1. Answer to you query: Dr. Roger Domingo-Roca has answered homogeneous and heterogeneous nucleation and has already addressed:- (a)  the thermodynamic barrier in terms of Gibbs free energy change, (b) the maximum in free energy function that defines the critical cavity size, beyond which they can further grow. Please note that the volume energy and the surface energy terms and their contribution to the total change in free energy with increasing radius (r) of the cavity is used to define the critical radius of the cavity as that radius (i.e., rc) at which the plot of (free energy change) vs. radius reaches maximum. 

I like to add on as my research interest for a long time is mainly 'Creep deformation, damage and fracture'.

Please refer to:- Rishi Raj, Acta Metallurgica, 1978, pp. 995-1006.

Intergranular creep fracture is a complex subject and it generally occurs by grain boundary (GB) cavitation. Cavities nucleate, grow and then the coalescence (or linkage) of cavities to form bigger cavities or micro-cracks, and further the crack propagation can cause failure with sufficient loss of creep ductility. Creep cavities nucleate (heterogeneously) at geometrical irregularities on grain boundaries (i.e., at specific sites) due to high stress concentration arising from grain boundary sliding (GBS) when the stress concentration is not relaxed by any accommodative processes (such as elasticity, diffusion or creep). Though cavities can nucleate on GBs at 450 to the stress axis, it is observed that the cavity distribution is maximum at 900 to the stress axis, and the driving force comes from the normal traction across the grain boundaries. Growth is also favored for cavities orthogonal (900) to stress axis. Hence cavitation, both nucleation and its growth is favored at specific sites and on those specifically oriented grain boundaries.  

Nucleating a cavity (of volume) involves work done by the stress, creation of a new surface and the grain boundary area consumed by the cavity. The Gibbs free energy change can accordingly be expressed and then one can get for a critical cavity, the expression for critical radius and the critical volume. Hence, the free energy barrier exists for nucleating  a cavity which is stable. Further, the  volume of a critical cavity = Fv (alpha) x rc3, where rc is the critical radius and Fv (alpha) is function of alpha and alpha is the di-hedral angle. Now, cavities can be nucleated at various possible sites (with suitable stress concentration) such as: - (i) triple points (or junctions) of high angle GBs, (ii) sub-boundary-GB intersections, (iii) grain boundary ledges and (iv) second phase particles. Various models for cavity nucleation are available in the literature (Please see Figure 1 in  L.-E. Svensson and G. L. Dunlop, Canadian Metallurgical Quarterly, Vol. 18, pp. 39-47, 1979).

Please refer to:- Rishi Raj, Acta Metallurgica, 1978, pp. 995-1006. Now, for heterogeneous nucleation of a cavity, nucleation at the triple junction formed by a second phase particle will have the smallest critical volume [i.e., Fv (alpha) x rc3 ] and the highest probability for cavity nucleation. Hence the specific cavity shape reduces the nucleation barrier. 

2) Nucleation and Growth of Creep Cavities: when the stress concentration developed at GBs is not relaxed by any accommodative processes, the r-type cavities or w-type cracks form on GBs. Cavitation is not connected with any crystal structure and can occur in high purity as well as single phase materials and in engineering alloys such as austenitic stainless steels, ferritic steels and in nickel base superalloys (J. Weertman, J.R. Weertman, Physical Metallurgy, Eds. R.W. Cahn and P. Haasen, Chapter 20, Elsevier Sci. Publ. company Inc., 1983). The critical assessment of various mechanisms for cavity nucleation by Argon (A.S. Argon, Sripta Metall., 1983, 17, p.5) suggests that the classical cavity nucleation, i.e., clustering of vacancies on Gb subjected to normal stress needs very high stresses (say E/100) and that by thermally activated breaking of bonds also requires very high stress concentration. So, it is well accepted that cavities nucleate due to stress concentrations developed at hard GB particles by the process of GBS. Cavities also can nucleate at (tensile) GB ledges as well as due to the intersection of slip band with a GB (B. F. Dyson, M. S. Loveday and M. J. Rodgers, Proc. R. Soc. 1976, 349A, p. 245). Cavities can also nucleate continuously [Dyson, B.F., 1983. Continuous cavity nucleation and creep fracture. Scripta Metall. 17 (1), 31–37]. The constained cavity growth was proposed by Dyson and later taken forward by other researchers (Dyson, B.F., 1976. Constraints on diffusional cavity growth rates. Metal. Sci. 10 (10), 349–353).

