TWI466714B - Method for making colloidal metal oxide particles - Google Patents

Description

Method for producing colloidal metal oxide particles

The present invention is directed to a method of making colloidal metal oxide particles.

Efforts continue to be made in the art to form colloidal metal oxide particles in a highly efficient manner.

There is a need in the art for a method of improving the formation of colloidal metal oxide particles in a highly efficient manner when utilizing an optimized device.

The present invention provides a novel method of forming colloidal metal oxide particles. It is disclosed that a method of forming colloidal metal oxide particles can form colloidal metal oxide particles under conditions close to the optimum method to form colloidal metal oxide particles in a very efficient manner. Furthermore, the disclosed method of forming colloidal metal oxide particles enables the reaction tank to be optimally utilized because it can reduce the reaction cycle required to form colloidal metal oxide particles.

A method of forming colloidal metal oxide particles comprising the step of adding one or more reactants to a reaction tank, wherein the step of adding one or more reactants takes into account different in situ reaction conditions, including but not limited (i) particle nucleation rate in the reaction tank, (ii) metal oxide deposition rate over the metal oxide particles present (eg, seed metal oxide particles and/or nucleating metal oxide particles, and/or Or (iii) at least one of the growth of metal oxide particles (for example, seed metal oxide particles and/or nucleating metal oxide particles) in the reaction tank.

In an exemplary embodiment, the method of making colloidal metal oxide particles includes the step of adding a reactive metal oxide to a reaction vessel at a metal oxide mass addition rate, the metal oxide mass addition rate being based on a mathematical model The mathematical model considers at least one of (i) particle nucleation rate, (ii) metal oxide particle deposition rate over the metal oxide present, and/or (iii) growth of metal oxide particles in the reaction cell, The metal oxide mass addition rate is increased as a function of reaction time. In another particular embodiment, during at least a portion of the reaction period, the addition rate of 1000 per hour per square meter of reactive metal oxide (m ²⁾ of total particle surface area of greater than 10.0 grams (g / 1000m ² - Hr). In even another exemplary embodiment, a method of making a colloidal metal oxide particle according to the present invention, comprising the step of adding a reactive metal oxide to a reaction vessel at a metal oxide mass addition rate according to a mathematical mode This mathematical mode provides the optimum metal oxide mass addition rate, q is represented by:

q=(3m ₀ G _r /D _PO ³ )(D _PO +G _r t) ²

among them:

(a) m ₀ represents the mass of the metal oxide particles in the reaction tank, and the measurement is in grams (g);

(b) G _r represents the growth rate of the metal oxide particles of the metal oxide particles in the reaction tank, which is determined by increasing the particle diameter and measured in nanometers per hour (nm/hr);

(c) D _PO represents the average metal oxide particle size measured in nanometers (nm);

(d) t represents the time in hours (hr).

It is disclosed that the method of producing colloidal metal oxide particles may include the steps of forming nucleating metal oxide particles and/or growing metal oxide seed particles. In an exemplary embodiment, the method of making colloidal metal oxide particles includes the step of adding one or more reactants to (i) a reaction vessel containing water and (ii) substantially free of any seed metal oxide particles. Wherein the one or more reactants can form nucleating metal oxide particles; forming nucleating metal oxide particles in the reaction tank; and growing the nucleating metal oxide particles in the reaction tank to form a colloidal metal oxide The particles, wherein the growing step comprises increasing the feed rate of the one or more reactants during the reaction cycle.

It is disclosed that a method of producing colloidal metal oxide particles may produce colloidal metal oxide particles in order to form colloidal metal oxide particles in an energy efficient manner having a reaction period much lower than a conventional reaction period. In an exemplary embodiment, a method of making colloidal metal oxide particles comprising the step of adding a reactive metal oxide to a reaction vessel at a metal oxide mass addition rate during a reaction period in excess of Forming colloidal metal oxide particles having an average final particle size ranging from about 10 nm to about 200 nm, wherein the reaction period is less than 50% using conventional techniques (eg, a constant reactive metal oxide feed rate) The similar reaction cycle is the same. For example, when conventional methods of forming colloidal metal oxide particles of similar size require a reaction cycle of at least 30 minutes, typically about 31 to 40 minutes, the process can be formed in a reaction cycle of about 21-28 minutes using the process. Colloidal metal oxide particles having an average particle size of about 20-30 nm.

