10 askeland chap

111

10Dispersion Strengthening andEutectic Phase Diagrams

10–22 A hypothetical phase diagram is shown in Figure 10–32. (a) Are any intermetalliccompounds present? If so, identify them and determine whether they are stoichio-metric or nonstoichiometric. (b) Identify the solid solutions present in the system. Iseither material A or B allotropic? Explain. (c) Identify the three-phase reactions bywriting down the temperature, the reaction in equation form, the composition ofeach phase in the reaction, and the name of the reaction.

Solution: (a) u � non-stoichiometric intermetallic compound.

(b) a, h, g, and b; material B is allotropic, existing in three different forms atdifferent temperatures

(c)

10–23 The Cu–Zn phase diagram is shown in Figure 10–33. (a) Are any intermetalliccompounds present? If so, identify them and determine whether they are stoichio-metric or nonstoichiometric. (b) Identify the solid solutions present in the system.(c) Identify the three-phase reactions by writing down the temperature, the reactionin equation form, and the name of the reaction.

u: 40% B h: 95% B 300°C: bS u � h; eutectoid; b: 90% B

b: 80% B u: 37% B 600°C: a � bS u; peritectoid; a: 5% B

a: 5% B b: 90% B 680°C: L S a � b; eutectic; L: 60% B

L 2: 50% B a: 5% B 900°C: L1 S L2 � a; monotectic; L1: 28% B

g: 97% B b: 90% B 1100°C: g � L S b; peritectic; L: 82% B

112 The Science and Engineering of Materials Instructor’s Solution Manual

Solution: (a) b, b�, g, d, e: all nonstoichiometric.

(b) a, u

(c)

10–24 A portion of the Al–Cu phase diagram is shown in Figure 10–34. (a) Determine theformula for the u compound. (b) Identify the three-phase reaction by writing downthe temperature, the reaction in equation form, the composition of each phase in thereaction, and the name of the reaction.

Solution: (a)

(b)

10–25 The Al–Li phase diagram is shown in Figure 10–35. (a) Are any intermetallic com-pounds present? If so, identify them and determine whether they are stoichiometricor nonstoichiometric. Determine the formula for each compound. (b) Identify thethree-phase reactions by writing down the temperature, the reaction in equationform, the composition of each phase in the reaction, and the name of the reaction.

Solution: (a) b is non-stoichiometric @ 21 wt% Li:

g, is stoichiometric @ 34 wt% Li:

(b)

10–26 An intermetallic compound is found for 10 wt% Si in the Cu–Si phase diagram.Determine the formula for the compound.

Solution: at% Si �10 g�28.08 g/mol

10�28.08 � 90�63.54� 0.20 or SiCu4

g: 34% Li a 1Li2: 99% Li170°C: L S g � a 1Li2 eutectic L: 98% Li

L: 47% Li g: 34% Li510°C: b � L S g peritectic b: 25% Li

a: 4% Li b: 20.4% Li600°C: L S a � b eutectic L: 9.9% Li

at% Li �34 g�6.94 g/mol

34�6.94 � 66�26.981� 100% � 66.7% Li ∴ AlLi2

at% Li �21 g�6.94 g/mol

21�6.94 � 79�26.981� 100% � 50 at% Li ∴ AlLi

a: 5.65% Cu, u: 52.5% Cu.548°C; L S a � u; eutectic; L: 33.2% Cu,

u at 54% Cu; 54 g�63.54 g/mol

54�63.54 � 46�26.981� 33 at% Cu; CuAl2

250°C: b¿ S a � g; eutectoid

420°C: e � L S u; peritectic

550°C: d S g � e; eutectoid

600°C: d � L S e; peritectic

700°C: g � L S d; peritectic

830°C: b � L S g; peritectic

900°C: a � L S b; peritectic

10–27 Using the phase rule, predict and explain how many solid phases will form in aneutectic reaction in a ternary (three-component) phase diagram, assuming that thepressure is fixed.

Solution: F � C � P � 1

At the eutectic, F � 0, C � 3 0 � 3 � P � 1 or P � 4

Therefore, L S a� b� g and 3 solid phases form.

