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On the solidification of nodular cast iron and its relation to the expansion and contraction
Materials Science and Engineering A 413–414 (2005) 363–372
On the solidification of nodular cast iron and its relation to the expansion and contraction H. Fredriksson a,∗ , J. Stjerndahl a , J. Tinoco b a
Casting of Metals, Royal Institute of Technology, SE-100 44 Stockholm, Sweden b Outokumpu Copper R&D, P.O. Box 594, SE-721 10 V¨ aster˚as, Sweden Received in revised form 8 September 2005
Abstract Directional solidification and quench-out thermal analysis experiments have been performed in Mg-treated cast iron alloys. The volume fraction of liquid, austenite and graphite was evaluated. It was observed that the volume fraction of austenite is much larger than expected from the equilibrium phase diagram at the beginning of the solidification process. It was also been observed that the last melt solidifies far below the equilibrium eutectic temperature. The solidification process was analyzed by non-equilibrium thermodynamic models. The theoretical treatment was supported by the observation that the latent heat decreases during the solidification process. The formation of small pores was observed at the very end of the solidification. An explanation for the formation of the small pores is given in terms of a vacancies creep model. The formation of macropores was related to the large fraction of austenite formed during the first part of the solidification process. © 2005 Elsevier B.V. All rights reserved. Keywords: Cast iron; Nodular; Shrinkage; Solidification; Metastable
1. Introduction In order to improve the properties of cast iron Morrogh and Williams [1,2] developed a method to give the graphite a spherical shape by adding cerium to the melt. Later on, the method was improved by using magnesium instead of cerium [3], cast iron with spherical shaped is today well known as ductile cast iron. The solidification process of nodular cast iron has since then been discussed extensively [4,5]. Today one has a good understanding of the solidification process. Graphite nodules are nucleated in the melt, grow to a certain size freely in the melt and are later on surrounded by an austenite shell. The growth is then continued by diffusion through the austenite shell. The growth mechanism is very different from the one in flake graphite cast iron, where both austenite and graphite grow side by side into the liquid. The difference in growth mechanism between nodular cast iron and flake cast iron give rise to a difference in solidification shrinkage or expansion. It is well known that nodular cast iron needs larger risers than flake graphite cast iron. This is puzzling. It was early shown both experimentally and theoretically [6,7] ∗
Corresponding author. Tel.: +46 8 790 7869; fax: +46 8 21 6557. E-mail address: [email protected] (H. Fredriksson).
0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2005.09.028
that as soon as the carbon content gets above 3.8 wt.% carbon for a binary alloy the larger specific volume of graphite compensates the smaller specific volume of austenite so that the specific volume of austenite and graphite in the eutectic structure gets larger than the one in the liquid. Then, shrinkage pores may not be formed. The shrinkage process for nodular cast iron has been extensively studied over the years [8–11]. According to Stefanescu [10] and others [11] two different types of pores are distinguished, one formed during the early stage of the solidification process and another type formed during the last part of the solidification process. It is also known that the fraction of pores depends on the cooling rate, the number of graphite nodules and the carbon content. Hummer [12,13] has shown in a series of experiments that a contraction occurs during the early stage of solidification of nodular cast iron and that an expansion occurs at the end. The contraction at the early stage was assumed to be an effect of precipitation of austenite, which also was proposed by Hummer [13]. If the solidification process would be described by conventional models for eutectic reactions the volume fraction of different phases should be the same and independent of the growth mechanisms. During the last 10 years it has been shown that the latent heat released during a solidification process is not a constant
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[14–17]. This indicates that the equilibrium phase diagram is not valid when describing a solidification process. In three recently published papers it has been shown that this is also the case for cast iron. A large difference in latent heat release has been observed between grey and nodular cast iron [18,19]. In order to explain this difference the equilibrium phase diagram can be changed to a metastable diagram. In this work new observations will be used to analyze the solidification process of nodular cast iron. Experimental results both from new investigations in directional solidified samples and from earlier presented results [5] will first be presented. The results are evaluated and analyzed. At the end, new models are used to discuss the shrinkage pore formation process in nodular cast iron.
Table 3 Conditions achieved during the TA experiments Alloy
Sample
dT/dt (K/min)
tsol (s)
2 2 2 2 2 2 3 3 3 4 4 4 4 4
1 2 3 4 5 6 1 2 3 1 2 3 4 5
130 126 126 131 121 131 10 10 10 10 10 10 10 10
18 52 60 86 152 244 18 163 204 23 63 146 217 252
2. Experimental procedure The directional solidification (DS) experiments were done in the following way: Mg-treated eutectic iron–nickel–carbon alloys were melted in an alumina crucible and then sucked up in an alumina tube with an inner diameter of 2.5 mm. The tube was placed in a furnace with a steep temperature gradient so the alloy in the tube was partly remelted. The tube was removed at a constant rate through the temperature gradient in which the solidification takes place. The experiments were suddenly interrupted by quenching the tube in brine. The remaining liquid thus solidified to a very fine structure, which easily could be distinguished from the solidification front structure by a normal metallographic analysis. The experiments were carried out in an atmosphere of argon. The temperature gradient in the furnace was determined by removing a Pt/Pt–10% Rh thermocouple at a low constant rate. The method has earlier been used in a number of investigations [8]. A number of experiments were performed with the same type of alloy. In this report it is only presented results from two of the experiments. The composition of the alloy and the experimental conditions are shown in Tables 1 and 2. Results from different types of thermal analysis (TA) experiments will also be discussed. Those experiments were presented
earlier by Wetterfall et al. [5] and Tinoco and Fredriksson [18]. Additional results will be presented here. In these experiments the solidification process was also interrupted by quenching the samples into brine. The composition of the alloys is included in Table 1. Information concerning the conditions achieved during the TA experiments is presented in Table 3. 3. Experimental results Lengthwise sections of the DS sample were metallographically prepared. The samples were etched in nital. In a sample with eutectic composition (sample 1) was observed that graphite and austenite started to precipitate at the same level or the same time. This indicates the start of the eutectic reaction. The volume fraction of liquid, austenite and graphite was evaluated as function of the distance from the start of the eutectic reaction. The results are presented in Fig. 1. The figure shows that the volume fraction of liquid continuously decreases with distance and that the volume fraction of graphite increases in steps. Two plateaus are observed, one at the level close to the front and one at a distance of 1.5 mm from the start of the solidification. It was
Table 1 Chemical composition of the alloys studied in wt.% Element
C
Ni
Mg
Si
C.E.
Alloy 1 (DS) Alloy 2 (DS, TA) Alloy 3 (TA) Alloy 4 (TA) Alloy 5 (TA)
4.05 3.89 3.84 4.00 3.88
3.02 2.94 1.33 1.12 0.06
0.14 0.14 0.235 0.033 0.001
0.03 0.02 1.88 0.12 2.21
4.06 3.89 4.46 4.04 4.61
Table 2 Conditions achieved during the directional solidification (DS) experiments Experiment #1 Alloy Drawing rate (mm/min) Temperature gradient (K/mm) dT/dt (K/min)
1 40 3 120
Experiment #2 2 10 2 20
Fig. 1. Volume fraction of graphite and liquid as function of the distance to the solidification front for sample alloy 1.