In another exemplary embodiment, a method of producing colloidal metal oxide particles, the method comprising the step of adding a reactive metal oxide to a reaction vessel at a metal oxide mass addition rate during a reaction period, To form colloidal metal oxide particles having an average final particle size ranging from about 20 nm to about 200 nm, the metal oxide mass addition rate is increased at least once during the reaction cycle. The increase in the rate of mass addition of the metal oxide, for example, can be a single step increase or a multi-step increase.

The invention further refers to a method of using colloidal metal oxide particles. In an exemplary method of using colloidal metal oxide particles, the method comprises applying a colloidal metal oxide particle composition on a substrate and drying the colloidal metal oxide particle composition to form a coating on the substrate. Floor.

These and other features and advantages of the present invention will become apparent from the Detailed Description of the appended claims.

The present invention has been described with respect to the specific embodiments and the specific embodiments are described in the specific language. However, it should be understood that the scope of the invention is not intended to be limited by the specific language. Variations, further modifications, and further applications of the principles of the invention discussed above are contemplated as would be apparent to those of ordinary skill in the art.

It must be noted that the singular forms "a", "and" and "the" are used in the <RTIgt; </ RTI> <RTIgt; </ RTI> <RTIgt; Thus, for example, reference to "an oxide" includes a plurality of such oxides, and reference to "oxide" includes reference to one or more oxides and equivalents known to those skilled in the art. Wait.

"About" is used to disclose the composition, concentration, volume, program temperature, procedure time, recovery or yield, flow rate, and similar enthalpy of the description of the specific embodiments, and the number of modified (for example) components in its range. , which refers to, for example, classical type measurement and processing procedures; mistakes that are not noticed in these procedures; components that are used differently for performing the method; and variations in the number of measurements that occur for similar reasons . The term "about" also encompasses different amounts of aging due to a particular initial concentration of the formulation or mixture, as well as different amounts of mixing and handling of the formulation or mixture with a particular initial concentration. Whether modified by the term "about", the scope of the patent application relates to this addition and includes the same quantities.

As used herein, "metal oxide" is defined as a binary oxygen compound wherein the metal is a cation and the oxide is an anion. The metal may also comprise a metalloid. The metal contains those elements on the left side of the periodic table that are diagonally drawn from boron to germanium. The quasi-metal or semi-metal contains those elements on this line. Examples of metal oxides include cerium oxide, aluminum oxide, titanium oxide, zirconium dioxide, and the like, and mixtures thereof.

The present invention is directed to a method of making colloidal metal oxide particles. The invention further refers to colloidal metal oxide particles, compositions comprising colloidal metal oxide particles, and methods of using colloidal metal oxide particles. The following is a description of examples of colloidal metal oxide particles, methods of making colloidal metal oxide particles, and methods of using colloidal metal oxide particles.

I. Method of producing colloidal metal oxide particles

The present invention is directed to a method of making colloidal metal oxide particles. The materials used to form the colloidal metal oxide particles of the present invention, as well as the method steps for forming the colloidal metal oxide particles of the present invention, are detailed below.

A. Raw materials

A method of making colloidal metal oxide particles that utilizes one or more of the following materials to produce colloidal ceria particles, but alternative materials can be utilized to form other types of colloidal metal oxide materials, such as glue Alumina particles, colloidal titanium dioxide particles, colloidal zirconium dioxide particles, and the like, and combinations thereof.