10–30 Consider a Pb–15% Sn alloy. During solidification, determine (a) the compositionof the first solid to form, (b) the liquidus temperature, solidus temperature, solvustemperature, and freezing range of the alloy, (c) the amounts and compositions ofeach phase at 260°C, (d) the amounts and compositions of each phase at 183°C, and(e) the amounts and compositions of each phase at 25°C.

Solution: (a) 8% Sn

(b) liquidus � 290°C, solidus � 240°C,solvus � 170°C, freezing range � 50°C

(c) L: 30% Sn a: 12% Sn;

(d) a: 15% Sn 100% a

(e) a: 2% Pb b: 100% Sn

10–31 Consider an Al–12% Mg alloy (Figure 10–36). During solidification, determine(a) the composition of the first solid to form, (b) the liquidus temperature, solidustemperature, solvus temperature, and freezing range of the alloy, (c) the amountsand compositions of each phase at 525°C, (d) the amounts and compositions of eachphase at 450°C, and (e) the amounts and compositions of each phase at 25°C.

Solution: (a) 2.5% Mg

(b) liquidus � 600°C, solidus � 470°C,solvus � 400°C, freezing range � 130°C

(c) L: 26% Mg a: 7% Mg;

(d) a: 12% Mg 100% a

(e)

%a �34 � 12

34 � 1� 100% � 67% %b � 33%

a: 1% Mg b: 34% Mg

%a �26 � 12

26 � 7� 100% � 74% %L � 26%

%a �100 � 15

100 � 2� 100 � 87% %b � 13%

%L �15 � 12

30 � 12� 100% � 17% %a � 83%

CHAPTER 10 Dispersion Strengthening and Eutectic Phase Diagrams 113


10–32 Consider a Pb–35% Sn alloy. Determine (a) if the alloy is hypoeutectic or hypereu-tectic, (b) the composition of the first solid to form during solidification, (c) theamounts and compositions of each phase at 184°C, (d) the amounts and composi-tions of each phase at 182°C, (e) the amounts and compositions of each microcon-stituent at 182°C, and (f) the amounts and compositions of each phase at 25°C.

Solution: (a) hypoeutectic (b) 14% Sn

(c)

(d)

(e) primary a: 19% Sn %primary a � 63%eutectic: 61.9% Sn %eutectic � 37%

(f)

10–33 Consider a Pb–70% Sn alloy. Determine (a) if the alloy is hypoeutectic or hypereu-tectic, (b) the composition of the first solid to form during solidification, (c) theamounts and compositions of each phase at 184°C, (d) the amounts and composi-tions of each phase at 182°C, (e) the amounts and compositions of each microcon-stituent at 182°C, and (f) the amounts and compositions of each phase at 25°C.

Solution: (a) hypereutectic (b) 98% Sn

(c)

(d)

(e) primary b: 97.5% Sn %primary b� 22.8%eutectic: 61.9% Sn %eutectic � 77.2%

(f)

10–34 Calculate the total % b and the % eutectic microconstituent at room temperature forthe following lead-tin alloys: 10% Sn, 20% Sn, 50% Sn, 60% Sn, 80% Sn, and 95%Sn. Using Figure 10–22, plot the strength of the alloys versus the % b and the % eutectic and explain your graphs.