H. Fredriksson et al. / Materials Science and Engineering A 413–414 (2005) 363–372
Fig. 2. Volume fraction of graphite and liquid as function of the distance to the solidification front for sample alloy 2.
also observed in the samples that at the end of each plateau a number of new graphite nodules were formed. From the figure it can be concluded that the volume fraction of liquid decreases faster than it is expected from the volume fraction of graphite. This can be explained as a precipitation of austenite larger than expected from the phase diagram. The figure also shows that the fraction of liquid slightly increases towards the end of solidification process. Similar results were obtained in sample 2. In this case solidification rate was lower (Table 2). Fig. 2 shows the volume fraction of graphite and liquid as function of the distance from the start of solidification for sample 2. Two deviations between the results presented in Figs. 1 and 2 can be noticed. There is a significant increase in the volume fraction of graphite at a certain distance from the solidification front. No increase in the fraction of liquid could be observed at the end. The fraction of liquid decreased continuously until the process was finished. The sudden increase of the volume fraction of graphite at 1.5 mm is related to nucleation of new graphite nodules. A number of small pores were observed a short distance behind the end of the solidification front. The volume fraction of these pores was measured as function of the distance to the front. The result is presented in Fig. 3. The figure shows that the fraction of pores first increases to a maximum and then decreases. The reason is that the formed pores are filled with graphite. Results from TA experiments will now be presented. The experimental methods have been described elsewhere [18]. The first results which will be discussed are from the work by Wetterfall et al. [5]. In both works series of TA experiments were interrupted by quenching the sample after a certain time. Typical temperature–time curves in the work by Tinoco and Fredriksson [18] are shown in Fig. 4. Micrographs of the quenched samples by Wetterfall et al. are shown in Fig. 5(a)–(d). The alloy composition in this case is slightly hypoeutectic eutectic (alloy 1). The solidification starts with a primary precipitation of austenite dendrites, Fig. 5(a). Graphite nodules freely formed in the liquid are observed. The graphite nodules get later into contact with the austenite dendrites and are surrounded by an austenite
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Fig. 3. Fraction of liquid and pores as function of the distance to the solidification front for sample alloy 2.
shell. Then they continue to grow by carbon diffusion through the austenite shell as shown in Fig. 5(b)–(d). The figures illustrate that the austenite dendrites increase in size as well as the dendrite arm spacing. The structure presented in Fig. 5 was evaluated by measuring the volume fraction of graphite, austenite forming the shell and austenite dendrites. Results of those measurements are presented in Fig. 6. Figs. 7 and 8 present results from measurements performed for alloys 2 and 4, respectively. Fig. 6 shows that, as it was observed in the micrographs, the volume fraction of austenite precipitating as dendrites increases with time. The figure clearly shows that the fraction of austenite in the dendrites is much larger at the beginning of the solidification process. According to the equilibrium phase diagram at the eutectic point, the volume fraction ratio graphite/␥ should be equal to 0.115 (ρ␥ = 7.87 g/cm3 , ρgraphite = 1.5 g/cm3 ). This value is presented in Fig. 9 together with the measured values of volume fraction ratio. In the figure the ratio represents the total volume fraction of austenite (the sum of the austenite in the dendrites and in the shell) divided by the volume fraction of graphite. It is clear that the volume
Fig. 4. Typical cooling curves obtained (alloy 4).
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Fig. 5. Microstructure observe in the samples obtained from quench-out operations for samples of alloy 1: (a) 18 s, (b) 52 s, (c) 60 s, (d) 86 s, (e) 152 s, and (f) 244 s.
fraction of graphite/␥ is much lower at the beginning. At the end of the process it gets closer to the ideal value given by the phase diagram. Tinoco and Fredriksson [18] performed an evaluation of the latent heat as function of the fraction of solid formed during TA experiments at low cooling rates. Results of those measurements are presented in Fig. 10. The figure presents latent heat measurements as function of the total volume fraction of solid.
As it is possible to observed the latent heat released is lower than the tabulated value at the beginning of the solidification process and increases towards the end. In the figure the latent heat released during the solidification of the same alloy without Mg-addition is also presented. This alloy solidified to a grey structure. In this case no effect on the volume fraction could be observed. The explanation to this observation is the difference in growth mechanism. For nodular cast iron the growth of graphite
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Fig. 6. Measured volume fraction of austenite and graphite in the microstructure of samples (alloy 2).
Fig. 9. Volume fraction ratio (graphite/␥) as function of the solid fraction obtained from Figs. 6–8.
Fig. 7. Measured volume fraction of austenite and graphite in the microstructure of samples (alloy 3).
Fig. 10. Measured latent heat as function of the solid fraction for samples of alloys 3–5.
is controlled by diffusion of carbon through the austenite shell and for grey cast iron the growth is controlled by diffusion of carbon through the liquid ahead of the solidification front. This will be discussed further on. 4. Thermodynamics calculations
Fig. 8. Measured volume fraction of austenite and graphite in the microstructure of samples (alloy 4).
Fig. 10 illustrated that the latent heat varied with the growth mode of graphite. It has also been observed a variation of the latent heat with cooling rate for cast iron [19]. This variation has also been experimentally observed in other alloying systems as well [16,17]. It has been proposed that a decrease of the latent heat with the cooling rate is related to the number of lattice defects formed in the solid during the solidification process. The energy of the solid will thus be larger than under equilibrium conditions. This has earlier been used for a recalculation of the phase diagram where the liquidus and solidus lines change according to the volume fraction of lattice defects [16]. That
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model will now be briefly described. The molar Gibbs energy of the liquid phase can be described by ◦
◦
L L L L L L GLm = xFe GLFe + xC GLC + Rgas T [xFe ln xFe + xC ln xC ] L L xC + xFe
n
L n L LFe,C (xFe
k
L − xC )
(1)
i=0
where ◦ GLFe and ◦ GLC are the Gibbs energies of the pure components in its reference state, Rgas the gas constant, T the temperature, LLFe,C the interaction parameter between the components in the liquid and k = · · ·. The molar fraction of the component is defined as ni xi = (2) nFe + nC where ni is the number of atoms of i. If thermal vacancies are considered, the molar Gibbs energy of the austenite phase can be expressed as ␥◦
␥
␥◦
␥
Fe G␥m = xFe GFe + xC GC + xV GV ␥ ␥ + Rgas T [xFe ln xFe ␥
␥ ␥ + xC ln xC ␥
␥
␥ ␥ + xV ln xV ␥
− (1 + xV ) ln(1 + xV )] + xFe xC
n
␥ n ␥ LFe,C (xFe
␥ k
− xC )
i=0
␥ ␥ + xC xV LC,V
(3)
␥
␥
where xV is the fraction of vacancies in austenite, LFe,C the interaction parameter between in the solid and LC,V is the interaction parameter between the solute and vacancies. Fe GV is the Gibbs energy of vacancies in pure iron. The Gibbs energy of the graphite phase can be described as = ◦ GC Ggraphite m
graphite
C + xV (GC V + FC−V ) C C C C ln xV + Rgas T xV − (1 + xV ) ln(1 + xV )
(4)
The chemical potential of each component can be calculated using the Gibbs–Duhem relationship for any component A µA = Gm + (1 − xi )
∂Gm ∂Gm − xi ∂xi ∂xj
(5)
j=0
The chemical potentials of each component are equal at ␥ ␥ equilibrium (µA = µLA , µB = µLB ). The interaction parameters: ␥ L LFe,C , LFe,C , and LC,V where estimated from the system of equations formed by (4) and (5) using the experimental data on the phase diagram given by Chipman [20]. The calculated parameters are presented in Table 4. Table 4 Interaction parameters between Fe–C and C-vacancies Parameter
T-relationship (J/mol)
␥ LFe,C LLFe,C ␥ LC,V
−1.5345 × 105 + 76.78T −1.9245 × 105 + 89.10T −7.6078 × 104 − 412.18T
␥
␥ equilibrium
Fig. 11. Calculated Fe–C–V phase diagram (XV = XV 0.003). Volume fraction ratio (graphite/␥) = 0.114.
