solute distribution in casting of steel.pptx

Study on the Effect of the Localized Electrode Degradation on Weldability During an
Electrode Life Test in Resistance Spot Welding of Ultra-High Strength Steel

1. Steady State Solidification We can also consider another situation, where there is still no diffusion in the solid phase, but in the liquid phase there is now limited diffusion, rather than complete mixing. From Fick’s second law, we have: also: These combine to give: which can be evaluated with the appropriate boundary conditions to give us: This equation describes the solute profile in the liquid ahead of the interface, in the steady state.

2. This model describes the solute profile in the liquid ahead of the interface once the situation has reached a steady state. However, when solidifying a bar, for example; it will take a period of time for the solute ‘bow wave’ to build up ahead of the interface. During this initial transient solid will be forming with a concentration less than C , (initially it will be k C ) until a steady state is achieved. The solid then advances along the length of the bar with the solute bow wave ahead of the interface. When the bow wave, with a characteristic length, D L  / v, begins to run into the end of the bar, the solute ‘piles up’ giving a rapid increase in the solute concentration of the liquid. This final transient sees an increase in the concentration of the solid, and may result in the formation of some non-equilibrium eutectic at the end of the bar.

3. S teady State Solute Profile ahead of an Advancing Solidification Front During a time interval Δt , the solidification front advances by VΔt , where V is its velocity.  There is no change in the coordinates of physical matter in the fixed frame (neglecting the effect of the volume change on freezing).  Any particular slice of liquid, eg at x = x 1 , will therefore change in composition only as a result of diffusion (governed by the diffusion equation ). Since the profile is steady state, when referred to the moving frame x’(=x- Vt )., the composition must be constant for all values of x’. This requires

4. Where the constant = V C

5. This constant is found by setting C =C o /k at x’ = 0

6. Incidentally, it follows that the concentration gradient at any point head of the interface is given by The numerator in this expression is the composition difference across the interface, so the denominator must be the distance indicated in Fig.2.  It’s often termed the solute boundary layer (δ) and it can be defined either via the above equation or as the characteristic distance for the exponential decay (see Fig.2).  The above equation represents a simple and useful way of expressing the concentration gradient at the interface.

7. Constitutional Undercooling It is actually rare to have a negative temperature gradient ahead of the interface, yet it is observed that dendrites are very hard to avoid in practice. Consider the case of a binary eutectic alloy of elements A and B. An A-rich solid phase is forming from a liquid which is a solution of A and B. Liquid which has a higher concentration of B appears yellow. As the A-rich phase forms, element B is ejected into the liquid, creating a region of B-rich liquid ahead of the interface. If a random protuberance forms on the interface, the tip of the protuberance is in a liquid of lower B concentration than that at the interface, which means that the fusion temperature of the surrounding liquid is higher, giving a greater driving force for solidification at the tip.

8. Lateral growth of the tip results in partitioning of B perpendicular to the primary dendrite arms, so that secondary dendrite arms can form ( fig.a ) in the same way, resulting in a dendritic interface. Another growth front morphology, between the dendritic and planar extremes, is cellular growth, where only primary dendrite arms form (Fig B). As the cells grow, they partition solute laterally, which can cause secondary arms to form. However, if the B that is ejected cannot diffuse away from between the arms, there will be an approximately uniform distribution of B perpendicular to the main growth direction. This disfavours the formation of secondary arms.

9. The situation can be analysed quantitatively to determine whether a dendritic growth front is likely. If we assume a steady state diffusion profile ahead of the interface,   the concentration of solute at any distance, x’, ahead of the interface is given by: where T m is the melting temperature of the pure substance, and ∂ T L  / ∂x is the gradient of the liquidus on the phase diagram, which is usually negative if k < 1 .

10. The graph of liquidus temperature against distance, has a profile like the one shown below I f the temperature gradient, ∂T / ∂x, ahead of the interface is less than the gradient of the liquidus temperature at the interface, there will be an undercooled region ahead of the interface, in which a planar interface is unstable. To maintain a planar interface the relationship below must hold Differentiating Eq (1) w.r.t x, and setting x=0, we get

11. The critical gradient required to maintain a planar interface is given by: For temperature gradients only slightly below the critical gradient, the planar interface will break down, but the trapping of solute partitioned between the primary dendrite arms prevents growth of secondary arms, and a cellular growth front evolves . In order to ensure a planar interface, the growth front velocities need to be quite low, of the order of tens of microns per second. Hence it is often difficult to avoid dendritic growth during real solidification.

12. In dendritic growth, most of the solute is ejected between the dendrite arms. The structure of the dendrites serves to prevent mixing with the rest of the liquid, so that the solute partitioning occurs over the length scale of the secondary dendrite arm spacing. This is known as microsegregation In order to ensure a planar interface, the growth front velocities need to be quite low, of the order of 10s of microns per second. This is why it is often difficult to avoid dendritic growth in practical solidification. In dendritic growth, most of the solute is ejected between the dendrite arms. The structure of the dendrites serves to prevent mixing with the rest of the liquid, so that the solute partitioning described earlier occurs over the length scale of the secondary dendrite arm spacing. This is known as microsegregation