1. Silicate

A method of making colloidal cerium oxide particles that utilizes one or more cerium-containing materials. Suitable ruthenium containing materials include, but are not limited to, phthalates, such as alkali metal ruthenates. Desirably, one or more alkali metal silicates are used to form colloidal cerium oxide particles. Suitable alkali metal silicates include, but are not limited to, sodium citrate, potassium citrate, calcium citrate, lithium niobate, magnesium citrate, and combinations thereof.

Suitable commercially available phthalates include, but are not limited to, commercially available sodium citrate salts and potassium citrate from several sources including PQ Corporation (Valley Forge, PA) and Zaclon Corporation (Cleveland, OH). salt.

2. Ion exchange resin

In the disclosed method, any single citrate or citrate combination can be reacted with one or more cation exchange resins to form colloidal cerium oxide particles. Suitable cation exchange resins for use in the present invention include, but are not limited to, strong acid cation (SAC) resins, weak acid cation (WAC) resins, and combinations thereof.

Suitable commercially available cation exchange resins include, but are not limited to, commercially available cation exchange resins from several sources, including Purolite Ltd. (Bala Cynwyd, PA), such as Commercially designed to sell; and Dow Chemical (Midland, MI), like Sold by commodity design.

Typically, one or more positive particle exchange resins are added to the reaction tank at a resin addition rate to maintain the pH of the reaction tank between about 8.0 and about 10.0, desirably between about 9.2 and about 9.6.

3. Seed metal oxide particles

In some embodiments of the invention, seed metal oxide particles are used as starting materials. In these embodiments, a plurality of donor seed colloidal metal oxide particles can be used. Suitable seed colloidal metal oxide particles for use in the present invention include, but are not limited to, seed colloidal metal oxide particles, such as by Nissan Chemicals, Inc. (Houston, TX) and Eka Chemical Co., Ltd. (Marietta, GA) Commercially obtained colloidal cerium oxide particles.

B. Method steps

A method of making colloidal metal oxide particles is disclosed which includes a number of steps as detailed below.

1. Preparation of reaction tank

Disclosure of methods for making colloidal metal oxide particles, in order to form colloidal metal oxide particles, may result in colloidal metal oxide particles having a reaction cycle that is much lower than the conventional reaction cycle energy efficiency. In an exemplary embodiment, a method of making colloidal metal oxide particles comprising the step of adding one or more reactants to (i) a reaction vessel containing water and (ii) substantially free of any seed metal oxide particles Where the one or more reactants can form nucleating metal oxide particles. In this particular embodiment, the step of preparing the reaction vessel simply involves adding a predetermined amount of deionized (DI) water to the reaction vessel.

In another embodiment, a method of making colloidal metal oxide particles comprising adding one or more reactants to (i) deionized (DI) water and (ii) substantially free of any seed metal oxide A step of a reaction vessel of particles, wherein the one or more reactants are capable of forming nucleating metal oxide particles and/or growing seed metal oxide particles. In this particular embodiment, the step of preparing the reaction vessel includes the desired amount of (i) deionized (DI) water and (ii) seed metal oxide particles added to the reaction vessel. The seed metal oxide particles typically have an initial average particle size (i.e., maximum dimension) ranging from about 5 nm to about 15 nm when the seed metal oxide particles are utilized.

2. Add reactive metal oxides

A method of forming colloidal metal oxide particles comprising the step of adding one or more of the above reactants to a reaction tank, wherein the step of adding one or more reactants takes into account different in situ reaction conditions, including but not Limited to at least one of (i) particle nucleation rate in the reaction tank, and (ii) metal oxide particles (eg, seed metal oxide particles and/or nucleating metal oxide particles) present in the reaction tank The rate of metal oxide deposition thereon, and/or (iii) the growth of metal oxide particles (eg, seed metal oxide particles and/or nucleating metal oxide particles) in the reaction bath. A method of balancing the reactant feed rate to inhibit colloidal metallization particles formed by the rate of reactive metal oxide deposition over the metal oxide particles present is disclosed to facilitate control of the degree of supersaturation of the reactive metal oxide in the solution phase.

