 1 in. are now produced using a variety of processing techniques for compositions that require cooling rates on the order of only tens of degrees per second. Many bulk metallic glass compositions have been

336 CHAPTER 9

Principles of Solidification

discovered, including Pd 40 Ni 40 P 20 and Zr 41.2 Ti 13.8 Cu 12.5 Ni 10.0 Be 22.5 . Many metallic glasses have strengths in excess of 250,000 psi while retaining fracture toughnesses of more than 10,000 psi 1in. Excellent corrosion resistance, magnetic properties, and other physical properties make these materials attractive for a wide variety of applications.

Other examples of materials that make use of controlled nucleation are colored glass and photochromic glass (glass that can change color or tint upon exposure to sun- light). In these otherwise amorphous materials, nanocrystallites of different materials are deliberately nucleated. The crystals are small and, hence, do not make the glass opaque. They do have special optical properties that make the glass brightly colored or pho- tochromic.

Many materials formed from a vapor phase can be cooled quickly so that they do not crystallize and, therefore, are amorphous (i.e., amorphous silicon), illustrating that amorphous or non-crystalline materials do not always have to be formed from melts.

Glass-ceramics The term glass-ceramics refers to engineered materials that begin as amorphous glasses and end up as crystalline ceramics with an ultra-fine grain size. These materials are then nearly free from porosity, mechanically stronger, and often much more resistant to thermal shock. Nucleation does not occur easily in silicate glasses; how-

ever, we can help by introducing nucleating agents such as titania (TiO 2 ) and zirconia (ZrO 2 ). Engineered glass-ceramics take advantage of the ease with which glasses can be melted and formed. Once a glass is formed, we can heat it to deliberately form ultra-fine crystals, obtain- ing a material that has considerable mechanical toughness and thermal shock resistance. The crystallization of glass-ceramics continues until all of the material crystallizes (up to 99.9% crystallinity can be obtained). If the grain size is kept small (! 50–100 nm), glass- ceramics can often be made transparent. All glasses eventually will crystallize as a result of exposure to high temperatures for long lengths of times. In order to produce a glass-ceramic, however, the crystallization must be carefully controlled.

9-4 Growth Mechanisms

Once the solid nuclei of a phase form (in a liquid or another solid phase), growth begins to occur as more atoms become attached to the solid surface. In this discussion, we will concentrate on the nucleation and growth of crystals from a liquid. The nature of the growth of the solid nuclei depends on how heat is removed from the molten material. Let’s consider casting a molten metal in a mold, for example. We assume we have a nearly pure metal and not an alloy (as solidification of alloys is different in that in most cases, it occurs over a range of temperatures). In the solidification process, two types of heat must be removed: the specific heat of the liquid and the latent heat of fusion. The specific heat is the heat required to change the temperature of a unit weight of the material by one degree. The specific heat must be removed first, either by radiation into the surrounding atmos- phere or by conduction into the surrounding mold, until the liquid cools to its freezing temperature. This is simply a cooling of the liquid from one temperature to a temperature at which nucleation begins.

We know that to melt a solid we need to supply heat. Therefore, when solid crys- tals form from a liquid, heat is generated! This type of heat is called the latent heat of fusion (# H f ). The latent heat of fusion must be removed from the solid-liquid interface before solidification is completed. The manner in which we remove the latent heat of fusion determines the material’s growth mechanism and final structure of a casting.

9-4 Growth Mechanisms 337

Figure 9-3

When the temperature of the liquid is above the freezing temperature,

a protuberance on the solid-liquid interface will not grow, leading to maintenance of a planar interface. Latent heat is removed from the interface through the solid.