%a �100 � 70

100 � 2� 100% � 30% %b � 70%

a: 2% Sn b: 100% Sn

%a �97.5 � 70

97.5 � 19� 100% � 35% %b � 65%

a: 19% Sn b: 97.5% Sn

%b �70 � 61.9

97.5 � 61.9� 100% � 22.8% %L � 77.2%

b: 97.5% Sn L: 61.9% Sn

%a �100 � 35

100 � 2� 100% � 66% %b � 34%

a: 2% Sn b: 100% Sn

%a �97.5 � 35

97.5 � 19� 100% � 80% %b � 20%

a: 19% Sn b: 97.5% Sn

%a �61.9 � 35

61.9 � 19� 100% � 63% %L � 37%

a: 19% Sn L: 61.9% Sn

Solution: %b %eutectic

10% Sn 0%

20% Sn

50% Sn

60% Sn

80% Sn

95% Sn 97.5 � 95

97.5 � 61.9� 7.0%

95 � 2

99 � 2� 95.9%

97.5 � 80

97.5 � 61.9� 49.2%

80 � 2

99 � 2� 80.4%

60 � 19

61.9 � 19� 95.6%

60 � 2

99 � 2� 59.8%

50 � 19

61.9 � 19� 72.3%

50 � 2

99 � 2� 49.5%

20 � 19

61.9 � 19� 2.3%

20 � 2

99 � 2� 18.6%

10 � 2

99 � 2� 8.2%


10–35 Consider an Al–4% Si alloy. (See Figure 10–23.) Determine (a) if the alloy ishypoeutectic or hypereutectic, (b) the composition of the first solid to form duringsolidification, (c) the amounts and compositions of each phase at 578°C, (d) theamounts and compositions of each phase at 576°C, the amounts and compositionsof each microconstituent at 576°C, and (e) the amounts and compositions of eachphase at 25°C.

Solution: (a) hypoeutectic

(b) 1% Si

(c)

(d)

%a �99.83 � 4

99.83 � 1.65� 97.6% %b � 2.4%

a: 1.65% Si b: 99.83% Si

%a �12.6 � 4

12.6 � 1.65� 78.5% %L � 21.5%

a: 1.65% Si L: 12.6% Si

20 40 60 80 1004000

5000

6000

7000

8000

% β

Tens

ile s

tren

gth

(psi

)

20 40 60 80

5000

6000

7000

8000

% eutectic

Hypo-

Hyper-

Tens

ile s

tren

gth

(psi

)


(e) primary a: 1.65% Si %primary a� 78.5%eutectic: 12.6% Si %eutectic � 21.5%

10–36 Consider a Al–25% Si alloy. (See Figure 10–23.) Determine (a) if the alloy ishypoeutectic or hypereutectic, (b) the composition of the first solid to form duringsolidification, (c) the amounts and compositions of each phase at 578°C, (d) theamounts and compositions of each phase at 576°C, (e) the amounts and composi-tions of each microconstituent at 576°C, and (f) the amounts and compositions ofeach phase at 25°C.

Solution: (a) hypereutectic

(b) 100% Si

(c)

(d)

(e) primary b: 99.83% Si %primary b� 14.2%eutectic: 12.6% Si %eutectic � 85.8%

(f)

10–37 A Pb–Sn alloy contains 45% a and 55% b at 100°C. Determine the composition ofthe alloy. Is the alloy hypoeutectic or hypereutectic?

Solution:

10–38 An Al–Si alloy contains 85% a and 15% b at 500°C. Determine the composition ofthe alloy. Is the alloy hypoeutectic or hypereutectic?

Solution:

10–39 A Pb–Sn alloy contains 23% primary a and 77% eutectic microconstituent.Determine the composition of the alloy.

Solution:

10–40 An Al–Si alloy contains 15% primary b and 85% eutectic microconstituent.Determine the composition of the alloy.

%primary a � 23 �61.9 � x

61.9 � 19� 100 or x � 52% Sn

%a � 85 �100 � x

100 � 1� 100 or x � 15.85% Si Hypereutectic

%a � 45 �98.0 � x

98.0 � 5� 100 or x � 56.15% Sn Hypoeutectic

a: 0% Si b: 100% Si %a �100 � 25

100 � 0� 75% %b � 25%

%a �99.83 � 25

99.83 � 1.65� 76.2% %b � 23.8%

a: 1.65% Si b: 99.83% Si

%L �99.83 � 25

99.83 � 12.6� 85.8% %b � 14.2%

b: 99.83% Si L: 12.6% Si

a: 0% Si b: 100% Si %a �100 � 4

100 � 0� 96% %b � 4%

Solution:

10–41 Determine the maximum solubility for the following cases.