graphite
, XV
=
The experiments showed that the volume ratio graphite/␥ was lower than the predicted by the equilibrium phase diagram. By using the thermodynamic model presented previously an analysis of the experimental results can be made. There are two possible ways for the volume fraction ratio graphite/␥ to change, either the graphite liquidus line moves towards higher carbon content or the austenite solidus line move towards higher carbon contents. The first case can be illustrated by using Eq. (4). In this case an increase in the Gibbs energy of graphite can be achieved by increasing its vacancy concentration. An example of this case is presented in Fig. 11. The second case is illustrated by changing the interaction coefficient between lattice defects and carbon in austenite so the carbon solubility in austenite increases. Carbon is an interstitial soluble element in austenite and an increase of the number of lattice defects reduces the density of the phase. The extra free volume can then give a possibility for more carbon to be dissolved into the austenite. The results of this calculation are presented in Fig. 12. In this case both the fraction of vacancies was increased and the interaction parameter between carbon and vacancies was increased. This is a reasonable assumption considering that carbon is interstitially dissolved in austenite. Then it is likely that the austenite will be enriched in carbon atoms around a vacancy and therefore the solubility will be increased. An increase in solubility can be described by an increase in the interaction parameter C-vacancies. From Figs. 11 and 12 the volume fraction ratio between graphite/␥ was evaluated. The results are 0.114 for case 1 (Fig. 11) and 0.084 for case 2 (Fig. 12). The largest effect is given by an increase of the carbon content in the austenite due to an increase of the number of lattice defects in the austenite (Fig. 12). A comparison between the calculated phase diagram and the experimental results, Figs. 2, 3, 9 and 10, shows that the experimental observations can only be described by the case where the carbon solubility increases in austenite. At the early stages of the solidification process the ratio graphite/␥ was found to be between 0.02 and 0.09. If, for example the value 0.09 is to be explained the carbon content in the austenite must be around
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close to one in the austenite close to a growing nodule and it is higher in the austenite close to the liquid. The growth of nodules surrounded by an austenite shell is controlled by the deformation process of the austenite shell and by diffusion of carbon through this shell. These two processes are coupled in series. A theoretical analysis of this process will now be presented. The diffusion of carbon through the shell is described by the following relation: dR V gr ␥ X␥/l − X␥/gr 1 = ␥ DC gr ␥/gr 1 dt V X −X R2 −1 R
gr
␥
graphite
Fig. 12. Calculated Fe–C–V phase diagram (XV = 0.009, XV graphite equilibrium ), XV
in this case (graphite/␥) = 0.084.
␥ LCV
=
was increased 15%. Volume fraction ratio
2.58 wt.% which is an increase of 20% from the equilibrium value. This can be observed in Fig. 12. 4.1. Volume fraction of austenite The phase diagrams presented in Figs. 11 and 12 will be used for a qualitative discussion of the solidification process. Fig. 11 shows a case where a large number of lattice defects are formed in the graphite nodules and a fewer amount in the austenite. Fig. 12 shows the opposite case where the number of lattice defects is large in the austenite and low in the graphite. The figures as well as the thermodynamic calculations show that the activity of carbon decreases with increasing fraction of vacancies that means that the carbon solubility in austenite increases with increasing lattice defects as is shown by comparing, Figs. 11 and 12. This is the same effect as alloying steel or cast iron with silicon. The number of lattice defects is related to the growth rate and it can be expected a larger number of defects the larger the growth rate is. The experimental observations might be explained in the following way. The nodules are formed in the liquid they grow to a certain size and then they are surrounded by an austenite shell. The nodules will grow by diffusion of carbon through the austenite shell. The graphite formed is less dense than the liquid and the austenite. The shell will expand by a plastic deformation. The deformation process will take part by a creep process. This process is assumed to occur by formation of vacancies. When the nodules are formed in the liquid they are small and grow very fast. The austenite dendrites formed increase in fraction by thickening, the growth process is slow. A lower amount of lattice defects are formed. Thus Fig. 11 is valid for this situation where graphite and austenite are formed from the liquid. The deformation process of the austenite shell when the graphite is growing by diffusion of carbon through the shell gives a larger number of lattice defects. The creep rate is largest near the graphite nodule and the numbers of defects is largest there. This case is illustrated by Fig. 12. The activity of carbon is thus
(6)
S
␥
where Vm is the molar volume of graphite, Vm the molar volume of austenite, D the diffusion rate of carbon in austenite, R the nodule size of graphite, S the distance to the austenite shell border, X␥/L the molar fraction of carbon in austenite in contact with liquid, X␥/gr the molar fraction of carbon in austenite in contact with graphite and Xgr is the molar fraction of carbon in graphite. In the analysis given by Wetterfall et al. [5] the mole fraction values are given from the equilibrium phase diagram. In this case the relations derived in the previous section will be used so the mole fractions will thus be a function of the growth rate. The values will be given by creep laws as described below. The molar volume of graphite is larger than the partial molar volume of carbon dissolved in austenite and in the liquid. During the growth of graphite nodules the austenite shell must expand by a plastic deformation. The expansion is related to the growth rate of the graphite nodules. This process has been described by Nabarro and Herring as a diffusion process of vacancies from the border austenite/liquid to the border austenite/graphite. According to Herring [21] and Shewmon [22], the growth rate of the cavity filled with graphite can be described by the following relation: Gcreep dR DV =n dt Rgas T R2 1 − 1 R S
(7)
where DV is the diffusion coefficient of vacancies in austenite, Gcreep the driving force for the creep process and n is a constant which vary between 10 and 40 depending on the geometry [22]. In order to solve these relations and describe the growth rate as function of time or nodule size the driving force for the growth process is divided into two steps, one corresponding to the creep process and one to the diffusion of carbon. This can be written in the following way: GL→Gr = Gcreep + GC tot Diff
(8)
where GL→Gr is the total driving force for precipitation of tot graphite nodules, Gcreep the driving force for plastic deformation of the austenite shell and GC Diff is the driving force for diffusion of carbon through austenite. Fig. 13 shows a schematic representation of the free energy curves at a temperature below the eutectic one in the Fe–C system. At the right side of the figure is included the total driving force for precipitation of graphite from a melt in equilibrium with austenite and formation of graphite in equilibrium with austenite. This driving force can
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Fig. 13. Schematic representation of the Gibbs energy curves at a temperature below the eutectic in the Fe–C system.
be formulated by normal thermodynamic laws according to GL→Gr = Rgas T tot
␥/L aC ln ␥/gr aC
Fig. 14. Nodule growth rate as function of its radius calculated according to Eq. (13).