Planar Growth When a well-inoculated liquid (i.e., a liquid containing nucleating agents) cools under equilibrium conditions, there is no need for undercool-

ing since heterogeneous nucleation can occur. Therefore, the temperature of the liquid ahead of the solidification front (i.e., solid-liquid interface) is greater than the freezing temperature. The temperature of the solid is at or below the freezing temperature. During solidification, the latent heat of fusion is removed by conduction from the solid- liquid interface. Any small protuberance that begins to grow on the interface is surrounded by liquid above the freezing temperature (Figure 9-3). The growth of the protuberance then stops until the remainder of the interface catches up. This growth mechanism, known as planar growth, occurs by the movement of a smooth solid-liquid interface into the liquid.

Dendritic Growth When the liquid is not inoculated and the nucleation is poor, the liquid has to be undercooled before the solid forms (Figure 9-4). Under these

conditions, a small solid protuberance called a dendrite, which forms at the interface, is encouraged to grow since the liquid ahead of the solidification front is undercooled. The word dendrite comes from the Greek word dendron that means tree. As the solid dendrite grows, the latent heat of fusion is conducted into the undercooled liquid, raising the tem- perature of the liquid toward the freezing temperature. Secondary and tertiary dendrite arms can also form on the primary stalks to speed the evolution of the latent heat. Dendritic growth continues until the undercooled liquid warms to the freezing tempera- ture. Any remaining liquid then solidifies by planar growth. The difference between pla- nar and dendritic growth arises because of the different sinks for the latent heat of fusion. The container or mold must absorb the heat in planar growth, but the undercooled liquid absorbs the heat in dendritic growth.

In pure metals, dendritic growth normally represents only a small fraction of the total growth and is given by

c Dendritic fraction = f = ¢T

(9-3)

¢H f

where c is the specific heat of the liquid. The numerator represents the heat that the undercooled liquid can absorb, and the latent heat in the denominator represents the total heat that must be given up during solidification. As the undercooling #T increases,

338 CHAPTER 9

Principles of Solidification

Figure 9-4 (a) If the liquid is undercooled, a protuberance on the solid-liquid interface can grow rapidly as a dendrite. The latent heat of fusion is removed by raising the temperature of the liquid back to the freezing temperature. (b) Scanning electron micrograph of dendrites in steel (* 15). (Reprinted courtesy of Don Askeland.)

more dendritic growth occurs. If the liquid is well-inoculated, undercooling is almost zero and growth would be mainly via the planar front solidification mechanism.

9-5 Solidification Time and Dendrite Size

The rate at which growth of the solid occurs depends on the cooling rate, or the rate of heat extraction. A higher cooling rate produces rapid solidification, or short solidifica- tion times. The time t s required for a simple casting to solidify completely can be calcu- lated using Chvorinov’s rule:

t s =B a (9-4)

where V is the volume of the casting and represents the amount of heat that must be removed before freezing occurs, A is the surface area of the casting in contact with the mold and rep- resents the surface from which heat can be transferred away from the casting, n is a constant (usually about 2), and B is the mold constant. The mold constant depends on the properties and initial temperatures of both the metal and the mold. This rule basically accounts for the geometry of a casting and the heat transfer conditions. The rule states that, for the same con- ditions, a casting with a small volume and relatively large surface area will cool more rapidly.

9-5 Solidification Time and Dendrite Size 339

Example 9-2 Redesign of a Casting for Improved Strength

Your company currently is producing a disk-shaped brass casting 2 in. thick and

18 in. in diameter. You believe that by making the casting solidify 25% faster the improvement in the tensile properties of the casting will permit the casting to be made lighter in weight. Design the casting to permit this. Assume that the mold con-

stant is 22 min 2 >in. for this process and n = 2.

SOLUTION

One approach would be to use the same casting process, but reduce the thickness of the casting. The thinner casting would solidify more quickly and, because of the faster cooling, should have improved mechanical properties. Chvorinov’s rule helps us cal- culate the required thickness. If d is the diameter and x is the thickness of the casting, then the volume, surface area, and solidification time of the 2-in. thick casting are

V 2 = (%>4)d 2 x

= (%>4)(18) 3 (2) = 508.9 in.

A 2 2 = 2(%>4)d 2 + %dx = 2(%>4)(18) + %(18)(2) = 622 in.