(a) lithium in aluminum (Figure 10–35),(b) aluminum in magnesium (Figure 10–37),(c) copper in zinc (Figure 10–33), and(d) carbon in g-iron (Figure 10–38)

Solution: (a) 4% Li dissolves in aluminum

(b) 12.7% Al dissolves in magnesium

(c) 3% Cu dissolves in zinc

(d) 2.11% C dissolves in g-iron

10–42 Determine the maximum solubility for the following cases.

(a) magnesium in aluminum (Figure 10–36),(b) zinc in copper (Figure 10–33),(c) beryllium in copper (Figure 10–33), and(d) Al2O3 in MgO (Figure 10–39)

Solution: (a) 14.9% Mg dissolves in aluminum

(b) 40% Zn dissolves in copper

(c) 2.5% Be dissolves in copper

(d) 18% Al2O3 dissolves in MgO

10–43 Observation of a microstructure shows that there is 28% eutectic and 72% primaryb in an Al–Li alloy (Figure 10–35). (a) Determine the composition of the alloy andwhether it is hypoeutectic or hypereutectic. (b) How much a and b are in the eutec-tic microconstituent?

Solution: (a)

(b)

10–44 Write the eutectic reaction that occurs, including the compositions of the threephases in equilibrium, and calculate the amount of a and b in the eutectic microcon-stituent in the Mg–Al system, (Figure 10–36).

Solution:

∴ %aEut �40.2 � 32.3

40.2 � 12.7� 100% � 28.7% and %gEut � 71.3%

L32.3 S a12.7 � g40.2

%aEut �20.4 � 9.9

20.4 � 4� 100% � 64% and %bEut � 36%

28 �20.4 � x

20.4 � 9.9� 100 or x � 17.46% Li Hypereutectic

%eutectic � 85 �100 � x

100 � 12.6� 100 or x � 25.71% Si



10–45 Calculate the total amount of a and b and the amount of each microconstituent in aPb–50% Sn alloy at 182°C. What fraction of the total a in the alloy is contained inthe eutectic microconstituent?

Solution:

10–46 Figure 10–40 shows a cooling curve for a Pb–Sn alloy. Determine (a) the pouringtemperature, (b) the superheat, (c) the liquidus temperature, (d) the eutectic tempera-ture, (e) the freezing range, (f ) the local solidification time, (g) the total solidifica-tion time, and (h) the composition of the alloy.

Solution: (a)

(b)

(c)

(d)

(e)

(f)

(g)

(h) approximately 32% Sn

10–47 Figure 10–41 shows a cooling curve for an Al–Si alloy. Determine (a) the pouringtemperature, (b) the superheat, (c) the liquidus temperature, (d) the eutectic tempera-ture, (e) the freezing range, (f) the local solidification time, (g) the total solidifica-tion time, and (h) the composition of the alloy.

Solution: (a)

(b)

(c)

(d)

(e)

(f)

(g)

(h) approximately 45% Si

total solidification time � 11.5 min

local solidification time � 11.5 � 1 � 10.5 min

freezing range � 1000 � 577 � 423°C

eutectic temperature � 577°C

liquidus temperature � 1000°C

superheat � 1150 � 1000 � 150°C

pouring temperature � 1150°C

total solidification time � 600 s

local solidification time � 600 � 110 � 490 s

freezing range � 250 � 183 � 67°C

eutectic temperature � 183°C

liquidus temperature � 250°C

superheat � 360 � 250 � 110°C

pouring temperature � 360°C

f � 32.8�60.5 � 0.54

ain eutectic � 60.5 � 27.7 � 32.8%

aPrimary �61.9 � 50

61.9 � 19� 100% � 27.7% Eutectic � 72.3%

atotal �97.5 � 50

97.5 � 19� 100% � 60.5% bTotal � 39.5%

10–48 Draw the cooling curves, including appropriate temperatures, expected for thefollowing Al–Si alloys.

(a) Al–4% Si (b) Al–12.6% Si (c) Al–25% Si (d) Al–65% Si

Solution:


10–49 Based on the following observations, construct a phase diagram. Element A melts at850°C and element B melts at 1200°C. Element B has a maximum solubility of 5%in element A, and element A has a maximum solubility of 15% in element B. Thenumber of degrees of freedom from the phase rule is zero when the temperature is725°C and there is 35% B present. At room temperature 1% B is soluble in A and7% A is soluble in B.

