Nucleation and growth of a core-shell composite nucleus by diffusion
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r Nucleation and growth of a core-shell composite nucleus by diffusion
Masao Iwamatsu ∗ Department of Physics, Faculty of Liberal Arts and Sciences,Tokyo City University, Setagaya-ku, Tokyo 158-8557, Japan (Dated: September 7, 2018)The critical radius of a core-shell-type nucleus grown by diffusion in a phase-separated solution isstudied. A kinetic critical radius rather than the thermodynamic critical radius of standard classicalnucleation theory can be defined from the diffusional growth equations. It is shown that there existtwo kinetic critical radii for the core-shell-type nucleus, for which both the inner core radius andthe outer shell radius will be stationary. Therefore, these two critical radii correspond to a singlecritical point of the nucleation path with a single energy barrier even though the nucleation lookslike a two-step process. The two radii are given by formulas similar to that of classical nucleationtheory if the Ostwald-Freundlich boundary condition is imposed at the surface of the inner nucleusand that of the outer shell. The subsequent growth of a core-shell-type post-critical nucleus followsthe classical picture of Ostwald’s step rule. Our result is consistent with some of the experimentaland numerical results which suggest the core-shell-type critical nucleus.
PACS numbers: 68.55.A-
I. INTRODUCTION
Nucleation and growth are basic phenomena that playvital roles in the processing of various materials acrossmany industries and in various natural phenomena [1].In particular, the growth of fine particles in solution suchas semiconductor quantum dots [2], bio-minerals [3] andother molecular crystals [4, 5], has attracted considerableinterest recently, since such materials have many poten-tial applications spanning the electronics to the biomedi-cal industries. In nucleation and growth, material trans-port by diffusion and its effect on the size (radius) of thecritical nucleus often plays a fundamental roles in con-trolling the size of the products [6–11].In particular, growth by diffusion has been studied asit applies to various problems such as precipitation fromsolution [6], liquid droplet nucleation from a supersat-urated vapor [7], vapor bubble nucleation from a super-saturated solution [8], and colloidal particle formation [9]from solution. However, the nucleation and growth pro-cesses were studied separately in many cases [12, 13].Furthermore, nucleation by diffusion has not attractedmuch attention except for a recent attempt to bridge dif-fusion and thermodynamic evolution [14–16]. Indeed, itis well recognized [10] that the evolution equation knownas the Zeldovich relation [17, 18] from classical nucle-ation theory (CNT) and the diffusional growth equationare formally the same for the post-critical nucleus.In our previous paper [16], we pointed out that thetwo evolution equations based on the thermodynamicCNT and the kinetic diffusional growth equation leadto two different definition of critical radii (i.e., thermo-dynamic and kinetic ). However, except for the growthof bubbles in solution, the nucleation and growth of con-densed matter proceeds via complex processes. For ex- ∗ [email protected] ample, various biomaterials such as protein crystals arenot formed directly from the bulk mother solution, butrather, from within a phase-separated solution of an in-termediate phase [5, 19–21] to form a core-shell com-posite nucleus [22, 23]. In fact, such a core-shell typenucleus has bee observed by experiments [24] and bycomputer simulations [19, 25–28]. Various model calcu-lations [22, 29–32] also suggest the existence of core-shellstructure. Crystal C Solution L Solution L Crystal CSolution L Solution L Free energy G FIG. 1. A model of a core-shell-type composite nucleusin a three-phase system. The phase transition occurs frommetastable solution L to stable crystal C via the interme-diate metastable solution L (L → L → C) according to theOstwald’s step rule (left). The spherical crystal nucleus C isgrown from the solution L , which is phase-separated from theoriginal mother solution L to form a spherical shell aroundthe crystal C. Therefore, the critical nucleus is the compositenucleus of core-shell structure (right). In this communication, we consider the problem ofnucleation and growth of the core-shell-type nucleus insolution by diffusion. Therefore, we consider the two-step nucleation in a three-phase system [20] according tothe Ostwald’s step rule [31, 33] as shown in Fig. 1. Ini-tially, the pre-critical