V 2 508.9 2

t =B a

A b = (22) a 622 b = 14.73 min

The solidification time t r of the redesigned casting should be 25% shorter than the current time:

t r = 0.75t = (0.75)(14.73) = 11.05 min

Since the casting conditions have not changed, the mold constant B is unchanged. The V >A ratio of the new casting is

t r =B a

A b = (22) a r b A r = 11.05 min

A b = 0.502 in. r or A r = 0.709 in.

If x is the required thickness for our redesigned casting, then

V (p

>4)d x

(p

>4)(18) 2 (x)

2(p >4)d + pdx

2(p >4)(18) + p(18)(x)

2 = 0.709 in.

Therefore, x = 1.68 in. This thickness provides the required solidification time, while reducing the overall

weight of the casting by more than 15%.

Solidification begins at the surface, where heat is dissipated into the surrounding mold material. The rate of solidification of a casting can be described by how rapidly the thickness d of the solidified skin grows:

d =k solidification 1t - c 1 (9-5) where t is the time after pouring, k solidification is a constant for a given casting material and

mold, and c 1 is a constant related to the pouring temperature.

340 CHAPTER 9

Principles of Solidification

Figure 9-5 (a) The secondary dendrite arm spacing (SDAS). (b) Dendrites in an aluminum alloy (* 50). (From ASM Handbook, Vol. 9, Metallography and Microstructure (1985), ASM

International, Materials Park, OH 44073-0002.)

Effect on Structure and Properties The solidification time affects the size of the dendrites. Normally, dendrite size is characterized by measuring the distance between the secondary dendrite arms (Figure 9-5). The secondary dendrite arm spacing (SDAS) is reduced when the casting freezes more rapidly. The finer, more exten- sive dendritic network serves as a more efficient conductor of the latent heat to the under- cooled liquid. The SDAS is related to the solidification time by

(9-6) where m and k are constants depending on the composition of the metal. This relation-

SDAS = kt m

ship is shown in Figure 9-6 for several alloys. Small secondary dendrite arm spacings are associated with higher strengths and improved ductility (Figure 9-7).

Rapid solidification processing is used to produce exceptionally fine secondary dendrite arm spacings; a common method is to produce very fine liquid droplets that freeze into solid particles. This process is known as spray atomization. The tiny droplets

freeze at a rate of about 10 4 °C >s, producing powder particles that range from !5–100 (m. This cooling rate is not rapid enough to form a metallic glass, but does produce a fine

dendritic structure. By carefully consolidating the solid droplets by powder metallurgy processes, improved properties in the material can be obtained. Since the particles are

Figure 9-6

The effect of solidification time on the secondary dendrite arm spacings of copper, zinc, and aluminum.

9-5 Solidification Time and Dendrite Size 341

Figure 9-7

The effect of the secondary dendrite arm spacing on the mechanical properties of an aluminum casting alloy.

derived from a melt, many complex alloy compositions can be produced in the form of chemically homogenous powders.

The following three examples discuss how Chvorinov’s rule, the relationship between SDAS and the time of solidification, and the SDAS and mechanical properties can be used to design casting processes.

Example 9-3 Secondary Dendrite Arm Spacing for Aluminum Alloys

Determine the constants in the equation that describe the relationship between secondary dendrite arm spacing and solidification time for aluminum alloys (Figure 9-6).

SOLUTION

We could obtain the value of SDAS at two times from the graph and calculate k and m using simultaneous equations; however, if the scales on the ordinate and abscissa are equal for powers of ten (as in Figure 9-6), we can obtain the slope m from the log-log plot by directly measuring the slope of the graph. In Figure 9-6, we can mark five equal units on the vertical scale and 12 equal units on the horizontal scale. The slope is

The constant k is the value of SDAS when t s = 1 s, since log SDAS = log k + m log t s

If t s = 1 s, m log t s = 0, and SDAS = k, from Figure 9-6: -4 k = 7 * 10 cm

342 CHAPTER 9

Principles of Solidification

Example 9-4 Time of Solidification

A 4-in.-diameter aluminum bar solidifies to a depth of 0.5 in. beneath the surface in 5 minutes. After 20 minutes, the bar has solidified to a depth of 1.5 in. How much time is required for the bar to solidify completely?