In an exemplary embodiment, a method of making colloidal metal oxide particles comprising the step of adding a reactive metal oxide to a reaction vessel at a metal oxide mass addition rate, in a mathematical mode based metal oxide quality At the rate of addition, a reactive metal oxide is added to the reaction tank based on the rate of nucleation of at least one of (i) particles, (ii) metal oxidation over the metal oxide particles present. A mathematical mode of deposition rate of the material, and/or (iii) growth of metal oxide particles in the reaction vessel, wherein the metal oxide mass addition rate increases as a function of reaction time. In another embodiment, the rate of addition of the reactive metal oxide system per 1000 square meters (m ² ) of total particle surface area per hour is greater than 10.0 grams (g/1000 m ² -hr during at least a portion of the reaction cycle) ). In a still further embodiment, the method of colloidal metal oxide particles produced in accordance with the present invention comprises the step of adding a reactive metal oxide to the reaction vessel at a metal oxide mass addition rate according to a mathematical model , the mathematical mode provides the optimum metal oxide mass addition rate, q is represented by the following formula: q = (3m ₀ G _r / D _PO ³ ) (D _PO + G _r t) ²

among them:

(a) m ₀ represents the mass of the metal oxide particles in the reaction tank, and the measurement is in grams (g);

(b) G _r represents the growth rate of the metal oxide particles of the metal oxide particles in the reaction tank, which is determined by increasing the particle diameter and measured in nanometers per hour (nm/hr);

(c) D _PO represents the average metal oxide particle size measured in nanometers (nm);

(d) t represents the time in hours (hr).

In certain embodiments, during at least part of the reaction period, G _r is the range from about 10 to about 50nm / hr, and q ranges from about 10.6 to about 52.8g / 1000m ² -hr. In other embodiments, during at least part of the reaction period, G _r is the range from about 20 to about 40nm / hr, and q ranges from about 21.1 to about 42.3g / 1000m ² -hr.

Figure 1 is a graph depicting (i) the nucleation rate (R _N ) of the reactive metal oxide and (ii) the reactive metal oxide on the particles present as a function of the concentration of the reactive metal oxide A plot of deposition rate (D _R ). As shown in Fig. 1, nucleation occurs until (i) the reactive metal oxide concentration exceeds the saturation concentration (C _S ), and (ii) reaches the critical degree of supersaturation identified by C _C . At this point, nucleation proceeds at an exponential rate as the deposition rate continues along the linear path as the concentration of reactive metal oxide increases.

Figure 2 is a graphical representation of the process conditions that favor (i) the rate of reactive metal oxide deposition over the particles present (i.e., when the reactive metal oxide concentration is below C _C ), (ii) Nucleation of new colloidal metal oxide particles (ie, when the reactive metal oxide concentration is above C _C ), and (iii) both (i) and (ii) (ie, high concentration of reactive metal oxides) in C _C shown in FIG. 2 or below the concentration C _N) with the concentration of reactive metal oxide increases. When the concentration of the reactive metal oxide increases above the C _N shown in Figure 2, the process conditions significantly favor the nucleation of the new metal oxide particles during the deposition of the metal oxide on the particles present.

3. Finishing of the particle forming step

Once the predetermined metal oxide particle size is reached, the addition of reactants to the reaction tank is stopped, and a certain amount of deionized water is added to the reaction tank for quenching the reaction.

4. Filtering steps

A post-quenching step followed by a filtration step (eg, an ultrafiltration step) can be used to remove excess salts produced by one or more cation exchange resins and one or more metal oxide feedstocks.

C. Advantages of the method

A method of producing colloidal metal oxide particles is disclosed which is capable of producing colloidal metal oxide particles while optimizing reactor time and energy utilization. In certain exemplary embodiments, the method of making colloidal metal oxide particles enables the production of colloidal metal oxide particles having a final particle size ranging from about 30 to about 200 nm during the reaction cycle. The cycle represents a 50% reduction in the reaction cycle required to produce the same colloidal metal oxide particles using conventional methods.