nucleus (embryo) of intermediatemetastable phase (metastable dense solution L ) nucle-ates within the metastable mother solution L . Next,the stable crystal C nucleates from the intermediatemetastable solution L . However, there will be a singlecomposite core-shell critical nucleus with a single energybarrier [19, 22, 28, 30, 31]. In such a case, the core-shellcritical nucleus forms through the diffusion of materials,which must occur in two steps; bulk diffusion in the origi-nal mother solution L and the diffusion in the surround-ing shell of the intermediate metastable solution L .This work is complementary to our previous work [23],in which the same problem was considered from thestandpoint of dynamics governed by the Fokker-Planckequation within the framework of CNT. Here, we con-sider the same problem by diffusion in real space. Insection II, we show that the evolution equations for thenucleus and shell will afford two distinct critical radiusfor the core nucleus and the shell. Furthermore, both thenuclear radius and the shell radius become stationary si-multaneously at a single critical point, as if there were asingle activation process. This is consistent with previousthermodynamic arguments [19, 22], in which only a sin-gle saddle point that corresponded to the core-shell-typenucleus was identified. In section III, we further showthat by imposing the Ostwald-Freundlich boundary con-dition [15, 16] for this core-shell system, these two criticalradii are given by formulas similar to those of CNT. Fi-nally, in section IV, we discuss the results of our modeland conclude by pointing out that the scenario given bythis model would explain, qualitatively, some experimen-tal and numerical observations of the core-shell-type nu-cleus. II. NUCLEATION AND GROWTH OF ACORE-SHELL NUCLEUS FROM A TWO-PHASESOLUTION BY DIFFUSION
We consider a spherical core-shell-type composite nu-cleus composed of a stable crystal phase (C) surroundedby an intermediate metastable liquid solution (L ) nu-cleated in the original metastable mother liquid solution(L ), shown in Fig. 1. Of course, the formation of such acore-shell-type embryo (pre-critical nuclear) needs addi-tional thermodynamic discussion [19, 22, 30, 31]. Here,we pay most attentions to the critical and post criti-cal nucleus after forming such a core-shell-type embryo.We consider a two-step nucleation and growth processL → L → C instead of one-step direct nucleation L → C.The crystal nucleus C with radius R C is surrounded byan intermediate metastable solution L , which is sepa-rated at a radius R L by a boundary from the originalmetastable solution L (Fig. 1). The two liquid phasesL and L are assumed to be phase-separated.When the crystal nucleus C is grown by diffusion asshown in Fig. 2, the diffusional growth equation can bewritten as dR C dt = − v m j R C , (1) where v m is the molecular volume in the nucleus C, and j R C = − D (cid:18) ∂c ∂r (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) r = R C (2)is the diffusion flux ( j R C ) at the surface of the growingcrystal nucleus, where c is the concentration of solute(monomer), D is the solute diffusivity in the solutionL , and r is the radial coordinate from the center of thenucleus. Note that the incoming diffusional flux must benegative ( j R C <
0) in order to feed the nucleus to grow.In Eq. (1), we have neglected the transport term dueto the hydrodynamic flow [8, 13] because we are mainlyinterested in the problem near the critical point wherethe growth of nucleus becomes stationary ( dR C /dt = 0).Now, the radius R C of the crystal nucleus grows ac-cording to the incoming diffusion flux j R C . The concen-tration field c i ( r, t ), where i = 1 , and L , obeys the three-dimensional (3D)diffusion equation [6, 10, 34] ∂c i ( r, t ) ∂t = D i r ∂ ∂r ( rc i ( r, t )) (3)for spherical symmetry, where t denotes the time and D i denotes the solute diffusivity of the two surroundingphases, L and L . The steady state ( ∂c /∂t = 0) solu-tion for the mother solution L is given by c ( r ) = c , ∞ − ( c , ∞ − c ( R L )) R L r , R L < r, (4)where c ( R L ) is the concentration at the boundary of thetwo solutions L and L at R L and c , ∞ = c ( r → ∞ ) isthe bulk (oversaturated) concentration of the metastablemother phase L . Similarly, the concentration profile ofthe monomers in the surrounding solution L is given by c ( r ) = R L R C ( c ( R C ) − c ( R L )) R L − R C r + R L c ( R L ) − R C c ( R C ) R L − R C ,R C < r < R L , (5)where c ( R C ) is the concentration at the surface of thegrowing core nucleus at R C and c ( R L ) is the concentra-tion at