SOLUTION

From our measurements, we can determine the constants k solidification and c 1 in Equation 9-5:

0.5 in. = k solidification 1(5 min) - c 1 or c 1 = k15 - 0.5

1.5 in. = k solidification 1(20 min) - c 1 = k120 - ( k15 - 0.5)

1.5 = k solidification (120 - 15) + 0.5

4.472 - 2.236 = 0.447 1min

c 1 = (0.447)15 - 0.5 = 0.5 in.

Solidification is complete when d = 2 in. (half the diameter, since freezing is occur- ring from all surfaces):

2 = 0.4471t - 0.5 1t = 2 + 0.5 5.59 0.447 = t = 31.25 min

In actual practice, we would find that the total solidification time is somewhat longer than 31.25 min. As solidification continues, the mold becomes hotter and is less effective in removing heat from the casting.

Example 9-5 Design of an Aluminum Alloy Casting

Design the thickness of an aluminum alloy casting with a length of 12 in., a width of 8 in., and a tensile strength of 40,000 psi. The mold constant in Chvorinov’s rule

for aluminum alloys cast in a sand mold is 45 min 2 >in . Assume that data shown in Figures 9-6 and 9-7 can be used.

SOLUTION

In order to obtain a tensile strength of 42,000 psi, a secondary dendrite arm spac- ing of about 0.007 cm is required (see Figure 9-7). From Figure 9-6 we can deter- mine that the solidification time required to obtain this spacing is about 300 s or

5 minutes.

From Chvorinov’s rule

t s =B a

9-6 Cooling Curves 343

where B = 45 min>in. 2 and x is the thickness of the casting. Since the length is 12 in. and the width is 8 in.,

V = (8)(12)(x) = 96x

A = (2)(8)(12) + (2)(x)(8) + (2)(x)(12) = 40x + 192

96x

5 min = (45 min>in. 2 ) a b

40x + 192 96x

40x + 192 = 1(5>45) = 0.333 96x = 13.33x + 64

x = 0.77 in.

9-6 Cooling Curves

We can summarize our discussion at this point by examining cooling curves. A cooling curve shows how the temperature of a material (in this case, a pure metal) decreases with time [Figure 9-8 (a) and (b)]. The liquid is poured into a mold at the pouring temperature, point A. The difference between the pouring temperature and the freezing temperature is the superheat. The specific heat is extracted by the mold until the liquid reaches the freez- ing temperature (point B). If the liquid is not well-inoculated, it must be undercooled

B –C: Undercooling is necessary for homogeneous

nucleation to occur

Figure 9-8 (a) Cooling curve for a pure metal that has not been well-inoculated. The liquid cools as specific heat is removed (between points

A and B). Undercooling is thus necessary (between points B and C). As the

nucleation begins (point C), latent heat of fusion is released causing an increase in the temperature of the liquid. This process is known as recalescence (point

C to point D). The metal continues to solidify at a constant temperature ( T melting ). At point

E, solidification is complete. The solid casting continues to cool from this point.

(b) Cooling curve for a well-inoculated, but otherwise pure, metal. No undercooling is needed. Recalescence is not observed. Solidification begins at the melting temperature.

344 CHAPTER 9

Principles of Solidification

(point B to C). The slope of the cooling curve before solidification begins is the cooling rate ¢T . As nucleation begins (point C), latent heat of fusion is given off, and the tem-

¢t perature rises. This increase in temperature of the undercooled liquid as a result of nucle-

ation is known as recalescence (point C to D). Solidification proceeds isothermally at the melting temperature (point D to E) as the latent heat given off from continued solidifica- tion is balanced by the heat lost by cooling. This region between points D and E, where the temperature is constant, is known as the thermal arrest. A thermal arrest, or plateau, is produced because the evolution of the latent heat of fusion balances the heat being lost because of cooling. At point E, solidification is complete, and the solid casting cools from point E to room temperature.