Figure 3 is a graphical depiction of the use of (i) the optimum reactive ceria feed rate of the present invention and (ii) the conventional method using a constant reactive ceria feed rate to reduce the formation of an average particle size of 22 nm. The reaction time required for the colloidal ceria particles.

Figure 4 is a graphical depiction of the progressive addition of reactive ceria using the optimum method of the present invention to approximate the immediate optimum feed rate. As shown in FIG. 4, it is disclosed that the method of making colloidal ceria particles can include gradually increasing the reactive ceria feed rate one or more times during the given reaction cycle. Although Figure 4 has shown that there are only two-stage or three-stage processes, the reactive ceria feed rate is increased for any number of stages, which is used in the present invention to follow the Figure 4 The "Optimum" line describes the optimum feed rate.

II . Producing colloidal metal oxide particles

The colloidal metal oxide particles formed in the above method of the present invention have physical structures and properties similar to those of the colloidal metal oxide particles formed in the conventional method of forming colloidal metal oxide particles as described below.

A. Size of metal oxide particles

The colloidal metal oxide particles of the present invention have a spherical particle shape having an average maximum particle size (i.e., a maximum diameter size). Typically, the colloidal metal oxide particles of the present invention have an average maximum particle size of less than about 700 [mu]m, more typically less than about 100 [mu]m. In a desirable embodiment of the invention, the colloidal metal oxide particles have an average maximum particle size of from about 10.0 to about 100 μm, more desirably from about 10.0 to about 30 μm.

The colloidal metal oxide particles of the present invention typically have an aspect ratio of less than about 1.4, which is measured, for example, using transmission electron microscopy (TEM) techniques. The term "aspect ratio" as used herein is used to describe the ratio between (i) the average maximum particle size of the colloidal metal oxide particles and (ii) the average maximum cross-sectional particle size of the colloidal metal oxide particles, wherein the cross-sectional particle size The maximum particle size of substantially vertical colloidal metal oxide particles. In some embodiments of the invention, the colloidal metal oxide particles have an aspect ratio of less than about 1.3 (or less than about 1.2, or less than about 1.1, or less than about 1.05). Typically, the colloidal metal oxide particles have an aspect ratio of from about 1.0 to about 1.2.

B. Surface area of metal oxide particles

The colloidal metal oxide particles of the present invention have an average surface area similar to that of the colloidal metal oxide particles formed by conventional methods. Typical, colloidal metal oxide particles of the present invention have a range of g to about 180m ² / average surface area of about 90m ² / g of. Desirably, the colloidal metal oxide particles of the present invention have an average surface area ranging from about 100 m ² /g to about 160 m ² /g, more desirably from about 110 m ² /g to about 150 m ² /g.

Figure 5 is a graph comparing the colloidal metal oxide particles (in this case colloidal ceria particles) formed by the optimum method of the present invention with conventional methods (i.e., non-optimal methods, i.e., constant metal oxidation). The raw material feed rate) formed colloidal cerium oxide particles. As shown in Fig. 5, the colloidal ceria particles formed by the conventional method have an average particle size of about 27.6 nm and an average particle surface area of about 136 m ² /g, whereas the colloidal ceria formed by the optimum method of the present invention particles have an average particle size of about 28.7nm, and from about 142m ² / g, average particle surface area.

As shown in Fig. 5, the colloidal metal oxide (e.g., cerium oxide) particles formed by the optimum method of the present invention can substantially produce colloidal metal oxide particles similar to those formed by conventional methods. However, as noted above, the colloidal metal oxide particles formed by the optimum method of the present invention can be produced in a more efficient manner with up to 50% less reactor time and process energy.