Cavities can grow by pure diffusion, or by power-law creep or by coupled diffusion and power-law creep (i.e., growth is by local diffusion process which in turn is dictated by the creeping matrix). The early work of cavitation by  Greenwood (J. N. Greenwood, D. R. Miller and J. W. Suiter, Acta Metall., 1954, 2, 250) was reformed by others and the Hull-Rimmer model (D. Hull and D.E. Rimmer, Phil Mag. 1959, 4, p. 673) considers the growth of cavities on GBs subjected to a normal stress by pure diffusion process. Void grows by the stress assisted diffusion of vacancies along GB. Surface diffusion has to be invoked to support the observation of spherical cavities. If surface diffusivity is greater than the GB diffusivity, growing voids remains spherical in shape, otherwise the voids become crack like. Pure diffusional growth can be regarded as the limiting case of growth by coupled diffusion and power-law creep, and occurs at low stresses that power-law creep is negligible. 

3. A Few References: 

(i) "NUCLEATION OF CAVITIES AT SECOND PHASE PARTICLES IN GRAIN BOUNDARIES" by RISHI RAJ, Acta Metallurgical, 1978, pp. 995-1006.

(ii) "Creep crack propagation by cavitation near crack tips", R. Raj and S. Baik,  Metal Science, August-September, 1980, p. 385.

(iii) "THE ROLE OF INTERFACIAL DISLOCATIONS IN THE NUCLEATION OF INTERGRANULAR CREEP CAVITIES", L.-E. SVENSSON and G. L. DUNLOP, Canadian Metallurgical Quarterly, Vol. 18, pp. 39-47, 1979.

(iv) :View Point Set No. 5 on Creep Cavitation", Scripta Metall., 1983, 17, pp. 1-43.

(v) M.V. Beere, M.V. Speight, Metal Sci., 1978, 12. p. 172.

(vi) G.H. Edward, M.F. Ashby, Acta Metall., 1979, 27, p. 1505.

(vii) A.C.F. Cocks, M.F. Ashby, Progress in Mater. Sci., 1982, 27, p. 189.

(viii) M.F. Ashby, C, Gandhi, D.M.R. Taplin, Acta Metall., 1979, 27, p. 699.

(ix) "Perspectives in Creep Fracture", 1983  by M. F. Ashby (Editor), L. M. Brown (Series Editor), Pergamon (January 1, 1983).

(x) "Fracture at high temperatures", Hermann Riedel, Springer, 31-Dec-1987 - Technology & Engineering - 418 pages.

(xi) A. J. Perry, Review: Cavitation in creep, J. Mater. Sci., 9, 1016–1039 (1974).

(xii) A Review Paper: L. E. Svensson and G L. Dunlop, Growth of Intergranular Creep Cavities, In--ternational Metals Reviews, No.2, 109–131 (1981).

(xiii) A Review Paper: Creep cavitation in metals, M.E. Kassner, T.A. Hayes, International Journal of Plasticity, Volume 19, Issue 10, October 2003, Pages 1715–1748. 

(xiv) M. F. ASHBY and B. F. DYSON, “Creep Damage Mechanics and Micromechanics, in Advances in Fracture Research”, Vol. 1, 6th International Conference on Fracture (ICF 6), edited by S. R. Valluri, D. M. R. Taplin, P. Rama Rao, J. F. Knott, R. Dubey (Pergamon Press, Oxford, 1984) pp.3–30. 

Hi,

the nucleation (both the heterogeneous and the homogeneous) is a consequence of the variation in the Gibbs energy. Therefore, to grow new nuclei a work is needed and consequently, to overcome an energy barrier (generally, the heterogeneous nucleation is energetically more favorable than the homogeneous nucleation because the energy barrier is lower). This is what the nucleation barrier refers to, the work that you may do to overcome a critical value and start the nucleation. Your nuclei will become stable when its radius overcome a critical value (if you type nucleation barrier in google images you will find the typical curves for both homogeneous and heterogeneous nucleation).

So what the author means with the sentence '[...] the specific cavity shape may greatly reduce the nucleation barrier' is that the shape of the cavity you have in your material can reduce the energy barrier and, therefore, the work that you need in order to grow nuclei in that cavity is lower, encouraging the 'cavity nucleation at distinct nucleation sites in grain boundaries, rather than homogeneously in the crystal lattice' or, what is the same, encouraging the heterogeneous nucleation rather than the homogeneous nucleation. So, finally,  you can see that at the end everything is related: you need energy for nucleation, what implies applying a work, so you need to overcome an energy barrier (which is related with the radius, or what is the same, the cavity shape). As the work you need for heterogeneous nucleation is generally lower than that for homogeneous nucleation, the latest is the one that will predominate. So now can you find the sense at the sentence 'the cavity shape could influence the cavity sites by reducing the nucleation barrier'?

I hope I've answered your question. If you didn't refer to this just let me know and I'll be delighted to answer again. Also, if you are interested in the energy formulas just let me know, but maybe you can find them somewhere in the internet.