If the liquid is well-inoculated, the extent of undercooling and recalescence is usu- ally very small and can be observed in cooling curves only by very careful measurements. If effective heterogeneous nuclei are present in the liquid, solidification begins at the freezing temperature [Figure 9-8 (b)]. The latent heat keeps the remaining liquid at the freezing tem- perature until all of the liquid has solidified and no more heat can be evolved. Growth under these conditions is planar. The total solidification time of the casting is the time required to remove both the specific heat of the liquid and the latent heat of fusion. Measured from the time of pouring until solidification is complete, this time is given by Chvorinov’s rule. The local solidification time is the time required to remove only the latent heat of fusion at a par- ticular location in the casting; it is measured from when solidification begins until solidifi- cation is completed. The local solidification times (and the total solidification times) for liquids solidified via undercooled and inoculated liquids will be slightly different.

We often use the terms “melting temperature” and “freezing temperature” while discussing solidification. It would be more accurate to use the term “melting tempera- ture” to describe when a solid turns completely into a liquid. For pure metals and com- pounds, this happens at a fixed temperature (assuming fixed pressure) and without superheating. “Freezing temperature” or “freezing point” can be defined as the tempera- ture at which solidification of a material is complete.

9-7 Cast Structure

In manufacturing components by casting, molten metals are often poured into molds and permitted to solidify. The mold produces a finished shape, known as a casting. In other cases, the mold produces a simple shape called an ingot. An ingot usually requires exten- sive plastic deformation before a finished product is created. A macrostructure sometimes referred to as the ingot structure, consists of as many as three regions (Figure 9-9). (Recall that in Chapter 2 we used the term “macrostructure” to describe the structure of a mate- rial at a macroscopic scale. Hence, the term “ingot structure” may be more appropriate.)

Chill Zone The chill zone is a narrow band of randomly oriented grains at the sur- face of the casting. The metal at the mold wall is the first to cool to the freezing temperature. The mold wall also provides many surfaces at which heterogeneous nucleation takes place.

Columnar Zone The columnar zone contains elongated grains oriented in a particular crystallographic direction. As heat is removed from the casting by the mold material, the grains in the chill zone grow in the direction opposite to that of the heat

9-7 Cast Structure 345

Figure 9-9

Development of the ingot structure of a casting during solidification: (a) nucleation begins, (b) the chill zone forms, (c) preferred growth produces the columnar zone, and (d) additional nucleation creates the equiaxed zone.

flow, or from the coldest toward the hottest areas of the casting. This tendency usually means that the grains grow perpendicular to the mold wall.

Grains grow fastest in certain crystallographic directions. In metals with a cubic crystal structure, grains in the chill zone that have a 81009 direction perpendicular to the mold wall grow faster than other less favorably oriented grains (Figure 9-10). Eventually, the grains in the columnar zone have 81009 directions that are parallel to one another, giv- ing the columnar zone anisotropic properties. This formation of the columnar zone is

Figure 9-10 Competitive growth of the grains in the chill zone results in only those grains with favorable orientations developing into columnar grains.

346 CHAPTER 9

Principles of Solidification

influenced primarily by growth—rather than nucleation—phenomena. The grains may be composed of many dendrites if the liquid is originally undercooled. The solidification may proceed by planar growth of the columnar grains if no undercooling occurs.

Equiaxed Zone Although the solid may continue to grow in a columnar man- ner until all of the liquid has solidified, an equiaxed zone frequently forms in the center of the casting or ingot. The equiaxed zone contains new, randomly oriented grains, often caused by a low pouring temperature, alloying elements, or grain refining or inoculating agents. Small grains or dendrites in the chill zone may also be torn off by strong convec- tion currents that are set up as the casting begins to freeze. These also provide heteroge- neous nucleation sites for what ultimately become equiaxed grains. These grains grow as relatively round, or equiaxed, grains with a random orientation, and they stop the growth of the columnar grains. The formation of the equiaxed zone is a nucleation-controlled process and causes that portion of the casting to display isotropic behavior.