Solution:

T

tAl – 4% Si

630°577°

T

tAl – 12.6% Si

577°T

tAl – 25% Si

780°

577° T

tAl – 65% Si

1200°

577°

b + L

b

L

a + b

A B205 40 60 80 85% B

200

400

600

800

1000

1200

Tem

pera

ture

(°C

)

a + La


10–50 Cooling curves are obtained for a series of Cu–Ag alloys, (Figure 10–42). Use thisdata to produce the Cu–Ag phase diagram. The maximum solubility of Ag in Cu is7.9% and the maximum solubility of Cu in Ag is 8.8%. The solubilities at roomtemperature are near zero.

Solution: Tliq Tsol

0% Ag S 1085°C8% Ag S 1030°C 950°C20% Ag S 975°C 780°C50% Ag S 860°C 780°C71.9% Ag S 780°C 780°C90% Ag S 870°C 780°C100% Ag S 961°C

10–51 The SiO2–Al2O3 phase diagram is included in Figure 10–27(b). A refractory isrequired to contain molten metal at 1900°C. (a) Will pure Al2O3 be a potential can-didate? Explain. (b) Will Al2O3 contaminated with 1% SiO2 be a candidate? Explain.

Solution: (a) Yes. Tm � 2040°C � 1900°C No liquid will form.

(b) No. Some liquid will form.

This liquid will weaken the refractory.

%L �100 � 99

100 � 80� 100% � 5% L

700

800

900

1000

1100

Cu 29 40 60 80 Ag

Tem

pera

ture

(°C

)

L

%Ag

aa + L

b + L

a + bb

10–66 Consider the ternary phase diagram shown in Figures 10–30 and 10–31. Determinethe liquidus temperature, the first solid to form, and the phases present at room tem-perature for the following compositions.

(a) 30% B–20% C, balance A (b) 10% B–25% C, balance A(c) 60% B–10% C, balance A

Solution: (a)

(b)

(c)

10–67 Consider the ternary phase diagram shown in Figures 10–30 and 10–31. Determinethe liquidus temperature, the first solid to form, and the phases present at room tem-perature for the following compositions.

(a) 5% B–80% C, balance A (b) 50% B–5% C, balance A(c) 30% B–35% C, balance A

Solution: (a)

(b)

(c)

10–68 Consider the liquidus plot in Figure 10–30. (a) For a constant 20% B, draw a graphshowing how the liquidus temperature changes from 20% B–0% C, balance A to20% B–80% C, balance A, (b) What is the composition of the ternary eutectic in thissystem? (c) Estimate the temperature at which the ternary eutectic reaction occurs.

Solution: %A %B %C Tliquidus

80–20–0 390C70–20–10 355°C60–20–20 300°C50–20–30 210°C40–20–40 150°C30–20–50 210°C20–20–60 270°C10–20–70 320°C0–20–80 400°C

TLiq � 290°C; b; a � b � g

TLiq � 330°C; b; a � b

TLiq � 390°C; g; a � g

TLiq � 390°C; b; a � b

TLiq � 330°C; a; a � g

TLiq � 220°C; b; a � g � b


100

200

300

400

0 20 40 60 80

Tem

pera

ture

(°C

)

%C

L

a + L

g + L

B = 20%


(b) The composition of the ternary eutectic is about 40% 20% B–40% C,balance A

(c) The ternary eutectic temperature is about 150°C

10–69 From the liquidus plot in Figure 10–30, prepare a graph of liquidus temperatureversus percent B for a constant ratio of materials A and C (that is, from pure B to50% A–50% C on the liquidus plot). Material B melts at 600°C.

Solution: A B C

50– 0–50 200°C45–10–45 180°C40–20–40 150°C35–30–35 280°C30–40–30 330°C25–50–25 375°C20–60–20 415°C15–70–15 485°C0–100–0 580°C

100

200

300

400

500

600

0 20 40 60 80 100

Tem

pera

ture

(°C

)

% B

L

% A = % C a + L

b + L

10 askeland chap

Documents