Chapter 5 Fabrication and properties of PbS nanocrystals
5.3 Formation mechanism of Pb nanowires .1 Reactions between Pb and O 2
Casting begins with molten metal and many reactions that occur between molten metal and its surroundings can lead to defects in the casting process. Oxygen and molten metal often react to produce metal oxides. These oxides usually comprise the dross or slag with other impurities and can be trapped in the casting to impair surface finish and mechanical properties. The oxides would float to the surface of the liquid metal and raise more difficulties during pouring the mold especially for nano-scale casting.
Therefore, it is an important issue to minimize the oxidation of the molten metal before casting. Here we discuss the reactions between lead and oxygen first.
According to the available thermodynamic data [96], there are three metal oxides, PbO, PbO2 and Pb3O4, in this system. The possible reactions for lead oxide formation are:
)
In the aspect of thermodynamics, any reaction for which the change in Gibbs free energy is negative should be favorable or spontaneous. The Gibbs free energy expression is given by the following equations:
∆G0 =∆H0 −T∆S0 (5-4)
∆G0 =−RTlnK =−2.303RTlogK (5-5)
Where ∆G0is the standard state Gibbs free energy change of a reaction;
H0
∆ is the change in enthalpy; T is the absolute temperature; ∆ is the S0 change in entropy; R is the gas constant, and K is the equilibrium constant.
All the presently observed reactions and their relevant thermodynamic data are listed in table 5-1. The magnitude of ∆G0would tell us how far the standard state is to equilibrium. Figure 5-5 presented the relationship between the change in Gibbs free energy and the temperature reveals that the most stable reaction in this system is the reaction 5-3. In addition, the equilibrium constant K of the reaction 5-3 can be expressed as:
P is the partial pressure of the oxygen.
Since the activity of Pb is unity, the relationship between the partial pressure of the oxygen and temperature is shown in Fig. 5-6. As indicated in Fig. 5-6, the partial pressure of oxygen needed markedly grows as the temperature increases and the Pb3O4 can always exist until the partial pressure of oxygen is less than 10-21.26 atm at 600 K ( the melting point of lead). In other words,
avoiding the formation of the oxide is much more difficult to achieve.
Therefore, enhancing the reaction temperature and keeping the chamber in the high vacuum is necessary for reducing the occurrence of the oxidation.
Table 5-1 Thermodynamic data of the Pb-O reactions.
Reaction ∆G0(kJ) logK
300 600 900 1200 1500
-600
Fig. 5-5 Relationship between the change in Gibbs free energy and the reaction temperature.
300 400 500 600 700 800 -60
-50 -40 -30 -20 -10
log Po 2 (atm)
Temperature (K)
3Pb + 2O2→ Pb3O4
Fig. 5-6 Relationship between the partial pressure of the oxygen and the temperature in the reaction: 3Pb(s,l) +2O2(g) →Pb3O4(s).
Fig. 5-7 The curve of required force with pore diameter when Pb melt is injected into the nanochannel.
0 100 200 300 400
0 3000 6000 9000 12000 15000
Diameter (nm)
Force (Kg)
5.3.2 Casting process of Pb nanowires
In the casting process, the molten Pb metal is injected into the porous alumina membrane forming metallic nanowires. The required force for Pb melt into nanochanel is proportional to the surface tension of the melt. The pressure for melt injection into nanochannel can be evaluated as [97-98]:
r A
P= F = −(2γ cosθ)
∆ (5-7)
Where F is the normal force; A is the area of specimen; r is the radius of nanochannel; γ is the surface tension of Pb melt, and θ is the contact angle between the Pb melt and the porous alumina membrane. In this experiment, the surface tension of Pb melt is 468 dyne/cm [99], A is 1 cm2 and the contact angle assumes the least favorable case of complete nonwetting (θ = 1800).
This is because liquid metal/ oxide systems mostly have non-wetted (θ＞900) and non-reactive wetted conditions. Moreover, the contact angle would be greater in a rough surface than in a corresponding flat surface [100].
Therefore, the required force to inject the Pb melt into nanochannels with different diameters could be calculated. Figure 5-7 shows the relationship between the diameter of the nanochannel and the required force. It can be seen that the force is inversely proportional to the diameter of the nanochannel and getting smaller nanowires needs to apply larger hydraulic force. The critical force of forming the Pb nanowire with a diameter of 20 nm is counted as 955 kg. As discussed above (section 5.3.1), an oxide film is easily produced on the surface of Pb melt even if the vacuum maintains at 10-6 Torr. The oxide film would increase the difficulty of the casting process.
Furthermore, the metal oxide usually has well mechanical properties and high melting temperature. The oxide has to be broken when the Pb melt is poured
into nanochannel. In the meanwhile, it is essential to increase the filling ratio of Pb nanowires as far as possible. For these reasons, the force used in this experiment is several times as the theoretical value.
5.3.3 Growth model of Pb nanowires
In the solidification process of pure metals, the solid phase will nucleate and grow in the melt. The nucleation of the solid phase should be homogeneous, but in actuality it is usually heterogeneous because less energy is added to the system if nuclei form at an already existing solid-liquid interface. Nucleation is favored at the mold wall, because not only there is a solid-liquid interface presented, the liquid is usually cooler near the mold wall. Growth of the grain is by movement of the interface between solid and liquid. The grain structure and direction of grain growth are controlled by the temperature gradients at the solid-liquid interface. Therefore, the growth of high quality single crystals is quite difficult and requires a planar solid-liquid interface and a particular solidified direction which is anti-parallel to that of the heat flow. That is, the formation of cells or dendrites must be avoided, and a sufficiently high thermal gradient or a sufficiently low growth rate will lead to the plane front growth. The temperature gradient (G) at the interface defines as:
dZ dT
G= (5-8)
Here, Z is the coordinate with respect to the movement of the solid/liquid interface. It can be found that the solid-liquid interface of a pure metal will be more stable if the temperature gradient is positive (G>0).
Growth rate (V) means the movement rate of the solid-liquid interface and
dt dZ
V = / (5-9)
The ratio, G/V, determines the solidification morphology (planar, columnar and dendritic). The product of G and V is equal to the cooling rate which controls the scale of the microstructures [101-102].
In this experiment, after nanochannels are filled with Pb melt, a solidification process proceeds immediately using the water cooling method.
Heat loss from the bottom of the wire is often a sufficient heat sink.
Solidification proceeds from the bottom of the wire to the top region in this experiment. The schematic diagram of the solidification is shown in Fig. 5-8.
Fig. 5-8 Schematic illustration of the the Pb nanowire solidification model. Here, Z is the coordinate with respect to the moving solid-liquid interface and V is the rate of the movement of the solid-liquid interface.
One must suppose that the steady stable condition for the temperature gradient through the specimen is applicable. There is a constant temperature gradient, a uniform growth rate and a highly uniform microstructure