(9)
␥/L aC
where is the activity of carbon in austenite in equilibrium ␥/gr with liquid and aC is the activity of carbon in austenite in equilibrium with graphite. In our further analysis we will assume that the activity coefficient is the same at these two equilibrium points. Expressing Eq. (10) into a series of expansion ␥/L Xeq Gtot = Rgas T −1 (10) ␥/gr Xeq By transforming the concentration gradients to chemical potential differences Eq. (6) can be rewritten as dR = dt
gr Vm ␥ Vm
␥ DC
GDiff Rgas T (Xgr − X␥/gr )R2 1 − 1 R S
Combining Eqs. (7), (8) and (10) with Eq. (11) ␥/L 2 1 − 1 R Xeq dR R S −1 = ␥ ␥/gr dt nDV Xeq 1 gr ␥/gr ␥ 2 1 dR Vm R R − S (X − X ) + ␥ dt Vmgr DC
(11)
dR = dt
R2
1 R
−
1 S
1 nDV
−1 +
␥
Vm (Xgr −X␥/gr ) ␥ gr DC Vm
Gtot = Rgas T ln
(12)
(13)
A material balance can be used to obtain a relation between R and S in order to get an idea about the effect of different parameters on the growth process. In this case the value derived by Wetterfall et al. [5] from the equilibrium phase diagram will be used (R = 2.4S). Calculations of the growth rate as function of nodule size were then performed. In the calculation the n parameter has been varied from 1 to 100. Ten different curves
C a␥/L eq C a␥/gr eq
= Rgas T ln
def aV eq aV
(14)
According to thermodynamics of lattice defects the last term can be rewritten in the following way: Rgas T ln
Rearranging Eq. (12), ␥/L Xeq ␥/gr Xeq
are presented in Fig. 14. The figure shows that at high n values (close to 100) the growth rate becomes independent of n. The growth process is in this case controlled by the diffusion of carbon through the austenite shell. As it was mentioned the n parameter is a structural dependent constant and it varies between 10 and 40. The conclusion here will thus be that the process is controlled by the deformation of the austenite shell. From the total driving force it can now be estimated the fraction of vacancies needed for the plastic deformation by the following relation:
def def aV XV V/C = R T ln gas eq eq + Rgas T ln γ aV XV
(15)
where the activity coefficient for the vacancies at equilibrium, γ V/C , is normally included in the molar free energy of vacancies GV , according to GV = Fe GV + Rgas T ln γ V/C
(16)
where Fe GV = HV − TSV . In this case HV is the heat of formation and SV is the entropy of formation of vacancies in austenite. However in this case the vacancy concentration is larger than the equilibrium values, this will therefore have an influence. The last term in Eq. (16) can be described in terms of the interaction parameter LFe,V . This factor will be further discussed by analysing the possible fraction of vacancies created by the plastic deformation process. Fig. 10 shows that the latent heat was related to the fraction of solid formed. The variation of the latent heat with the fraction of vacancies can be estimated by the following relation: eq
Hnc = Htab − (xV − xV )HV
(17)
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Fig. 15. Graphite nodule without a shell growing in contact with an austenite dendrite.
where Hnc is the measured latent heat of nodular cast iron, eq Htab the tabulated value of latent heat and (xV − xV ) is the difference in mole fraction of vacancies between nodular cast iron and grey cast iron. The heat balance given by Eq. (17) can be used to calculate the fraction of vacancies as function of the measured value on the latent heat. By using the data presented in Fig. 10 about the latent heat the resulting fraction of vacancies lies between 0.05 and equilibrium. The total driving force is derived from the phase diagram by extrapolating the solidus line of austenite into the two-phase region ␥/graphite. At a temperature of 10 K below the eutectic the values gives an increase of the fraction of vacancies of 1% from the equilibrium value, which is much smaller than was given by the energy balance. This indicates that the heat of formation of vacancies increases with increasing fraction of vacancies, or it can also be a strong concentration dependence on the heat of mixing of vacancies. This can be expressed in another way; the interaction parameter LFe,V is divided into an enthalpy term and an entropy term. The enthalpy term is strongly positive. It is further on suggested that the entropy for formation of vacancies, SV , is also concentration dependent. This dependence can be related to an atomic structure change of the austenite lattice [23]. 4.2. Growth of graphite The graphite is nucleated in the melt between the dendrites arms and it grows freely in the melt until it reaches a certain size. It was proposed by Wetterfall et al. [5] that the shell is formed when the graphite nodules gets contact with the austenite due to its flotation. However, this seems to be a too simple explanation. It has been observed that the graphite nodules can have contact with austenite without being grown around (Fig. 15). The reason for this might be the fact that the growth behaviour is such that graphite and austenite have the same chemical potential, which is necessary for the growth of austenite around graphite. The difference in chemical potential can be due to the plastic deformation of the austenite. But it can also be an effect of the interface kinetics, when graphite is precipitated from the liquid.
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This reaction can also produce lattice defects in the graphite which change its chemical potential during the growth. This was partly discussed in the chapter about the calculation of the phase diagram. When the nodules are small the growth rate will be large and the interface kinetics will be large. In this case it will not be possible to establish a growth of austenite around the graphite. The remaining driving force will be too small to cause the plastic deformation of the austenite. This can be established when the nodules has reached a certain size. In the case when austenite shells are formed the growth rate of graphite nodules decreases and the interface kinetics for growth of graphite is decreased further on, and the assumption that there is equilibrium between ␥/graphite when deriving the driving force might not be far from the reality. When the growth of austenite ceases the vacancies start to condense at the same time as the graphite starts to grow due to the decrease of the carbon content in the austenite. The remaining vacancies will now condense. At this process the austenite will shrink, pores are formed. Those pores can be filled with liquid if they are in contact with the liquid or will remain in the structure as shrinkage pores, Figs. 1 and 2. 5. Discussion Taylor and co-workers [9] performed a large number of experiments for various iron base alloys with carbon content close to the eutectic. It was found that above a certain volume fraction of free graphite in the matrix no solidification shrinkage occurs. According to a carbon balance in the binary Fe–C system of equilibrium this will correspond to 3.8 wt.%. This value can then be adjusted by changing of the eutectic line due to the silicon content. This is as Taylor showed valid for normal flake cast iron but as has been shown not valid for nodular cast iron. The reason is the difference in the growth process which produces a larger precipitation of austenite during an early stage of the solidification process for nodular cast iron compared to the precipitated for grey cast iron. The volume fraction of graphite is less than the critical value given by Taylor. Solidification shrinkage will then occur. This shrinkage occurs in an earlier stage of the solidification process where the solid fraction is low. It is thus possible to compensate for the shrinkage by filling in new liquid from a riser or from parts of casting solidifying with a lower rate. If no risers are used macropores will be formed. In a later stage of the solidification process the precipitation of graphite would increase. The austenite will thus expand and the volume of the casting will increase. However, the distance to the macropores becomes then too large therefore it is difficult to transport melt through a network of austenite with a small volume fraction of liquid. The macropores will then remain; the total casting will instead expand. The expansion will take part by a creep process. The lattice defects taking part in this process together with those ones formed during the solidification process will condense during the end of the solidification process or just after. Micropores are then formed in this case. The volume fraction ratio graphite/␥ was observed to be affected by the nucleation of new graphite nodules, Figs. 1 and 2.
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Therefore it will be possible by an efficient inoculation procedure to decrease the volume of a riser. 6. Conclusion The formation of shrinkage pores is analyzed by directional experiments. It was shown that the pores were formed after the solidification process. The formation of these pores is a result of a creep process due to vacancies formation and diffusion. The fraction of vacancies formed during the creep process is larger than the equilibrium fraction. These vacancies will condense and pores are formed. The fraction of vacancies is larger the smaller the nodules are. References [1] [2] [3] [4] [5]
M. Morrogh, W.J. Williams, JISI 155 (1947) 321. H. Morrogh, W.J. Williams, JISI 158 (1948) 306. K.D. Millis, US Patent 2,485,760. E. Scheil, J.D. Sch¨obel, Arch. Eisenhwes. (1953) 4. S.-E. Wetterfall, H. Fredriksson, M. Hillert, JISI 210 (1972) 323.