the (inner) boundary of the two solutions L andL at R L . Note that these steady state concentrations donot depend on the diffusion coefficient D and D .Then, the diffusion flux in the surrounding solution L is given by j = − D (cid:18) ∂c ∂r (cid:19) = D R L R C ( c ( R C ) − c ( R L )) R L − R C r , R C ≤ r ≤ R L . (6)Similarly, the diffusion flux in the surrounding solutionL , outside the surrounding shell of the solution L , is Diffusion flux R LSolution L1 Solution L2 R C Crystal C ( R L ) c ( R C ) c FIG. 2. A diffusion flux into the core-shell composite nucleus.A diffusion flux in the solution L is absorbed into the shellof the solution L and continues to diffuse in the solution L into the stable crystal C. Two fluxes in solution L and L must connect continuously at the boundary of radius R L . given by j = − D (cid:18) ∂c ∂r (cid:19) = − D ( c , ∞ − c ( R L )) R L r , R L ≤ r. (7)Since the two fluxes in Eqs. (6) and (7) must be contin-uous at r = R L , two alternative forms for the flux j R L at R L are obtained: j R L = D R C ( c ( R C ) − c ( R L )) R L ( R L − R C )= − D c , ∞ − c ( R L ) R L . (8)Therefore, the fluxes j R C , at the inner radius r = R C isalso given by two alternative forms, using either c ( r ) or c ( r ): j R C = D R L ( c ( R C ) − c ( R L )) R C ( R L − R C )= − D (cid:18) R L R C (cid:19) c , ∞ − c ( R L ) R C . (9)These two fluxes naturally satisfy the continuity of totalflux: 4 πR j R C = 4 πR j R L . (10)Since these fluxes j R C and j R L must be negative for thepost critical nucleus, two inequalities c , ∞ > c ( R L ) and c ( R L ) > c ( R C ) hold from Eqs. (8) and (9).Since the diffusion flux j R C at the crystal nuclear sur-face is incorporated into the growing nucleus, the radius R C of the nucleus grows according to Eq. (1). On theother hand the diffusion flux j R L at the surface of theL -L boundary at R L will be incorporated into both the surrounding solution L and the nucleus C, so we mustconsider the conservation of material given by dn dt + dn c dt = − πR j R L , (11)where n c = 4 π v m R (12)is the number of monomers in the nucleus C, and n = Z R L R C πr c ( r ) dr (13)is the number of monomers in the surrounding solutionL . Then, Eq. (11), combined with Eqs. (12) and (13)leads to the evolution equation for R L dR L dt = − c ( R L ) j R L + R (1 − c ( R C ) v m ) R c ( R L ) j R C , (14)where we have replaced dR C /dt by Eq. (1). The firstterm on the right-hand side denotes the increase in thenumber of monomers in solution L due to the incomingflux j R L , and the second term denotes the decrease in thenumber of monomers in L which are incorporated intonucleus C. Note that c ( R C ) v m ≪
1, as the molecularvolume 1 /c ( R C ) in solution L at the surface of thenucleus at R C is much larger than the molecular volume v m in the nucleus C.Using Eq. (10), Eqs. (1) and (14) can be written as dR C dt = − v m j R C = − v m (cid:18) R R (cid:19) j R L , (15) dR L dt = − v m (cid:18) c ( R C ) c ( R L ) (cid:19) j R L = − v m (cid:18) R R (cid:19) (cid:18) c ( R C ) c ( R L ) (cid:19) j R C . (16)Therefore, the stationary conditions dR C /dt = 0 and dR L /dt = 0 for the critical radius R ∗ C and R ∗ L will be sat-isfied simultaneously when j R C = j R L = 0, which leadsto c ( R ∗ L ) = c , ∞ , (17a) c ( R ∗ L ) = c ( R ∗ C ) = c , ∞ , (17b)where c , ∞ is a fictitious concentration when the crystalnucleus C is surrounded only by the solution L and thesolute concentration c ( r ) is given by a formula similarto Eq. (4). These simultaneous equations define the ki-netic critical radius R ∗ C and R ∗ L . Therefore, there is a sin-gle core-shell-type critical nucleus which corresponds toa single saddle point [19, 22]. The critical nucleus havingthe radii R ∗ C and R ∗ L is surrounded by a wetting layer ofuniform solution L with the concentration c ( r ) = c , ∞ from Eq. (5), which is surrounded further by a uniformsolutions L with the concentrations c ( r ) = c , ∞ of themetastable mother solution from Eq. (4).We will not consider another solution R ∗ L → ∞ derivedfrom Eq. (17a) because this solution corresponds to thedirect nucleation from the metastable intermediate so-lution L rather than the indirect nucleation from themother solution L . Note that the stationary condition dR C /dt = 0 or j R C = 0 inevitably leads to the stationar-ity of another radius dR L /dt = 0 or j R L = 0 because ofthe conservation of material in Eq. (10). When the innerradius R C becomes stationary, the outer radius R L mustalso becomes stationary. Therefore, the critical nucleusis always a composite nucleus of core-shell structure.