Creep cavities (cracks) often nucleation at grain boundary precipitates, which are "high energy" sites, either because of the local stress distribution or dislocation pile up due to the local heterrogenity.

Cavity nuclii are like cancer cells in human body, they are there by the making of the material (body). They spread only under conditions when the energy overflows the "barrier". Perhaps, it is more energytically favorable for a cavity to take penney shape as on grain boundaries than spherical shape as in crystal lattice. But, I believe it has more to do with the distribution or density of defects, which are apparently high along grain boundaries. 

1. Answer to you query: Dr. Roger Domingo-Roca has answered homogeneous and heterogeneous nucleation and has already addressed:- (a)  the thermodynamic barrier in terms of Gibbs free energy change, (b) the maximum in free energy function that defines the critical cavity size, beyond which they can further grow. Please note that the volume energy and the surface energy terms and their contribution to the total change in free energy with increasing radius (r) of the cavity is used to define the critical radius of the cavity as that radius (i.e., rc) at which the plot of (free energy change) vs. radius reaches maximum. 

I like to add on as my research interest for a long time is mainly 'Creep deformation, damage and fracture'.

Please refer to:- Rishi Raj, Acta Metallurgica, 1978, pp. 995-1006.

Intergranular creep fracture is a complex subject and it generally occurs by grain boundary (GB) cavitation. Cavities nucleate, grow and then the coalescence (or linkage) of cavities to form bigger cavities or micro-cracks, and further the crack propagation can cause failure with sufficient loss of creep ductility. Creep cavities nucleate (heterogeneously) at geometrical irregularities on grain boundaries (i.e., at specific sites) due to high stress concentration arising from grain boundary sliding (GBS) when the stress concentration is not relaxed by any accommodative processes (such as elasticity, diffusion or creep). Though cavities can nucleate on GBs at 450 to the stress axis, it is observed that the cavity distribution is maximum at 900 to the stress axis, and the driving force comes from the normal traction across the grain boundaries. Growth is also favored for cavities orthogonal (900) to stress axis. Hence cavitation, both nucleation and its growth is favored at specific sites and on those specifically oriented grain boundaries.  

Nucleating a cavity (of volume) involves work done by the stress, creation of a new surface and the grain boundary area consumed by the cavity. The Gibbs free energy change can accordingly be expressed and then one can get for a critical cavity, the expression for critical radius and the critical volume. Hence, the free energy barrier exists for nucleating  a cavity which is stable. Further, the  volume of a critical cavity = Fv (alpha) x rc3, where rc is the critical radius and Fv (alpha) is function of alpha and alpha is the di-hedral angle. Now, cavities can be nucleated at various possible sites (with suitable stress concentration) such as: - (i) triple points (or junctions) of high angle GBs, (ii) sub-boundary-GB intersections, (iii) grain boundary ledges and (iv) second phase particles. Various models for cavity nucleation are available in the literature (Please see Figure 1 in  L.-E. Svensson and G. L. Dunlop, Canadian Metallurgical Quarterly, Vol. 18, pp. 39-47, 1979).

Please refer to:- Rishi Raj, Acta Metallurgica, 1978, pp. 995-1006. Now, for heterogeneous nucleation of a cavity, nucleation at the triple junction formed by a second phase particle will have the smallest critical volume [i.e., Fv (alpha) x rc3 ] and the highest probability for cavity nucleation. Hence the specific cavity shape reduces the nucleation barrier. 

2) Nucleation and Growth of Creep Cavities: when the stress concentration developed at GBs is not relaxed by any accommodative processes, the r-type cavities or w-type cracks form on GBs. Cavitation is not connected with any crystal structure and can occur in high purity as well as single phase materials and in engineering alloys such as austenitic stainless steels, ferritic steels and in nickel base superalloys (J. Weertman, J.R. Weertman, Physical Metallurgy, Eds. R.W. Cahn and P. Haasen, Chapter 20, Elsevier Sci. Publ. company Inc., 1983). The critical assessment of various mechanisms for cavity nucleation by Argon (A.S. Argon, Sripta Metall., 1983, 17, p.5) suggests that the classical cavity nucleation, i.e., clustering of vacancies on Gb subjected to normal stress needs very high stresses (say E/100) and that by thermally activated breaking of bonds also requires very high stress concentration. So, it is well accepted that cavities nucleate due to stress concentrations developed at hard GB particles by the process of GBS. Cavities also can nucleate at (tensile) GB ledges as well as due to the intersection of slip band with a GB (B. F. Dyson, M. S. Loveday and M. J. Rodgers, Proc. R. Soc. 1976, 349A, p. 245). Cavities can also nucleate continuously [Dyson, B.F., 1983. Continuous cavity nucleation and creep fracture. Scripta Metall. 17 (1), 31–37]. The constained cavity growth was proposed by Dyson and later taken forward by other researchers (Dyson, B.F., 1976. Constraints on diffusional cavity growth rates. Metal. Sci. 10 (10), 349–353).