[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
[19]
[20] [21] [22] [23]
W.A. Schmidt, H.F. Taylor, AFS Trans. 101 (1952) 211. W.A. Schmidt, H.F. Taylor, AFS Trans. 43 (1953) 131. H. Fredriksson, Metall. Trans. 3 (1972) 2989. W.A. Schmidt, E. Sullivan, H.F. Taylor, AFS Trans. 9 (1954) 70. D.M. Stefanescu, H.Q. Qiu, C.H. Chen, AFS Trans. 57 (1995) 189. R. Hummer, Mater. Res. Symp. Proc. 34 (1985) 213. R. Hummer, Giesserei Praxis 17/18 (1985) 241. R. Hummer, Inst. Met. 421 (1989) 283. S. Berg, J. Dalhstr¨om, H. Fredriksson, ISIJ Int. 35 (1995) 876. H. Fredriksson, Adv. Mater. Res. 4 (1997) 505. T. ElBenawy, H. Fredriksson, Mater. Trans. JIM 41 (2000) 507. J. Mahmoudi, H. Fredriksson, Mater. Sci. Eng. A 22 (1997) 226. J. Tinoco, H. Fredriksson, Proceedings of the International Conference on the Science of Casting and Solidification, Brasov, Rumania, May, 2001, pp. 305–311. J. Tinoco, P. Delvasto, O. Quintero, H. Fredriksson, Proceedings of the International Conference on Solidification and Processing of Cast Iron VII, Barcelona, Spain, September, 2002. J. Chipman, Metall. Trans. 3 (1972) 55. C. Herring, J. Appl. Phys. 21 (1950) 437. P. Shewmon, Diffusion of Solids, TMS Publishing, New York, 1989, p. 212. H. Fredriksson, T. Emi, Mater. Trans. JIM 39 (1998) 292.
On the solidification of nodular cast iron and its relation to the expansion and contraction H. Fredriksson a,∗ , J. Stjerndahl a , J. Tinoco b a
Casting of Metals, Royal Institute of Technology, SE-100 44 Stockholm, Sweden b Outokumpu Copper R&D, P.O. Box 594, SE-721 10 V¨ aster˚as, Sweden Received in revised form 8 September 2005
Abstract Directional solidification and quench-out thermal analysis experiments have been performed in Mg-treated cast iron alloys. The volume fraction of liquid, austenite and graphite was evaluated. It was observed that the volume fraction of austenite is much larger than expected from the equilibrium phase diagram at the beginning of the solidification process. It was also been observed that the last melt solidifies far below the equilibrium eutectic temperature. The solidification process was analyzed by non-equilibrium thermodynamic models. The theoretical treatment was supported by the observation that the latent heat decreases during the solidification process. The formation of small pores was observed at the very end of the solidification. An explanation for the formation of the small pores is given in terms of a vacancies creep model. The formation of macropores was related to the large fraction of austenite formed during the first part of the solidification process. © 2005 Elsevier B.V. All rights reserved. Keywords: Cast iron; Nodular; Shrinkage; Solidification; Metastable
1. Introduction In order to improve the properties of cast iron Morrogh and Williams [1,2] developed a method to give the graphite a spherical shape by adding cerium to the melt. Later on, the method was improved by using magnesium instead of cerium [3], cast iron with spherical shaped is today well known as ductile cast iron. The solidification process of nodular cast iron has since then been discussed extensively [4,5]. Today one has a good understanding of the solidification process. Graphite nodules are nucleated in the melt, grow to a certain size freely in the melt and are later on surrounded by an austenite shell. The growth is then continued by diffusion through the austenite shell. The growth mechanism is very different from the one in flake graphite cast iron, where both austenite and graphite grow side by side into the liquid. The difference in growth mechanism between nodular cast iron and flake cast iron give rise to a difference in solidification shrinkage or expansion. It is well known that nodular cast iron needs larger risers than flake graphite cast iron. This is puzzling. It was early shown both experimentally and theoretically [6,7] ∗
Corresponding author. Tel.: +46 8 790 7869; fax: +46 8 21 6557. E-mail address: [email protected] (H. Fredriksson).
0921-5093/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2005.09.028
that as soon as the carbon content gets above 3.8 wt.% carbon for a binary alloy the larger specific volume of graphite compensates the smaller specific volume of austenite so that the specific volume of austenite and graphite in the eutectic structure gets larger than the one in the liquid. Then, shrinkage pores may not be formed. The shrinkage process for nodular cast iron has been extensively studied over the years [8–11]. According to Stefanescu [10] and others [11] two different types of pores are distinguished, one formed during the early stage of the solidification process and another type formed during the last part of the solidification process. It is also known that the fraction of pores depends on the cooling rate, the number of graphite nodules and the carbon content. Hummer [12,13] has shown in a series of experiments that a contraction occurs during the early stage of solidification of nodular cast iron and that an expansion occurs at the end. The contraction at the early stage was assumed to be an effect of precipitation of austenite, which also was proposed by Hummer [13]. If the solidification process would be described by conventional models for eutectic reactions the volume fraction of different phases should be the same and independent of the growth mechanisms. During the last 10 years it has been shown that the latent heat released during a solidification process is not a constant
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[14–17]. This indicates that the equilibrium phase diagram is not valid when describing a solidification process. In three recently published papers it has been shown that this is also the case for cast iron. A large difference in latent heat release has been observed between grey and nodular cast iron [18,19]. In order to explain this difference the equilibrium phase diagram can be changed to a metastable diagram. In this work new observations will be used to analyze the solidification process of nodular cast iron. Experimental results both from new investigations in directional solidified samples and from earlier presented results [5] will first be presented. The results are evaluated and analyzed. At the end, new models are used to discuss the shrinkage pore formation process in nodular cast iron.
Table 3 Conditions achieved during the TA experiments Alloy
Sample
dT/dt (K/min)
tsol (s)
2 2 2 2 2 2 3 3 3 4 4 4 4 4
1 2 3 4 5 6 1 2 3 1 2 3 4 5
130 126 126 131 121 131 10 10 10 10 10 10 10 10
18 52 60 86 152 244 18 163 204 23 63 146 217 252
2. Experimental procedure The directional solidification (DS) experiments were done in the following way: Mg-treated eutectic iron–nickel–carbon alloys were melted in an alumina crucible and then sucked up in an alumina tube with an inner diameter of 2.5 mm. The tube was placed in a furnace with a steep temperature gradient so the alloy in the tube was partly remelted. The tube was removed at a constant rate through the temperature gradient in which the solidification takes place. The experiments were suddenly interrupted by quenching the tube in brine. The remaining liquid thus solidified to a very fine structure, which easily could be distinguished from the solidification front structure by a normal metallographic analysis. The experiments were carried out in an atmosphere of argon. The temperature gradient in the furnace was determined by removing a Pt/Pt–10% Rh thermocouple at a low constant rate. The method has earlier been used in a number of investigations [8]. A number of experiments were performed with the same type of alloy. In this report it is only presented results from two of the experiments. The composition of the alloy and the experimental conditions are shown in Tables 1 and 2. Results from different types of thermal analysis (TA) experiments will also be discussed. Those experiments were presented
earlier by Wetterfall et al. [5] and Tinoco and Fredriksson [18]. Additional results will be presented here. In these experiments the solidification process was also interrupted by quenching the samples into brine. The composition of the alloys is included in Table 1. Information concerning the conditions achieved during the TA experiments is presented in Table 3. 3. Experimental results Lengthwise sections of the DS sample were metallographically prepared. The samples were etched in nital. In a sample with eutectic composition (sample 1) was observed that graphite and austenite started to precipitate at the same level or the same time. This indicates the start of the eutectic reaction. The volume fraction of liquid, austenite and graphite was evaluated as function of the distance from the start of the eutectic reaction. The results are presented in Fig. 1. The figure shows that the volume fraction of liquid continuously decreases with distance and that the volume fraction of graphite increases in steps. Two plateaus are observed, one at the level close to the front and one at a distance of 1.5 mm from the start of the solidification. It was
Table 1 Chemical composition of the alloys studied in wt.% Element
C
Ni
Mg
Si
C.E.
Alloy 1 (DS) Alloy 2 (DS, TA) Alloy 3 (TA) Alloy 4 (TA) Alloy 5 (TA)
4.05 3.89 3.84 4.00 3.88
3.02 2.94 1.33 1.12 0.06
0.14 0.14 0.235 0.033 0.001
0.03 0.02 1.88 0.12 2.21
4.06 3.89 4.46 4.04 4.61
Table 2 Conditions achieved during the directional solidification (DS) experiments Experiment #1 Alloy Drawing rate (mm/min) Temperature gradient (K/mm) dT/dt (K/min)
1 40 3 120
Experiment #2 2 10 2 20
Fig. 1. Volume fraction of graphite and liquid as function of the distance to the solidification front for sample alloy 1.
