October 2002 Hydrothermal Synthesis of Nanocrystalline Cerium( /V) Oxide Powders 250524q(B) S05449b 40 nm 505242 S054244 (C) (D 40 nm 40 nm Fig. 5. TEM photographs of the hydrothermal powders synthesized in the neutral medium for 72(A)and 120 h(B), and in the acidic medium for 72(C) and 120 h(D). The reaction temperature was fixed at 200'C smaller grains dissolve far more slowly, and in addition, grains of where Au is the driving force of the crystallization, k is the precipitated Ce(OH)a cluster together to form agglomerates, so the Boltzmann constant, and T is the absolute temperature. In the solute diffuses more quickly and the larger grains grow more solution systems Au can be expressed as quickly. Therefore, it is assumed that the growth kinetics are controlled by dissolution △μ= kT In C/c Ce(OH)a is a basic pr [OH I will lead to n apparent decrease of solubility of Ce(OH)a, and increasing where C is the concentration of the supersaturated solution, and Co [HI will lead to a sizable increase of solubility of Ce(OH)a is the concentration of the saturated solution From Eqs. (1)to (3). For Ostwald ripening, there is an equilibrium value for grain the following formula can be obtained size r. When r is smaller than r, smaller grains will dissolve and eventually disappear. When r is larger than r", larger grains will R= B(C-Co)/Co grow. The relationship between grain size and solubility may be expressed by the Gibbs-Thomson equation: formula shows that, with increasing C, R will increase ally; and with decreasing C, R will decrease gradually unti C(r)=C, exp(2yV/vRTr equal to Co and a new dissolution-recrystallization equilib- rium is established. This can be represented as where C(r) is the solubility of a grain with radius r, Ca is the normal equilibrium solubility of the substance (r- oo), y is the 2CeOH=[CeOCe]+H,O (Ks= 16.5) (5) interfacial tension, V is the molar volume of the solute, v is the number of ions in the formula unit, r is the grain radius. R is the In this paper, the concentration of the species CeOH is taken for gas constant, and T is the absolute temperature. the concentration of the solute and is equal to C. In Eq.(4), Cis According to the maximal growth law of crystal growth. the qual to C(r) and Co is equal to C(the solubility while r is equal growth rate of the crystal can be written as to r). So Eq (4)can be represented as R= B[C, exp(2yV/vRTr)/C-11 where R is the growth rate of the crystal. B is a kinetic coefficient. This formula indicates that R is associated with C and o is the relative supersaturation. It is assumed that the maximal Strictly speaking, the formula of the reactant or product in growth law holds true in this paper. o is defined as he hydrothermal medium should be written as Ce, O,(OH)(H,O), (SO4),(CO). However, in this paper, the o=exp(△p/k7 2) reactants or products are respectively represented as CeOctober 2002 Hydrothermal Synthesis of Nanocrystalline Ccrium(lV) Oxide Ponders 2465 Fig. 5. TEM photographs of the hydrothermal powders synthesized in the neutral medium lor 72 (A) and 120 h (B), and in t!ie acidic medium for 72 (C) and 120 h |D). The reaction lemperature was fixed al 200°C. smaller grains dissolve far more slowly, and in addition, grains of precipitated Ce(0H)4 cluster together to form agglomerates, so the solute diffuses more quickly and the larger grains grow more quickly. Therefore, it is assumed that the growth kinetics are controlled by dissolution. CefOHjj is a basic precipitate, so increasing |0H~] will lead to an apparent decrease of solubility of CXOHjj. and increasing [H"*"] will lead to a sizable increase of solubility of CXOHjj. For Ostwald ripening, there is an equilibrium value for grain size r*. When r is smaller than r*. smaller grains will dissolve and eventually disappear. When ;• is larger than /•*. larger grains will grow. The relationship between grain size and solubility may be expressed by the Gibbs-Thomson equation:'*^ C{r) = C. e\p{2yVJvRTr) where C(r) is the solubility of a grain with radius r. C^. is the normal equilibrium solubility of the substance (r —> oc), -y is the interfacial tension. U,,, is the molar volume of tbe solute. \' is the number of ions in the formula unit, r is the grain radius. R Is the gas constant, and T is the absolute temperature. According to the maximal growth law of crystal growth,^" the growth rate of the crystal can be written as R-^u , (1) where R is the growth rate of the crystal. ^ is a kinetic coefficient, and (T is the relative supersaturation. It is assumed that the maximal growth law holds true in this paper, a is defined as where Afi is the driving force of the crystallization, k is the Boltzniann constant, and T is the absolute temperature. In the solution systems Ap. can be expressed as = kT In C/Q, (3) where C is the concentration of the supersaturated solution, and C(, is the concentration of the saturated solution. Prom Eqs. (I) to (3). the following formula can be obtained: (4) This formula shows that, with increasing C, R will increase gradually; and with decreasing C R will decrease gradually until C is equal to C,, and a new dissolution-recrystallization equilib￾rium is established. This can be represented as"' = [CeOCe]"' + 16.5) (5) In this paper, the concentration of the species CeOH"''^ is taken for the concentration of the solute and is equal to C,. In Eq. (4). C is equal to C{r] and C^, is equal to C* (the solubility while r is equal to /•*). So Eq. (4) can be represented as R = exp{2yVJvRTr)/C* - I] - 1 (2) This formula indicates that R is associated with C^, Strictly speaking, the formula of the reactant or product in the hydrothermal medium should be written as Ce,,O,(0H),,(H.0),(SO4),(C0,)- However, in this paper, the reactants or products are respectively represented as Ce''