1 2 3 4 5 6 7 8 9 100 c ( r ) r /R C * R C R C * R L R L * c c c
1, sat c
2, sat c ( r ) c ( r ) c ( r ) c ( r ) c ( ) R C c ( ) R L ; R C FIG. 3. Concentrations c ( r ) and c ( r ) of monomersaround the growing nucleus as a function of distance r fromthe center of the nucleus. The critical nucleus having theradii R ∗ C and R ∗ L is surrounded by a wetting layer of uni-form concentration c ( r ) = c ( R ∗ C ) = c ( R ∗ L ) = c , ∞ , whichis surrounded further by a uniform solutions with the con-centrations c ( r ) = c , ∞ of the metastable mother solutionto form a core-shell critical nucleus. As the post-critical nu-cleus begins to grow ( R ∗ C → R C , R ∗ L → R L ), the concentra-tions c , eq ( R C ) and c , eq ( R L ; R C ) at the surface of both thecore nucleus at R C and outer boundary at R L given by theOstwald-Freundlich boundary condition in Eqs. (30) and (32)start to decrease. Then, the concentrations c ( r ) of the so-lution L and c ( r ) of the solution L are no longer uniformand concentration gradients appears around the nucleus, ac-cording to Eqs. (6) and (7), and diffusional flux ensures thatthe growing nucleus is fed. In Fig. 3, we show a schematic concentration profile c ( r ) = c , ∞ = constant and c ( r ) = c , ∞ = constantfor the core-shell critical nucleus with critical radii R ∗ C and R ∗ L . We also show a schematic concentration profile c ( r ) and c ( r ), which are not uniform, for a post-criticalnucleus with R C > R ∗ C and R L > R ∗ L . The concentrationprofile of the post-critical nucleus will be discussed in thenext section using the thermodynamic argument. III. OSTWALD-FREUNDLICH BOUNDARYCONDITION IN A TWO-PHASE SOLUTION
In order to find the boundary values c ( R C ) and c ( R L ), we will consider the chemical equilibrium of thiscomposite core-shell nucleus. Within the capillarity ap-proximation, the Gibbs free energy of the composite nu-cleus in Fig. 2 is given by G ( n c , n ) = − n C ∆ µ − n ∆ µ + 4 πR γ c, + 4 πR γ , , (18)where γ c, and γ , are the surface tensions of the C-L interface at R C and of L -L at R L , respectively. Thenumber of monomers n C in the core and n in the shellare given by Eqs. (12) and (13), respectively. The chem-ical potential differences ∆ µ and ∆ µ of the two solu-tions L and L relative to that of the stable core nucleusC are related to the bulk solute (oversaturated) concen-trations c , ∞ and c , ∞ of the two solutions L and L through ∆ µ = β − ln ( c , ∞ /c , sat ) , (19)∆ µ = β − ln ( c , ∞ /c , sat ) , (20)with β − = k B T and c i, sat ( i = 1 ,
2) being the satura-tion concentration of the two solutions which are equi-librium at the C-L and L -L flat interfaces. Note that c , ∞ = c ( r → ∞ ) (Eq. (4)) is the concentrations of theoversaturated solution, which is larger than the satura-tion concentration c ( R L → ∞ ) = c , sat . However, c , ∞ does not corresponds to c ( r → ∞ ) (Eq. (5)), but it islarger than c ( R C → ∞ ) = c , sat .From Eq. (18) together with Eqs. (12) and (13), thechemical potential µ C of the core nucleus C with radius R C is given by µ C ( R C ) = ∂ ∆ G∂n C = − v m ∆ µ ( c ( R ∗ C ) − c ( R C )) − v m γ c , (cid:18) R ∗ C − R C (cid:19) , (21)where the critical radius R ∗ C is given by an implicit equa-tion ∆ µ − c ( R ∗ C ) v m ∆ µ = 2 γ c , v m R ∗ C (22)and is written formally as the classical CNT critical