III. Method of using metal oxide particles

The invention further refers to the use of colloidal metal oxide particles formed by the above process. In an exemplary method of using colloidal metal oxide particles, the method comprises applying a colloidal metal oxide particle composition on a substrate and drying the colloidal metal oxide particle composition to form on a substrate coating. Suitable substrates include, but are not limited to, paper, polymeric films, polymeric foams, glass, metals, ceramics, and fabrics.

In an exemplary embodiment, a method of using colloidal metal oxide particles includes utilizing colloidal metal oxide particles as a grinding/polishing composition for microelectronics or other articles. In other exemplary embodiments, the method of using colloidal metal oxide particles includes utilizing colloidal metal oxide particles as an additive to the coating to improve the mechanical properties of the dry coating film.

Example

The invention is further illustrated by the following examples, which are not to be construed as limiting the scope of the invention. Rather, it will be apparent that, after reading the description herein, the skilled artisan will be able to obtain various methods in accordance with the present invention without departing from the scope of the invention and/or the scope of the appended claims. Embodiments, changes, and equivalents thereof.

Example 1

Preparation of colloidal cerium oxide particles using seed cerium oxide particles and optimum cerium oxide addition rate

Add 28.4 kg (kg, 62 lbs (1b)) of deionized (DI) water to a 113.5 liter (1) (30 gallon (gal)) heated stirred tank that has been added to 4.9 kg (10.91b) A 40% by weight solid suspension of a 12 nm colloidal ceria material was used as the seed material. While stirring, the mixture was heated and maintained at a temperature between 90 and 96 °C. Then 167.8 g equivalent to _{(g) SiO 2 /min(0.37lbSiO 2 /} min) of the initial silicate addition rate while silicic acid sodium salt (29 wt% SiO _2, 9 wt% Na ₂ O) and a strong acid ion exchange The resin is added to the tank. After 10 minutes, the silicate addition rate was increased to _{317.5g SiO 2 /min(0.70lbSiO 2 / min)} , and maintained at a higher rate in a further 11 minutes.

Throughout the process, the resin addition rate was controlled to maintain the pH of the cell between 9.2 and 9.6. After 21 minutes of citrate addition, both additions were stopped and the reaction was quenched by the addition of DI water.

The resulting product was measured to have a particle size of 22+2 nm, the finest reading of additional nucleation with small particles.

Comparative Example 1

Preparation of colloidal cerium oxide particles using seed cerium oxide particles and citrate feed rate maintained at a constant rate

In addition to maintaining the overall process equivalents 167.8 grams _{(g) SiO 2 /min(0.371bSiO 2 /} min) of silicate addition rate, the procedure of Example 1 was repeated. The rate of resin addition was controlled to maintain the pH of the cell between 9.2 and 9.6. This method lasted for 31 minutes after the addition of the citrate and ion exchange resin was stopped and the growth reaction was quenched by the addition of DI water.

The resulting product was measured to have a particle size of 22 + 2 nm.

While the invention has been described with respect to a particular embodiment of the invention, the specific embodiments are not to It will be apparent that the exemplary embodiments of the present invention may be further modified, equivalents, and variations. All and percentages in the examples, as well as the remainder of the specification, are by weight unless otherwise indicated. In addition, any range of numbers, such as properties, units of measurement, conditions, physical states, or percentages, which are recited in the specification or the scope of the claims, are considered to be fully incorporated herein by reference. In contrast, any number falling within the range includes any subset of the number in any range described. For example, whenever a range of numbers 下限 having a lower limit R _L and an upper limit R _U is revealed, any number R that is specifically disclosed in the range is disclosed. In particular, the following numbers in this range are specifically disclosed: R = R _L + k(R _{U -} R _L ), where k is a range change from 1% to 100% with a 1% increase, for example: k It is 1%, 2%, 3%, 4%, 5%...50%, 51%, 52%...95%, 96%, 97%, 98%, 99% or 100%. In addition, any two enthalpy of R calculated as described above is also specifically disclosed to represent any range of numbers. Any changes to the present invention will become apparent to those skilled in the art from the foregoing description and appended claims. Such changes are considered to fall within the scope of the appended claims. All publications cited herein are incorporated by reference in their entirety.