By understanding the factors that influence solidification in different regions, it is possible to produce castings that first form a “skin” of a chill zone and then dendrites. Metals and alloys that show this macrostructure are known as skin-forming alloys. We also can control the solidification such that no skin or advancing dendritic arrays of grains are seen; columnar to equiaxed switchover is almost at the mold walls. The result is a cast- ing with a macrostructure consisting predominantly of equiaxed grains. Metals and alloys that solidify in this fashion are known as mushy-forming alloys since the cast material seems like a mush of solid grains floating in a liquid melt. Many aluminum and magne- sium alloys show this type of solidification. Often, we encourage an all-equiaxed structure and thus create a casting with isotropic properties by effective grain refinement or inocu- lation. In a later section, we will examine one case (turbine blades) where we control solid- ification to encourage all columnar grains and hence anisotropic behavior.

Cast ingot structure and microstructure are important particularly for compo- nents that are directly cast into a final shape. In many situations though, as discussed in Section 9-1, metals and alloys are first cast into ingots, and the ingots are subsequently subjected to thermomechanical processing (e.g., rolling, forging etc.). During these steps, the cast macrostructure is broken down and a new microstructure will emerge, depending upon the thermomechanical process used (Chapter 8).

9-8 Solidification Defects

Although there are many defects that potentially can be introduced during solidification, shrinkage and porosity deserve special mention. If a casting contains pores (small holes), the cast component can fail catastrophically when used for load-bearing applications (e.g., turbine blades).

Shrinkage Almost all materials are more dense in the solid state than in the liquid state. During solidification, the material contracts, or shrinks, as much as 7% (Table 9-2).

Often, the bulk of the shrinkage occurs as cavities, if solidification begins at all surfaces of the casting, or pipes, if one surface solidifies more slowly than the others (Figure 9-11). The presence of such pipes can pose problems. For example, if in the pro- duction of zinc ingots a shrinkage pipe remains, water vapor can condense in it. This water can lead to an explosion if the ingot gets introduced in a furnace in which zinc is being remelted for such applications as hot-dip galvanizing.

9-8 Solidification Defects 347

TABLE 9-2 ■ Shrinkage during solidification for selected materials Material

2.7 Ga +3.2 (expansion)

+8.3 (expansion)

Low-carbon steel

High-carbon steel

White Cast Iron

Gray Cast Iron

+1.9 (expansion)

Note: Some data from DeGarmo, E. P., Black, J. T., and Koshe, R. A. Materials and Processes in Manufacturing, Prentice Hall, 1997.

Figure 9-11

Several types of macroshrinkage can occur, including cavities and pipes. Risers can be used to help compensate for shrinkage.

A common technique for controlling cavity and pipe shrinkage is to place a riser, or an extra reservoir of metal, adjacent and connected to the casting. As the casting solid- ifies and shrinks, liquid metal flows from the riser into the casting to fill the shrinkage void. We need only to ensure that the riser solidifies after the casting and that there is an internal liquid channel that connects the liquid in the riser to the last liquid to solidify in the casting. Chvorinov’s rule can be used to help design the size of the riser. The follow- ing example illustrates how risers can be designed to compensate for shrinkage.

Example 9-6 Design of a Riser for a Casting

Design a cylindrical riser, with a height equal to twice its diameter, that will compen- sate for shrinkage in a 2 cm * 8 cm * 16 cm, casting (Figure 9-12).

Figure 9-12

The geometry of the casting and riser (for Example 9-6).