ra-dius for the core nucleus R ∗ C = 2 γ c , v m ∆ µ , eff , (23)with ∆ µ , eff = ∆ µ − c , ∞ v m ∆ µ , (24)where we have used c , ∞ = c ( R ∗ C ), is the effective su-persaturation of the solution phase L for the criticalnucleus. Note that c , ∞ v m ≪ µ , eff ≃ ∆ µ .Then, Eq. (23) becomes exactly the CNT critical radiuswhen the crystal C is directly nucleated from the inter-mediate metastable solution L .Similarly, the chemical potential µ of the sphericalshell of the surrounding solution L with the outer radius R L and the inner radius R C is given by µ ( R L ; R C ) = ∂ ∆ G ( n C , n ) ∂n = ∆ µ c ( R L ) ( c ( R ∗ L ) − c ( R L )) − γ , c ( R L ) (cid:18) R ∗ L − R L (cid:19) + ∆ µ c ( R C ) ( c ( R ∗ C ) − c ( R C ))+ 2 γ c , c ( R C ) (cid:18) R ∗ C − R C (cid:19) , (25)where, again, the critical radius R ∗ L is given by R ∗ L = 2 γ , c , ∞ ∆ µ , (26)where we have used c ( R ∗ L ) = c , ∞ from Eq. (17b). Thecritical radius in Eq. (26) is given exactly by the CNT for-mula when a solution L is nucleated from the mother so-lution L . Note that the chemical potential µ in Eq. (25)depends indirectly on the core radius R C . Then Eq. (21)is written as µ C ( R C ) = − v m γ , R ∗ L c , ∞ ( c , ∞ − c ( R C )) − v m γ c , (cid:18) R ∗ C − R C (cid:19) , (27)and Eq. (25) is written as µ ( R L ; R C ) = 2 γ , R ∗ L (cid:18) c ( R L ) − c , ∞ (cid:19) − γ , c ( R L ) (cid:18) R ∗ L − R L (cid:19) + 2 γ , R ∗ L (cid:18) c ( R C ) − c , ∞ (cid:19) + 2 γ c , c ( R C ) (cid:18) R ∗ C − R C (cid:19) , (28)where we have used Eq. (17b). Equations (27) and (28)are the thermodynamic driving force of the growing nu-cleus.Therefore, if the surrounding solution L is alwaysin chemical equilibrium with the core nucleus at theC-L interface at R C , the concentration at the inter-face c ( R C ) is given by the equilibrium concentration c , eq ( R C ), which is related to the chemical potential µ C of the core nucleus C given by Eq. (27) through µ C ( R C ) = β − ln ( c , eq ( R C ) /c , ∞ ) , (29) which leads to the Ostwald-Freundlich (OF) or Gibbs-Thomson equation given by c , eq ( R C ) = c , ∞ exp ( βµ C ( R C )) . (30)Then, c , eq ( R ∗ C ) = c , ∞ at the critical point of nucleationfrom Eq. (17b) because µ C ( R ∗ C ) = 0. The concentration c , eq ( R C ) is expected to decreases monotonically as afunction of R C from Eq. (27).Similarly, if the solution L is in chemical equilibriumwith a droplet shell of the solution L with the outer ra-dius R L and inner radius R C , the concentration at the in-terface c ( R L ) is given by the equilibrium concentration c , eq ( R L ; R C ), which is related to the chemical potential µ ( R L ; R C ) of the shell given by Eq. (28) through µ ( R L ; R C ) = β − ln ( c , eq ( R L ; R C ) /c , ∞ ) , (31)which leads to the OF equation given by c , eq ( R L ; R C ) = c , ∞ exp ( βµ ( R L ; R C )) . (32)Note that the concentration c ( R L ) = c , eq ( R L ; R C ) atthe boundary R L depends on the radius R C . The equi-librium concentration c , eq ( R L ; R C ) is also expected todecrease monotonically as a function of R L for a fixed R C from Eq. (28). At the critical point with R ∗ L and R ∗ C , c , eq ( R ∗ L ; R ∗ C ) = c , ∞ because µ ( R ∗ L ; R ∗ C ) = 0 fromEq. (28).When the core nucleus C and the surrounding solu-tion L are always in chemical equilibrium, c ( R C ) = c , eq ( R C ). Similarly, when the surrounding sphericalshell of solution L and the mother solution L are alwaysin chemical equilibrium, c ( R L ) = c , eq ( R L ; R C ). Theconcentration c ( R L ) = c , eq ( R L ) is determined fromthe continuity condition of Eqs. (8) and (9). Two equi-librium concentrations c , eq ( R C ) and c , eq ( R L ; R C ) aredetermined from the implicit equations Eqs. (27), (30),(28), and (32) by replacing c ( R C ) by c , eq ( R C ) and c ( R L ) by c , eq ( R L ; R C )In Fig. 3, we show a schematic concentration pro-file c ( r ) and c ( r ) given by Eqs. (5) and (4) as wellas c , eq ( R C ) and c , eq ( R L ; R C ) (for a fixed R C ) at theboundaries for the post-critical core-shell critical nucleuswith R C > R ∗ C and R L > R ∗ L .For the post-critical nucleus, the concentrations pro-files c ( r ) and c ( r ) given by Eqs. (5) and (4) arenot uniform anymore and the depletion zone increases.Since the two interfaces at R C and R L become flat as R C → ∞ and R L → ∞ , two concentrations c , eq ( R C )and c , eq ( R L ; R C ) at the boundaries must approach theirsaturation concentration at the flat interface c , sat and c , sat . Therefore, we have c , eq ( R C ) → c , sat (33) c , eq ( R L ; R C ) → c , sat (34)as shown in Fig. 3When the radii R C and R L are close to the critical radii R ∗ C and R ∗ L , we can replace c ( R L ) by c , eq ( R L ; R C ) inEqs. (8) and (9), and expand exponential in Eq. (32).Then the driving forces j R C and j R L of growing nucleusradii are given by j R C = D (cid:18) R L R C (cid:19) βc , ∞ µ ( R L ; R C ) R C (35) j R L = D βc , ∞ µ ( R L ; R C ) R L (36)Not only the growth velocity dR C /dt but also dR L /dt vanish at the critical point when R C = R ∗ C and R L = R ∗ L because µ ( R ∗ C , R ∗ L ) = 0 from Eq. (28).Equations (15) and (16) combined with Eqs. (35)and (36) are similar in form to the Zeldovich rela-tion [16, 18] derived from the thermodynamic evolutionequation [10, 16, 18]. Therefore, the Zeldovich relationcan be equivalent to the diffusional growth equation inour core-shell nucleus as well. However, this is valid onlynear the thermodynamic critical point with critical radii R ∗ C and R ∗ L , as it is recognized that the expansion of theexponential in Eq. (32) is valid only near the critical ra-dius, and the full expression in Eqs. (30) and (32) willbe necessary to describe the subsequent growth of thenucleus [35].Now, we can discuss the evolution of this compos-ite core-shell nucleus from Eqs. (35) and (36). Whenthe concentration is uniform and oversaturated ( c ( r ) = c , ∞ > c , sat and c ( r ) = c , ∞ > c , sat ), the core ra-dius R C and the outer shell radius R L reach the criticalradius R ∗ C and R ∗ L simultaneously. Therefore, there isonly a single critical point which corresponds to a sin-gle critical nucleus characterized by two critical radii R ∗ C and R ∗ L . After passing through this single critical pointcharacterized by a single energy barrier , two radii R C and R L of the core-shell nucleus start to increase. Thegrowth of the nucleus follows exactly the same scenariogiven by CNT, by regarding the R C and R L as the radiusof nucleus.In the limit of R C → ∞ and R L → ∞ , we have j R C → − D c , ∞ − c , sat R C , (37) j R L → − D c , ∞ − c , sat R L , (38)from Eqs. (8), (9) and (34). Then the evolution equationsEqs. (15) and (16) are written as R C dR C dt = v m D ( c , ∞ − c , sat ) , (39) R L dR L dt = v m D ( c , ∞ − c , sat ) . (40)Therefore, two radii grow according to the classicallaw [6, 34] R C ∝ √ t and R L ∝ √ t . Furthermore R − R → constant. Then, the radius R C eventuallycatches up R L as R C → ∞ and the surrounding shellof solution L disappears as the Ostwald’s step rule pre-dicted [31, 33]. IV. DISCUSSION AND CONCLUSION