Figure 1 is a graph depicting (i) the rate of nucleation of the reactive metal oxide and (ii) the rate of deposition of the reactive metal oxide on the particles present as the concentration of reactive metal oxide changes;

Figure 2 is a diagram depicting the benefits of (i) reactive metal oxide deposition rates over existing particles, (ii) nucleation of new colloidal metal oxide particles, and (iii) (i) and (ii) Both change with the concentration of reactive metal oxides;

Figure 3 is a graph depicting the use of (i) the optimum reactive metal oxide feed rate of the present invention and (ii) the conventional method to maintain a constant feed metal oxide particle feed rate while reducing the formation of an average particle size of 22 nm. The reaction time required for the colloidal metal oxide particles of the diameter;

Figure 4 is a diagram depicting the stepwise addition of a reactive metal oxide using the optimum method of the present invention to follow the optimum feed rate;

Figure 5 is a graph depicting particle size and surface area and colloidal ceria particles formed by the optimum method of the present invention on colloidal ceria particles formed by conventional methods (i.e., maintaining a reactive ceria feed rate constant) ).

Claims (23)

A method of producing colloidal metal oxide particles, the method comprising the steps of: (a) calculating a mass metal addition mass rate q based on a mathematical mode, in grams per hour of metal oxide (g/ Hr) measurement, which considers (i) particle nucleation rate, (ii) metal oxide deposition rate over the metal oxide particles present, and (iii) growth of metal oxide particles in the trench; a metal oxide solution is added to the tank at a mass addition rate q of the optimum metal oxide during the reaction in the case of growing the colloidal metal oxide particles; (c) during the reaction, according to the mathematical mode Calculating the rate of increase as a function of time, increasing the optimum metal oxide mass addition rate q that is greater than the rate of addition that results in nucleation of the metal oxide particles, wherein the metal oxide particles of the metal oxide in the bath G _r particle growth rate is determined by the increase in particle size, and by using nanotechnology hour (nm / hr) measured, in the range from about 10 to about 50nm / hr. The method of claim 1, wherein the mathematical mode provides a mass addition rate of the optimum metal oxide, q is represented by the following formula: q = (3m ₀ G _r / D _PO ³ ) (D _PO + G _r t ) ² wherein: (a) m ₀ represents before the growth of the metal oxide particles in the mass of the groove of the metal oxide particles, which is measured in grams (g) the count; (b) G _r representing the in the groove The growth rate of the metal oxide particles of the metal oxide particles, which is determined by increasing the particle diameter and measured in nanometers per hour (nm/hr); (c) D _PO represents the growth of the metal oxide particles before The average metal oxide particle size measured in nanometers (nm); and (d) t represents the time in hours (hr). The method of claim 2, wherein the G _r ranges from about 20 to about 40 nm/hr. The method of claim 1, wherein the step of adding a metal oxide comprises gradually increasing the mass addition rate of the metal oxide one or more times. The method of claim 1, further comprising the step of introducing seed metal oxide particles into the tank prior to the step of adding the metal oxide solution. The method of claim 5, wherein the seed metal oxide particles have an initial average particle size ranging from about 5 nm to about 15 nm. The method of claim 1, wherein the adding step results in the formation of nucleating metal oxide particles in the bath during at least a portion of the reaction. The method of claim 7, further comprising the step of: initially adding an aqueous solution before the step of adding the metal oxide solution To the tank, the aqueous solution is substantially free of metal oxides. The method of claim 1, wherein the metal oxide solution is one or more citrate salts. The method of claim 1, wherein the time required to form the colloidal metal oxide particles is at least 50% less than the method of forming metal oxide particles by using a metal oxide mass addition rate constant to form a colloidal metal oxide. The time required for the particles. The method of claim 1, wherein G _r is constant during the reaction period. A method of producing