Cavities can grow by pure diffusion, or by power-law creep or by coupled diffusion and power-law creep (i.e., growth is by local diffusion process which in turn is dictated by the creeping matrix). The early work of cavitation by  Greenwood (J. N. Greenwood, D. R. Miller and J. W. Suiter, Acta Metall., 1954, 2, 250) was reformed by others and the Hull-Rimmer model (D. Hull and D.E. Rimmer, Phil Mag. 1959, 4, p. 673) considers the growth of cavities on GBs subjected to a normal stress by pure diffusion process. Void grows by the stress assisted diffusion of vacancies along GB. Surface diffusion has to be invoked to support the observation of spherical cavities. If surface diffusivity is greater than the GB diffusivity, growing voids remains spherical in shape, otherwise the voids become crack like. Pure diffusional growth can be regarded as the limiting case of growth by coupled diffusion and power-law creep, and occurs at low stresses that power-law creep is negligible. 

3. A Few References: 

(i) "NUCLEATION OF CAVITIES AT SECOND PHASE PARTICLES IN GRAIN BOUNDARIES" by RISHI RAJ, Acta Metallurgical, 1978, pp. 995-1006.

(ii) "Creep crack propagation by cavitation near crack tips", R. Raj and S. Baik,  Metal Science, August-September, 1980, p. 385.

(iii) "THE ROLE OF INTERFACIAL DISLOCATIONS IN THE NUCLEATION OF INTERGRANULAR CREEP CAVITIES", L.-E. SVENSSON and G. L. DUNLOP, Canadian Metallurgical Quarterly, Vol. 18, pp. 39-47, 1979.

(iv) :View Point Set No. 5 on Creep Cavitation", Scripta Metall., 1983, 17, pp. 1-43.

(v) M.V. Beere, M.V. Speight, Metal Sci., 1978, 12. p. 172.

(vi) G.H. Edward, M.F. Ashby, Acta Metall., 1979, 27, p. 1505.

(vii) A.C.F. Cocks, M.F. Ashby, Progress in Mater. Sci., 1982, 27, p. 189.

(viii) M.F. Ashby, C, Gandhi, D.M.R. Taplin, Acta Metall., 1979, 27, p. 699.

(ix) "Perspectives in Creep Fracture", 1983  by M. F. Ashby (Editor), L. M. Brown (Series Editor), Pergamon (January 1, 1983).

(x) "Fracture at high temperatures", Hermann Riedel, Springer, 31-Dec-1987 - Technology & Engineering - 418 pages.

(xi) A. J. Perry, Review: Cavitation in creep, J. Mater. Sci., 9, 1016–1039 (1974).

(xii) A Review Paper: L. E. Svensson and G L. Dunlop, Growth of Intergranular Creep Cavities, In--ternational Metals Reviews, No.2, 109–131 (1981).

(xiii) A Review Paper: Creep cavitation in metals, M.E. Kassner, T.A. Hayes, International Journal of Plasticity, Volume 19, Issue 10, October 2003, Pages 1715–1748. 

(xiv) M. F. ASHBY and B. F. DYSON, “Creep Damage Mechanics and Micromechanics, in Advances in Fracture Research”, Vol. 1, 6th International Conference on Fracture (ICF 6), edited by S. R. Valluri, D. M. R. Taplin, P. Rama Rao, J. F. Knott, R. Dubey (Pergamon Press, Oxford, 1984) pp.3–30. 

To add on further.

The complex subject of creep cavitation has evolved over time but at the end the various models try to obtain the experimentally observed stress dependence of time to failure. Monkman-Grant (MG) relationship is generally obeyed when the materials fail by intergranular creep fracture, i.e., fail by creep cavitation. The MG relationship between steady-state creep rate and time to failure says that time to failure (tF) is proportional to (stress) -n, where n is much higher than 1.  

The classic Hull-Rimmer model dealt with isolated cavity and GB diffusion control (spherical cavity) and got in the the stress dependence of tF; n = 1. The later versions were put forward and these approaches dealt the problem by taking in to account other factors for cavity growth. The following may be of help.

1) The various cavity growth models can be classified as (i) constrained growth and (ii) unconstrained growth. 

2) Then comes the growth (i) controlled by GB diffusion (ii) controlled by surface diffusion. That is whether the surface diffusion is faster than GB diffusion to keep cavities spherical in shape.  Or, GB is faster than surface diffusion such that the cavities are lenticular (non-spherical shape).

3) Then comes cavity growth by combined diffusion and power-law: This is nothing but the constrained growth model originally suggested by Dyson. 

Best wishes.