In this paper, we considered the evolution of a core-shell-type nucleus. The diffusional growth equations bothfor the core nucleus and the surrounding shell lead totwo evolution equations similar to the Zeldovich equa-tion of classical nucleation theory (CNT). Therefore, thekinetic critical radii for both the core radius and the sur-rounding shell radius can be defined, and there must bea single core-shell critical nucleus which corresponds toa single critical point and a single energy barrier. Whendiffusion is so fast that the concentrations at the twosurfaces are maintained at the value given by the equi-librium Ostwald-Freundlich boundary condition, simpleexpressions similar to those for the thermodynamic crit-ical radius of CNT can be derived. Our formulation fur-ther predicts a growing post critical nucleus with a corenucleus surrounded by a metastable intermediate solu-tion, whose thickness will be decreased during the courseof evolution in accordance with the Ostwald’s step rule.Although our discussion takes into account the two-step diffusion of a composite nucleus, a more thoroughconsideration not only of the diffusion processes but alsoof the reaction and the reorganization processes withinthe bulk crystal nucleus is required to understand thegrowth of near-spherical solid nucleus, since simple dif-fusional attachment is known to lead to non-sphericalfractal structures [36, 37]. Numerical simulations such asthe kinetic Monte Carlo method [2, 38] hold the greatestpotential for the study of such reorganization processes.Finally, since we considered only the material diffusionand neglected the heat flow, the instability and fractalgrowth [39] of the nucleus caused by thermal diffusioncannot be discussed within our present formalism.
ACKNOWLEDGMENTS
This work was partially supported under a project forstrategic advancement of research infrastructure for pri-vate universities, 2015-2020, operated by MEXT, Japan.
Appendix
Here, we have provided the results for the two-dimensional (2D) circular core-shell nucleus. Main dif-ferences from the three-dimensional (3D) spherical nu-cleus of the main text are the mathematical forms ofthe solution of diffusion equation and the thermodynamicOstwald-Freundlich condition.The solution of the 2D diffusion equation Eq. (3) arewritten as [40] c ( r ) = c ( R L ) − c ( R C )ln R L − ln R C ln r + c ( R C ) ln R L − c ( R L ) ln R C ln R L − ln R C R C < r < R L (A.1) c ( r ) = c , ∞ − c ( R L )ln R ∞ − ln R L ln r + c ( R L ) ln R ∞ − c , ∞ ln R L ln R ∞ − ln R L R L < r < R ∞ (A.2)where R ∞ is the large radius from the nucleus and c , ∞ is the concentration at this distance.The free energy of the core-shell nucleus in the 2D casethat corresponds to Eq. (18) is written as G ( n c , n ) = − n C ∆ µ − n ∆ µ + 2 πR C γ c, + 2 πR L γ , , (A.3)with n C = πv m R (A.4) and n = Z R L R C πrc ( r ) dr (A.5)Following the same procedure as that of Section III, thecritical