Dear ajit,

Generally dendrite structure formation takes place when a cast metal cooling rate is very high and it occurs in non-equilibrium cooling.

The development of dendritic structure is a very complicated story.

If you like to learn more about it, please have a look at this paper from

M.E. Glicksman:

Materials Science and Engineering, Elsevier, Vol. 65, Issue 1, July 1984, Pages 45-55.

doi: 10.1016/0025-5416(84)90198-8

Sorry, the title of the paper by M.E.Glicksman is:

'Free dendritic growth'

Dendrite structure develops under both equilibrium and non-equilibrium conditions. However, if the the cooling rate is high enougn and a large undercooling of the melt takes place, dendrite growth could be suppressed resulting in amorphous metal as in the case of rapid solidification. 

Indeed, it is not a very straigtforward matter but certainly not an enigma either. Dendritic structure forms NOT when the solidification rate is very high (unlike Jeenager, one of the followers of this question, suggested). The competing mode of solidification is solidification with a planar front. If the removal of the latent heat occurs through the liquid phase then we observe dentritic morphology. If the heat flow is through the solid behind the solidification front, then the planar mode is observed. This indicates a critical solidification rate. With regard to non-equilibrium solidification the following can be said: If a protrusion from the solidification front finds itself in a liquid that has a temperature higher than its melting temperature then it cannot survive, leading to planar mode. If however, a protrusion attempts to grow into a constitutionally supercooled liquid it can develop into a dendrite. All dendritic solidification structures are associated with some degree of micro segregation (if within an equiaxed grain we would appropriately call it 'coring'). Therefore we may say dendritic solidification is observed under non-equilibrium conditions. One should follow all these also on relevant drawings but... 

dendrite structure can be form in slow or high cooling rate (equil or non-equil) depends on the studied alloy. However, most studied alloys forms dendrite at high cooling rate.

One more thing, the shape and the direction of dendrite growth depend on the cooling flow.

Dendritic growth is a typical nonequilibrium process, which takes under finite and negative Gibbs free energy of solidification (i.e., positive internal entropy production).  At  any stage of the dendritic growth if the rate  of branching and the tip displacement speeds become constant then you may talk about the nonequilibrium Stationary State according to the definition of Prigogine ''Thermodynamics of Irreversible Processes CH VI.'' 

The displacement velocity  of the dendritic front (interface) in single component systems is solely control by how fast the local heat generated at the front  by solidification process is removed by the melt as well the dendrite body itself. This a moving boundary value heat transfer problem with convective and source terms. It  should be solved numerically but it is highly trouble some because of its inherent numerical instabilities  due to the convectıve term!

Hi,

If you mean by equilibrium cooling true equilibrium as defined by the phase-diagram (or minimum Gibbs-Energy), you are talking about cooling rates low enough for complete removal of any segregation in the solid. This takes place by solid state diffusion and is this quite slow. Under these conditions you will not see any dendrites. For directional solidification you will see a planar front, for equiaxed growth globular structures. However, passing the solidification (temperature) interval will take hours to days, if not weeks! For any technically relevant cooling rate, as observed for example in foundries, and typical alloy conditions you will see dendrites.

Dear Dr. Warnken; I am sorry to say that  you haven't bother in  reading  my comments carefully. Who talks about the equilibrium phase diagram?  I have a single component system such as H2O.  Therefore I am not dealing with constitutional dendritic  formation phenomena, which means the removal of the rejected solutes  by diffusion and convection is not a problem. I am strictly concerned with the faceted dendritic growth of anisotropic crystals having preferred orientations for growth. That means specific Helmholtz surface free energy is orientation dependent.[1]  This a heat transfer problem, which is  highly nonlinear Stefan type [2]  Butt there is  some approach to the stability  problem ad hoc  fashion  by   Cahn et. al.. See Ref.  [1].

One should also know that  the reversible process is a hypothetical case (zero internal entropy production), and it can  never take place in nature, since it  consist of the passage in either direction towards the stable equilibrium state through a continuous series of equilibrium states!

[1]Solidification Processing By Merton C. Flemings, 1974,  McGraw-Hill

[2] Free and Moving Boundary Problems By John Crank, 1999, Clarendon Press

Hi when G /R to be lesser than mc (1-k)/(Dk), denderitic solidification occur. Here G and R are thermal gradient in the liquid  and solidification growth rate respectively. M snd k can be obtained by phase diagram.  C and D are composition and diffusion coefficient of the liquid respectively.  More information 

Dear Dr. Khazaei,  this formula, which appears in Flemings (Eq.3-5)  is valid for the formation of cellular interface. For the stability of the flat interface it reads as  EG than, for the instability  as you mentioned  L less than.!  The formula applied for the faceted dendrites or hopper crystals   is given by  Eq. 9-54, which looks somewhat different and it involves  r  the radius of the faceted crystal,  or the diameter of the branches.  But a little manipulation of terms I could put it into a form, which is very similar to Eq.3-5 where only gradient is replaced by     DeT/ r , which has the dimension of the gradient!!  But this is completely superficial !!

Best Regards

Dear prof. Tark,

Thank you for your sufficient description. Unfortunately my comments was not  sent completely. 

Regards