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Fig. 2. Volume fraction of graphite and liquid as function of the distance to the solidification front for sample alloy 2.
also observed in the samples that at the end of each plateau a number of new graphite nodules were formed. From the figure it can be concluded that the volume fraction of liquid decreases faster than it is expected from the volume fraction of graphite. This can be explained as a precipitation of austenite larger than expected from the phase diagram. The figure also shows that the fraction of liquid slightly increases towards the end of solidification process. Similar results were obtained in sample 2. In this case solidification rate was lower (Table 2). Fig. 2 shows the volume fraction of graphite and liquid as function of the distance from the start of solidification for sample 2. Two deviations between the results presented in Figs. 1 and 2 can be noticed. There is a significant increase in the volume fraction of graphite at a certain distance from the solidification front. No increase in the fraction of liquid could be observed at the end. The fraction of liquid decreased continuously until the process was finished. The sudden increase of the volume fraction of graphite at 1.5 mm is related to nucleation of new graphite nodules. A number of small pores were observed a short distance behind the end of the solidification front. The volume fraction of these pores was measured as function of the distance to the front. The result is presented in Fig. 3. The figure shows that the fraction of pores first increases to a maximum and then decreases. The reason is that the formed pores are filled with graphite. Results from TA experiments will now be presented. The experimental methods have been described elsewhere [18]. The first results which will be discussed are from the work by Wetterfall et al. [5]. In both works series of TA experiments were interrupted by quenching the sample after a certain time. Typical temperature–time curves in the work by Tinoco and Fredriksson [18] are shown in Fig. 4. Micrographs of the quenched samples by Wetterfall et al. are shown in Fig. 5(a)–(d). The alloy composition in this case is slightly hypoeutectic eutectic (alloy 1). The solidification starts with a primary precipitation of austenite dendrites, Fig. 5(a). Graphite nodules freely formed in the liquid are observed. The graphite nodules get later into contact with the austenite dendrites and are surrounded by an austenite
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Fig. 3. Fraction of liquid and pores as function of the distance to the solidification front for sample alloy 2.
shell. Then they continue to grow by carbon diffusion through the austenite shell as shown in Fig. 5(b)–(d). The figures illustrate that the austenite dendrites increase in size as well as the dendrite arm spacing. The structure presented in Fig. 5 was evaluated by measuring the volume fraction of graphite, austenite forming the shell and austenite dendrites. Results of those measurements are presented in Fig. 6. Figs. 7 and 8 present results from measurements performed for alloys 2 and 4, respectively. Fig. 6 shows that, as it was observed in the micrographs, the volume fraction of austenite precipitating as dendrites increases with time. The figure clearly shows that the fraction of austenite in the dendrites is much larger at the beginning of the solidification process. According to the equilibrium phase diagram at the eutectic point, the volume fraction ratio graphite/␥ should be equal to 0.115 (ρ␥ = 7.87 g/cm3 , ρgraphite = 1.5 g/cm3 ). This value is presented in Fig. 9 together with the measured values of volume fraction ratio. In the figure the ratio represents the total volume fraction of austenite (the sum of the austenite in the dendrites and in the shell) divided by the volume fraction of graphite. It is clear that the volume
Fig. 4. Typical cooling curves obtained (alloy 4).
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Fig. 5. Microstructure observe in the samples obtained from quench-out operations for samples of alloy 1: (a) 18 s, (b) 52 s, (c) 60 s, (d) 86 s, (e) 152 s, and (f) 244 s.
fraction of graphite/␥ is much lower at the beginning. At the end of the process it gets closer to the ideal value given by the phase diagram. Tinoco and Fredriksson [18] performed an evaluation of the latent heat as function of the fraction of solid formed during TA experiments at low cooling rates. Results of those measurements are presented in Fig. 10. The figure presents latent heat measurements as function of the total volume fraction of solid.
As it is possible to observed the latent heat released is lower than the tabulated value at the beginning of the solidification process and increases towards the end. In the figure the latent heat released during the solidification of the same alloy without Mg-addition is also presented. This alloy solidified to a grey structure. In this case no effect on the volume fraction could be observed. The explanation to this observation is the difference in growth mechanism. For nodular cast iron the growth of graphite
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Fig. 6. Measured volume fraction of austenite and graphite in the microstructure of samples (alloy 2).
Fig. 9. Volume fraction ratio (graphite/␥) as function of the solid fraction obtained from Figs. 6–8.
Fig. 7. Measured volume fraction of austenite and graphite in the microstructure of samples (alloy 3).
Fig. 10. Measured latent heat as function of the solid fraction for samples of alloys 3–5.
is controlled by diffusion of carbon through the austenite shell and for grey cast iron the growth is controlled by diffusion of carbon through the liquid ahead of the solidification front. This will be discussed further on. 4. Thermodynamics calculations
Fig. 8. Measured volume fraction of austenite and graphite in the microstructure of samples (alloy 4).
Fig. 10 illustrated that the latent heat varied with the growth mode of graphite. It has also been observed a variation of the latent heat with cooling rate for cast iron [19]. This variation has also been experimentally observed in other alloying systems as well [16,17]. It has been proposed that a decrease of the latent heat with the cooling rate is related to the number of lattice defects formed in the solid during the solidification process. The energy of the solid will thus be larger than under equilibrium conditions. This has earlier been used for a recalculation of the phase diagram where the liquidus and solidus lines change according to the volume fraction of lattice defects [16]. That
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model will now be briefly described. The molar Gibbs energy of the liquid phase can be described by ◦
◦
L L L L L L GLm = xFe GLFe + xC GLC + Rgas T [xFe ln xFe + xC ln xC ] L L xC + xFe
n
L n L LFe,C (xFe
k
L − xC )
(1)
i=0
where ◦ GLFe and ◦ GLC are the Gibbs energies of the pure components in its reference state, Rgas the gas constant, T the temperature, LLFe,C the interaction parameter between the components in the liquid and k = · · ·. The molar fraction of the component is defined as ni xi = (2) nFe + nC where ni is the number of atoms of i. If thermal vacancies are considered, the molar Gibbs energy of the austenite phase can be expressed as ␥◦
␥
␥◦
␥
Fe G␥m = xFe GFe + xC GC + xV GV ␥ ␥ + Rgas T [xFe ln xFe ␥
␥ ␥ + xC ln xC ␥
␥
␥ ␥ + xV ln xV ␥
− (1 + xV ) ln(1 + xV )] + xFe xC
n
␥ n ␥ LFe,C (xFe
␥ k
− xC )
i=0
␥ ␥ + xC xV LC,V
(3)
␥
␥
where xV is the fraction of vacancies in austenite, LFe,C the interaction parameter between in the solid and LC,V is the interaction parameter between the solute and vacancies. Fe GV is the Gibbs energy of vacancies in pure iron. The Gibbs energy of the graphite phase can be described as = ◦ GC Ggraphite m
graphite
C + xV (GC V + FC−V ) C C C C ln xV + Rgas T xV − (1 + xV ) ln(1 + xV )
(4)
The chemical potential of each component can be calculated using the Gibbs–Duhem relationship for any component A µA = Gm + (1 − xi )
∂Gm ∂Gm − xi ∂xi ∂xj
(5)
j=0
The chemical potentials of each component are equal at ␥ ␥ equilibrium (µA = µLA , µB = µLB ). The interaction parameters: ␥ L LFe,C , LFe,C , and LC,V where estimated from the system of equations formed by (4) and (5) using the experimental data on the phase diagram given by Chipman [20]. The calculated parameters are presented in Table 4. Table 4 Interaction parameters between Fe–C and C-vacancies Parameter
T-relationship (J/mol)
␥ LFe,C LLFe,C ␥ LC,V
−1.5345 × 105 + 76.78T −1.9245 × 105 + 89.10T −7.6078 × 104 − 412.18T
␥
␥ equilibrium
Fig. 11. Calculated Fe–C–V phase diagram (XV = XV 0.003). Volume fraction ratio (graphite/␥) = 0.114.