colloidal metal oxide particles, comprising: introducing seed metal oxide particles into a bath; after the introducing step, optimizing during a reaction period in the case of forming the colloidal metal oxide particles The metal oxide mass addition rate q, the metal oxide solution is added to the tank, and the optimum metal oxide mass addition rate q is measured in grams per hour of metal oxide (g/hr) and is represented by the following formula: q=(3m ₀ G _r /D _PO ³ )(D _PO +G _r t) ² wherein: m ₀ represents the mass of the metal oxide particles in the bath before the growth of the metal oxide particles, and the measurement is in grams ( g); G _r represents the growth rate of the metal oxide particles of the metal oxide particles in the bath, which is determined by increasing the particle diameter, and is measured in nanometers per hour (nm/hr); _PO represents the average metal oxide particle size measured in nanometers (nm) before the growth of the metal oxide particles; and t represents the time in hours (hr), and as a function of time during the reaction period and Increasing the optimum metal oxide mass addition rate q according to the formula During at least a portion of the reaction, the optimum metal oxide mass addition rate q is greater than the rate of addition of metal oxide particles results in nucleation. A method for producing colloidal cerium oxide particles, the method comprising the steps of: (a) adding a rate q to an optimum cerium oxide mass during the reaction in the case of forming the colloidal metal oxide particles, and oxidizing The cerium solution is added to the tank, wherein the optimum cerium oxide mass addition rate q is measured in grams per hour of metal oxide (g/hr), and based on a mathematical model (i) particle nucleation rate, (ii) presence The rate of deposition of cerium oxide on the cerium oxide particles, and (iii) the growth of cerium oxide particles in the tank, which is represented by the following formula: q = (3m ₀ G _r / D _PO ³ ) (D _PO +G _r t) ² where: m ₀ represents the mass of the cerium oxide particles in the bath before the growth of the cerium oxide particles, the measurement being in grams (g); G _r representing the cerium oxide in the tank The growth rate of the cerium oxide particles of the particles, which is determined by increasing the particle diameter, and is measured in nanometers per hour (nm/hr); D _PO represents nanometers before the growth of the cerium oxide particles (nm) Measuring the average ceria particle size; and t representing the time in hours (hr), and (b) calculating the time according to the formula Rate function, increasing the addition rate q optimum quality silicon dioxide, and during at least a portion of the reaction, the optimum mass of silicon dioxide addition rate q is greater than silicon dioxide particles to cause nucleation rate of addition. The method of claim 13, wherein G _r ranges from about 10 to about 50 nm/hr. The method of claim 13, wherein G _r ranges from about 20 to about 40 nm/hr. The method of claim 13, wherein the step of adding the cerium oxide solution comprises gradually increasing the mass addition rate of the cerium oxide one or more times. The method of claim 13, further comprising the step of introducing seed cerium oxide particles into the tank before the step of adding the cerium oxide solution. The method of claim 17, wherein the seed cerium oxide particles have an initial average particle size ranging from about 5 nm to about 15 nm. The method of claim 13, wherein the adding step results in the formation of nucleating cerium oxide particles in the tank during at least a portion of the reaction period. The method of claim 19, further comprising the step of initially adding an aqueous solution to the tank prior to the step of adding the cerium oxide solution, the aqueous solution being substantially free of cerium oxide. The method of claim 13, wherein the time required to form the colloidal cerium oxide particles is less than 50% less than the method of forming the cerium oxide particles by using the cerium oxide mass increasing rate to form a colloidal second The time required for cerium oxide particles. The method of claim 13, wherein G _r is constant during the reaction period. The method of claim 13, wherein the method produces colloidal ceria particles having a particle size of about 22 nm in about 14 to about 21 minutes, the colloidal ceria particles before the reaction period It has an initial particle size of about 12 nm.