348 CHAPTER 9

Principles of Solidification

SOLUTION

We know that the riser must freeze after the casting. To be conservative, we typ- ically require that the riser take 25% longer to solidify than the casting. Therefore,

t r = 1.25t c or B a

A b = 1.25 B r a A b c

The subscripts r and c stand for riser and casting, respectively. The mold constant

B is the same for both casting and riser, so

A b r = 11.25 a A b c

The volume of the casting is

V c = (2 cm)(8 cm)(16 cm) = 256 cm 3

The area of the riser adjoined to the casting must be subtracted from the total surface area of the casting in order to calculate the surface area of the casting in contact with the mold:

c = (2)(2 cm)(8 cm) + (2)(2 cm)(16 cm) + (2)(8 cm)(16 cm) - 4 = 352 cm - 4 where D is the diameter of the cylindrical riser. We can write equations for the vol-

ume and area of the cylindrical riser, noting that the cylinder height H = 2D:

4 (2D) =

4 + pDH = 4 + pD(2D) = 4

where again we have not included the area of the riser adjoined to the casting in the area calculation. The volume to area ratio of the riser is given by

V (pD 3 >2)

(9pD 2 >4) 9

and must be greater than that of the casting according to

a b = A D r 9 7 11.25 a A b c

Substituting,

2 256 cm 3

9 D 7 11.25 a 352 cm 2 - pD 2 >4 b

Solving for the smallest diameter for the riser:

D = 3.78 cm

Although the volume of the riser is less than that of the casting, the riser solidifies more slowly because of its compact shape.

9-8 Solidification Defects 349

Figure 9-13 (a) Shrinkage can occur between the dendrite arms. (b) Small secondary dendrite arm spacings result in smaller, more evenly distributed shrinkage porosity. (c) Short primary arms can help avoid shrinkage. (d) Interdendritic shrinkage in an aluminum alloy is shown (* 80). (Reprinted courtesy of Don Askeland.)

Interdendritic Shrinkage This consists of small shrinkage pores between dendrites (Figure 9-13). This defect, also called microshrinkage or shrinkage porosity, is difficult to prevent by the use of risers. Fast cooling rates may reduce problems with interdendritic shrinkage; the dendrites may be shorter, permitting liquid to flow through the dendritic network to the solidifying solid interface. In addition, any shrink- age that remains may be finer and more uniformly distributed.

Gas Porosity Many metals dissolve a large quantity of gas when they are molten. Aluminum, for example, dissolves hydrogen. When the aluminum solidifies, how- ever, the solid metal retains in its crystal structure only a small fraction of the hydrogen since the solubility of the solid is remarkably lower than that of the liquid (Figure 9-14). The excess hydrogen that cannot be incorporated in the solid metal or alloy crystal struc- ture forms bubbles that may be trapped in the solid metal, producing gas porosity. The amount of gas that can be dissolved in molten metal is given by Sievert’s law:

Percent of gas = K1p gas (9-7) where p gas is the partial pressure of the gas in contact with the metal and K is a con-

stant which, for a particular metal-gas system, increases with increasing temperature. We can minimize gas porosity in castings by keeping the liquid temperature low, by adding materials to the liquid to combine with the gas and form a solid, or by ensur- ing that the partial pressure of the gas remains low. The latter may be achieved by plac- ing the molten metal in a vacuum chamber or bubbling an inert gas through the metal. Because p gas is low in the vacuum, the gas leaves the metal, enters the vacuum, and is

350 CHAPTER 9

Principles of Solidification

Figure 9-14

The solubility of hydrogen gas in aluminum when the partial pressure

of H 2 = 1 atm.

carried away. Gas flushing is a process in which bubbles of a gas, inert or reactive, are injected into a molten metal to remove undesirable elements from molten metals and alloys. For example, hydrogen in aluminum can be removed using nitrogen or chlorine. The following example illustrates how a degassing process can be designed.

Example 9-7 Design of a Degassing Process for Copper

After melting at atmospheric pressure, molten copper contains 0.01 weight percent oxygen. To ensure that your castings will not be subject to gas porosity, you want to reduce the weight percent to less than 0.00001% prior to pouring. Design a degassing process for the copper.

SOLUTION

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