graphite
, XV
=
The experiments showed that the volume ratio graphite/␥ was lower than the predicted by the equilibrium phase diagram. By using the thermodynamic model presented previously an analysis of the experimental results can be made. There are two possible ways for the volume fraction ratio graphite/␥ to change, either the graphite liquidus line moves towards higher carbon content or the austenite solidus line move towards higher carbon contents. The first case can be illustrated by using Eq. (4). In this case an increase in the Gibbs energy of graphite can be achieved by increasing its vacancy concentration. An example of this case is presented in Fig. 11. The second case is illustrated by changing the interaction coefficient between lattice defects and carbon in austenite so the carbon solubility in austenite increases. Carbon is an interstitial soluble element in austenite and an increase of the number of lattice defects reduces the density of the phase. The extra free volume can then give a possibility for more carbon to be dissolved into the austenite. The results of this calculation are presented in Fig. 12. In this case both the fraction of vacancies was increased and the interaction parameter between carbon and vacancies was increased. This is a reasonable assumption considering that carbon is interstitially dissolved in austenite. Then it is likely that the austenite will be enriched in carbon atoms around a vacancy and therefore the solubility will be increased. An increase in solubility can be described by an increase in the interaction parameter C-vacancies. From Figs. 11 and 12 the volume fraction ratio between graphite/␥ was evaluated. The results are 0.114 for case 1 (Fig. 11) and 0.084 for case 2 (Fig. 12). The largest effect is given by an increase of the carbon content in the austenite due to an increase of the number of lattice defects in the austenite (Fig. 12). A comparison between the calculated phase diagram and the experimental results, Figs. 2, 3, 9 and 10, shows that the experimental observations can only be described by the case where the carbon solubility increases in austenite. At the early stages of the solidification process the ratio graphite/␥ was found to be between 0.02 and 0.09. If, for example the value 0.09 is to be explained the carbon content in the austenite must be around
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close to one in the austenite close to a growing nodule and it is higher in the austenite close to the liquid. The growth of nodules surrounded by an austenite shell is controlled by the deformation process of the austenite shell and by diffusion of carbon through this shell. These two processes are coupled in series. A theoretical analysis of this process will now be presented. The diffusion of carbon through the shell is described by the following relation: dR V gr ␥ X␥/l − X␥/gr 1 = ␥ DC gr ␥/gr 1 dt V X −X R2 −1 R
gr
␥
graphite
Fig. 12. Calculated Fe–C–V phase diagram (XV = 0.009, XV graphite equilibrium ), XV
in this case (graphite/␥) = 0.084.
␥ LCV
=
was increased 15%. Volume fraction ratio
2.58 wt.% which is an increase of 20% from the equilibrium value. This can be observed in Fig. 12. 4.1. Volume fraction of austenite The phase diagrams presented in Figs. 11 and 12 will be used for a qualitative discussion of the solidification process. Fig. 11 shows a case where a large number of lattice defects are formed in the graphite nodules and a fewer amount in the austenite. Fig. 12 shows the opposite case where the number of lattice defects is large in the austenite and low in the graphite. The figures as well as the thermodynamic calculations show that the activity of carbon decreases with increasing fraction of vacancies that means that the carbon solubility in austenite increases with increasing lattice defects as is shown by comparing, Figs. 11 and 12. This is the same effect as alloying steel or cast iron with silicon. The number of lattice defects is related to the growth rate and it can be expected a larger number of defects the larger the growth rate is. The experimental observations might be explained in the following way. The nodules are formed in the liquid they grow to a certain size and then they are surrounded by an austenite shell. The nodules will grow by diffusion of carbon through the austenite shell. The graphite formed is less dense than the liquid and the austenite. The shell will expand by a plastic deformation. The deformation process will take part by a creep process. This process is assumed to occur by formation of vacancies. When the nodules are formed in the liquid they are small and grow very fast. The austenite dendrites formed increase in fraction by thickening, the growth process is slow. A lower amount of lattice defects are formed. Thus Fig. 11 is valid for this situation where graphite and austenite are formed from the liquid. The deformation process of the austenite shell when the graphite is growing by diffusion of carbon through the shell gives a larger number of lattice defects. The creep rate is largest near the graphite nodule and the numbers of defects is largest there. This case is illustrated by Fig. 12. The activity of carbon is thus
(6)
S
␥
where Vm is the molar volume of graphite, Vm the molar volume of austenite, D the diffusion rate of carbon in austenite, R the nodule size of graphite, S the distance to the austenite shell border, X␥/L the molar fraction of carbon in austenite in contact with liquid, X␥/gr the molar fraction of carbon in austenite in contact with graphite and Xgr is the molar fraction of carbon in graphite. In the analysis given by Wetterfall et al. [5] the mole fraction values are given from the equilibrium phase diagram. In this case the relations derived in the previous section will be used so the mole fractions will thus be a function of the growth rate. The values will be given by creep laws as described below. The molar volume of graphite is larger than the partial molar volume of carbon dissolved in austenite and in the liquid. During the growth of graphite nodules the austenite shell must expand by a plastic deformation. The expansion is related to the growth rate of the graphite nodules. This process has been described by Nabarro and Herring as a diffusion process of vacancies from the border austenite/liquid to the border austenite/graphite. According to Herring [21] and Shewmon [22], the growth rate of the cavity filled with graphite can be described by the following relation: Gcreep dR DV =n dt Rgas T R2 1 − 1 R S
(7)
where DV is the diffusion coefficient of vacancies in austenite, Gcreep the driving force for the creep process and n is a constant which vary between 10 and 40 depending on the geometry [22]. In order to solve these relations and describe the growth rate as function of time or nodule size the driving force for the growth process is divided into two steps, one corresponding to the creep process and one to the diffusion of carbon. This can be written in the following way: GL→Gr = Gcreep + GC tot Diff
(8)
where GL→Gr is the total driving force for precipitation of tot graphite nodules, Gcreep the driving force for plastic deformation of the austenite shell and GC Diff is the driving force for diffusion of carbon through austenite. Fig. 13 shows a schematic representation of the free energy curves at a temperature below the eutectic one in the Fe–C system. At the right side of the figure is included the total driving force for precipitation of graphite from a melt in equilibrium with austenite and formation of graphite in equilibrium with austenite. This driving force can
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Fig. 13. Schematic representation of the Gibbs energy curves at a temperature below the eutectic in the Fe–C system.
be formulated by normal thermodynamic laws according to GL→Gr = Rgas T tot
␥/L aC ln ␥/gr aC
Fig. 14. Nodule growth rate as function of its radius calculated according to Eq. (13).