radius R ∗ C which corresponds to Eq. (23) is givenby R ∗ C = γ c , v m ∆ µ , eff , (A.6)where ∆ µ , eff is given by Eq. (24). The critical radius R ∗ L of the intermediate metastable solution is give by aformula R ∗ L = γ , c , ∞ ∆ µ . (A.7)similar to Eq. (26). Therefore, Eqs. (A.6) and (A.7) canbe derived from Eqs. (23) and (26) by replacing 2 γ c, by γ c, and 2 γ , by γ , . Using the same replacement inEqs. (27) and (28), we can derive the formula for thechemical potentials µ C ( R C ) and µ L ( R C , R L ) in two di-mensional case. The evolution of the nucleus is describedby Eqs. (15) and (16) with Eqs. (35) and (36). There-fore, the scenario of nucleation and growth of the two-dimensional core-shell nucleus is the same as that of thethree-dimensional nucleus. [1] K. F. Kelton and A. L. Greer, Nucleation in CondensedMatter, Applications in Materials and Biology, Perga-mon, Oxford, 2010, Chapter 6.[2] V. Gorshkov and V. Privman, Physica E , 1 (2010).[3] F. C. Meldrum and H. C¨ofen, Chem. Rev. , 4332(2008).[4] R. P. Sear, Int. Mater. Rev. , 328 (2012).[5] P. G. Vekilov, J. Phys.: Condens Matter , 193101(2012).[6] C. Zener, J. Appl. Phys. , 950 (1949).[7] F. C. Frank, Proc. R. Soc. London Ser. A , 586(1950).[8] P. S. Epstein and M. S. Plesset, J. Chem. Phys. , 1505(1950).[9] H. Reiss, J. Chem. Phys. , 482 (1951).[10] V. V. Slezov, Kinetics of First-Order Phase Transition,Wiley-VCH, Weinheim, 2009, Chapter 5.[11] T. Wen, L. N. Brush, and K. M. Krishnan, J. Colloid.Interface Sci. , 79 (2014).[12] D. T. Robb and V. Privman, Langmuir , 26 (2008).[13] A. P. Grinin, G. Yu. Gor, and F. M. Kuni, Atoms. Res. , 503 (2011).[14] J. F. Lutsko, J. Chem. Phys. , 161101 (2011).[15] B. Peters, J. Chem. Phys. , 044107 (2011).[16] M. Iwamatsu, J. Chem. Phys. , 064702 (2014).[17] Ya. B. Zeldovich, Acta Physicochim URSS , 1 (1943).[18] D. S. van Putten and V. I. Kalikmanov, J. Chem. Phys. , 164508 (2009).[19] P. R. ten Wolde and D. Frenkel, Science , 1975 (1997).[20] M. C. R. Heijna, W. J. P. van Enckevort, and E. Vlieg,Phys. Rev. E , 011604 (2007).[21] M. Sleutela, and A. E. S. Van Driesscheb, Proc. Natl.Acad. Sci. , E546 (2014).[22] M. Iwamatsu, J. Chem. Phys. , 164508 (2011).[23] M. Iwamatsu, Phys. Rev. E , 041604 (2012).[24] W. L. Wang, Y. H. Wu, L. H. Li, D. L. Geng, and B.Wei, Phys. Rev. E , 032603 (2016).[25] C. Desgranges and J. Delhommelle, J. Am. Chem. Soc. , 7012 (2007).[26] J. A. van Meel, A. J. Page, R. P. Sear, and D. Frenkel,J. Chem. Phys. , 204505 (2008).[27] M. Iwamatsu, J. Alloy. Comp. , S538 (2010).[28] W. Qi, Y. Peng, Y. Han, R. K. Bowles, and M. Dijkstra,Phys. Rev. Lett. , 185701 (2015).[29] K. F. Kelton, Acta Mater. , 1967 (2000).[30] L. Granasy and D. W. Oxtoby, J. Chem. Phys. , 2410(2000).[31] M. Santra, R. S. Singh, and B. Bagchi, J. Phys. Chem.B , 13154 (2013).[32] J. F. Lutsko, J. Phys.: Condens. Matter , 244020(2016).[33] W. Ostwald, Z. Phys. Chem. , 289 (1897).[34] A. P. Grinin, A. K. Shchekin, F. M. Kuni, E. A. Grinina,and H. Reiss, J. Chem. Phys. , 387 (2004).[35] N. V. Mantzaris, Chem. Eng. Sci. , 4749 (2005).[36] P. Meakin, Phys. Rep. , 189 (1993).[37] F. Liu and N. Goldenfeld, Phys. Rev. A , 895 (1990). [38] V. Gorshkov, A. Zavalov, and V. Privman, Langmuir ,7940 (2009).[39] W. W. Mullins and R. F. Sekerka, J. Appl. Phys. , 323 (1963).[40] B. K. Chakraverty, J. Phys. Chem. Sol.28