(9)
␥/L aC
where is the activity of carbon in austenite in equilibrium ␥/gr with liquid and aC is the activity of carbon in austenite in equilibrium with graphite. In our further analysis we will assume that the activity coefficient is the same at these two equilibrium points. Expressing Eq. (10) into a series of expansion ␥/L Xeq Gtot = Rgas T −1 (10) ␥/gr Xeq By transforming the concentration gradients to chemical potential differences Eq. (6) can be rewritten as dR = dt
gr Vm ␥ Vm
␥ DC
GDiff Rgas T (Xgr − X␥/gr )R2 1 − 1 R S
Combining Eqs. (7), (8) and (10) with Eq. (11) ␥/L 2 1 − 1 R Xeq dR R S −1 = ␥ ␥/gr dt nDV Xeq 1 gr ␥/gr ␥ 2 1 dR Vm R R − S (X − X ) + ␥ dt Vmgr DC
(11)
dR = dt
R2
1 R
−
1 S
1 nDV
−1 +
␥
Vm (Xgr −X␥/gr ) ␥ gr DC Vm
Gtot = Rgas T ln
(12)
(13)
A material balance can be used to obtain a relation between R and S in order to get an idea about the effect of different parameters on the growth process. In this case the value derived by Wetterfall et al. [5] from the equilibrium phase diagram will be used (R = 2.4S). Calculations of the growth rate as function of nodule size were then performed. In the calculation the n parameter has been varied from 1 to 100. Ten different curves
C a␥/L eq C a␥/gr eq
= Rgas T ln
def aV eq aV
(14)
According to thermodynamics of lattice defects the last term can be rewritten in the following way: Rgas T ln
Rearranging Eq. (12), ␥/L Xeq ␥/gr Xeq
are presented in Fig. 14. The figure shows that at high n values (close to 100) the growth rate becomes independent of n. The growth process is in this case controlled by the diffusion of carbon through the austenite shell. As it was mentioned the n parameter is a structural dependent constant and it varies between 10 and 40. The conclusion here will thus be that the process is controlled by the deformation of the austenite shell. From the total driving force it can now be estimated the fraction of vacancies needed for the plastic deformation by the following relation:
def def aV XV V/C = R T ln gas eq eq + Rgas T ln γ aV XV
(15)
where the activity coefficient for the vacancies at equilibrium, γ V/C , is normally included in the molar free energy of vacancies GV , according to GV = Fe GV + Rgas T ln γ V/C
(16)
where Fe GV = HV − TSV . In this case HV is the heat of formation and SV is the entropy of formation of vacancies in austenite. However in this case the vacancy concentration is larger than the equilibrium values, this will therefore have an influence. The last term in Eq. (16) can be described in terms of the interaction parameter LFe,V . This factor will be further discussed by analysing the possible fraction of vacancies created by the plastic deformation process. Fig. 10 shows that the latent heat was related to the fraction of solid formed. The variation of the latent heat with the fraction of vacancies can be estimated by the following relation: eq
Hnc = Htab − (xV − xV )HV
(17)
H. Fredriksson et al. / Materials Science and Engineering A 413–414 (2005) 363–372
Fig. 15. Graphite nodule without a shell growing in contact with an austenite dendrite.
where Hnc is the measured latent heat of nodular cast iron, eq Htab the tabulated value of latent heat and (xV − xV ) is the difference in mole fraction of vacancies between nodular cast iron and grey cast iron. The heat balance given by Eq. (17) can be used to calculate the fraction of vacancies as function of the measured value on the latent heat. By using the data presented in Fig. 10 about the latent heat the resulting fraction of vacancies lies between 0.05 and equilibrium. The total driving force is derived from the phase diagram by extrapolating the solidus line of austenite into the two-phase region ␥/graphite. At a temperature of 10 K below the eutectic the values gives an increase of the fraction of vacancies of 1% from the equilibrium value, which is much smaller than was given by the energy balance. This indicates that the heat of formation of vacancies increases with increasing fraction of vacancies, or it can also be a strong concentration dependence on the heat of mixing of vacancies. This can be expressed in another way; the interaction parameter LFe,V is divided into an enthalpy term and an entropy term. The enthalpy term is strongly positive. It is further on suggested that the entropy for formation of vacancies, SV , is also concentration dependent. This dependence can be related to an atomic structure change of the austenite lattice [23]. 4.2. Growth of graphite The graphite is nucleated in the melt between the dendrites arms and it grows freely in the melt until it reaches a certain size. It was proposed by Wetterfall et al. [5] that the shell is formed when the graphite nodules gets contact with the austenite due to its flotation. However, this seems to be a too simple explanation. It has been observed that the graphite nodules can have contact with austenite without being grown around (Fig. 15). The reason for this might be the fact that the growth behaviour is such that graphite and austenite have the same chemical potential, which is necessary for the growth of austenite around graphite. The difference in chemical potential can be due to the plastic deformation of the austenite. But it can also be an effect of the interface kinetics, when graphite is precipitated from the liquid.
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This reaction can also produce lattice defects in the graphite which change its chemical potential during the growth. This was partly discussed in the chapter about the calculation of the phase diagram. When the nodules are small the growth rate will be large and the interface kinetics will be large. In this case it will not be possible to establish a growth of austenite around the graphite. The remaining driving force will be too small to cause the plastic deformation of the austenite. This can be established when the nodules has reached a certain size. In the case when austenite shells are formed the growth rate of graphite nodules decreases and the interface kinetics for growth of graphite is decreased further on, and the assumption that there is equilibrium between ␥/graphite when deriving the driving force might not be far from the reality. When the growth of austenite ceases the vacancies start to condense at the same time as the graphite starts to grow due to the decrease of the carbon content in the austenite. The remaining vacancies will now condense. At this process the austenite will shrink, pores are formed. Those pores can be filled with liquid if they are in contact with the liquid or will remain in the structure as shrinkage pores, Figs. 1 and 2. 5. Discussion Taylor and co-workers [9] performed a large number of experiments for various iron base alloys with carbon content close to the eutectic. It was found that above a certain volume fraction of free graphite in the matrix no solidification shrinkage occurs. According to a carbon balance in the binary Fe–C system of equilibrium this will correspond to 3.8 wt.%. This value can then be adjusted by changing of the eutectic line due to the silicon content. This is as Taylor showed valid for normal flake cast iron but as has been shown not valid for nodular cast iron. The reason is the difference in the growth process which produces a larger precipitation of austenite during an early stage of the solidification process for nodular cast iron compared to the precipitated for grey cast iron. The volume fraction of graphite is less than the critical value given by Taylor. Solidification shrinkage will then occur. This shrinkage occurs in an earlier stage of the solidification process where the solid fraction is low. It is thus possible to compensate for the shrinkage by filling in new liquid from a riser or from parts of casting solidifying with a lower rate. If no risers are used macropores will be formed. In a later stage of the solidification process the precipitation of graphite would increase. The austenite will thus expand and the volume of the casting will increase. However, the distance to the macropores becomes then too large therefore it is difficult to transport melt through a network of austenite with a small volume fraction of liquid. The macropores will then remain; the total casting will instead expand. The expansion will take part by a creep process. The lattice defects taking part in this process together with those ones formed during the solidification process will condense during the end of the solidification process or just after. Micropores are then formed in this case. The volume fraction ratio graphite/␥ was observed to be affected by the nucleation of new graphite nodules, Figs. 1 and 2.
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Therefore it will be possible by an efficient inoculation procedure to decrease the volume of a riser. 6. Conclusion The formation of shrinkage pores is analyzed by directional experiments. It was shown that the pores were formed after the solidification process. The formation of these pores is a result of a creep process due to vacancies formation and diffusion. The fraction of vacancies formed during the creep process is larger than the equilibrium fraction. These vacancies will condense and pores are formed. The fraction of vacancies is larger the smaller the nodules are. References [1] [2] [3] [4] [5]
M. Morrogh, W.J. Williams, JISI 155 (1947) 321. H. Morrogh, W.J. Williams, JISI 158 (1948) 306. K.D. Millis, US Patent 2,485,760. E. Scheil, J.D. Sch¨obel, Arch. Eisenhwes. (1953) 4. S.-E. Wetterfall, H. Fredriksson, M. Hillert, JISI 210 (1972) 323.
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