Application No.: 11/520,479
`
`1
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`Docket No.: 638772000109
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`IN THE UNITED STATES PATENT AND TRADEMARK OFFICE
`
`In re Patent Application of:
`Neil P. DESAI et al.
`
`Application No.: 111520,479
`
`Confirmation No.: 8972
`
`Filed: September 12, 2006
`
`For: NOVEL FORMULATIONS OF
`PHARMACOLOGICAL AGENTS, METHODS
`FOR THE PREPARATION THEREOF AND
`METHODS FOR THE USE THEREOF
`
`Art Unit: 1611
`
`Examiner: T. Love
`
`DECLARATION OF NEIL P. DESAI PURSUANT TO 37 C.F.R § 1.132
`
`Commissioner for Patents
`
`P.O. Box 1450
`
`Alexandria, VA 22313-1450
`
`Dear Sir:
`
`I, Neil P. Desai, declare as follows:
`
`1.
`
`I was formerly Senior Vice President of Global Research and Development at
`
`Abraxis BioScience, LLC ("Abraxis"), assignee of the above-referenced patent application, and am
`
`currently Vice President at Celgene Corporation, which acquired Abraxis. A copy of my biography
`
`is attached hereto as Exhibit 1.
`
`2.
`
`I have more than 20 years of experience in the research and development of drug
`
`delivery systems and biocompatible polymers. I was one of the individuals responsible for the
`
`development of Abraxis' nanoparticle-albumin bound (nab™) drug delivery platform and its
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`Application No.: 11/520,479
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`Docket No.: 638772000109
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`product Abraxane®, one of the leading drugs for treating metastatic breast cancer in the United
`
`States.
`
`3.
`
`I am one of the named inventors of the above-referenced patent application and am
`
`familiar with the technical features of the invention and the claims proposed to be amended.
`
`4.
`
`I have reviewed the Office Action dated December 29, 2010. I understand that
`
`claims in the present patent application are rejected as being obvious over one of Abraxis' earlier
`
`patents, U.S. Pat. No. 5,439,686 ("Desai"), for which I am also a named inventor, in view of U.S.
`
`Pat. No. 5,407,683 ("Shively").
`
`5.
`
`The claims as amended recite a pharmaceutical formulation comprising: paclitaxel at
`
`a concentration between 5 mg/ml and 15 mg/ml, wherein the pharmaceutical formulation is an
`
`aqueous suspension that is stable for at least 3 days under at least one of room temperature or
`
`refrigerated conditions, wherein the pharmaceutical formulation comprises nanoparticles comprising
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`a solid core ofpaclitaxel and an albumin coating, and wherein the size of the nanoparticles in the
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`composition is less than 400 nm. In the paragraphs below, I discuss stability considerations for
`
`nanoparticle formulations, the Shively and the Desai references, and the advantageous stability of
`
`the nanoparticle formulations recited in the amended claims.
`
`Stability considerations for nanoparticle formulations and the cited references
`
`6.
`
`Physical stability is a key consideration for ensuring safety and efficacy of
`
`nanoparticle drug products. Precipitation and particle size increase (for example by aggregation)
`
`are key parameters for evaluating physical stability of a nanoparticle formulation. Precipitation of
`
`drug nanoparticles in an aqueous suspension could decrease the effective amount of drugs being
`
`administered to a patient. The presence of large particles in an intravenously administered
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`nanoparticle product could lead to capillary blockage and embolism.
`
`7.
`
`The tendency ofnanoparticles to precipitate and increase in size (for example by
`
`aggregation) increases as the drug concentration increases. Specifically, an increase in the
`
`concentration of drug in the nanoparticle formulation could increase the size of the particles, the
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`Application No.: 11/520,479
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`Docket No.: 638772000109
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`density ofthe particles, and/or the number of particles per volume in a suspension. Under the well(cid:173)
`
`known Stoke's law, the settling velocity of a particle composition is proportionally related to the
`
`size and density of the particles. An increase in particle size and/or density could thus increase the
`
`tendency of the particles to precipitate. Further, an increase in the number of particles per unit
`
`volume could increase the collision of the particles and thus increase the tendency of the particles to
`
`increase in size and/or precipitate. It would therefore have been expected that a nanoparticle
`
`formulation having a solid core of paclitaxel and an albumin coating would be unstable at a high
`
`paclitaxel concentration, for example between 5 mg/ml and 15 mg/ml.
`
`8.
`
`Shively does not teach a nanoparticle formulation comprising a solid core of
`
`paclitaxel and an albumin coating having a paclitaxel concentration of between 5 mg/ml and 15
`
`mg/ml. Shively states that "[f]or therapeutic use, emulsions containing between about 0.5 and about
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`5 mg/ml [paclitaxel] are prepared by the foregoing methods and administered orally or
`
`intravenously." Column 9, lines 51-54. Shively thus teaches emulsions instead of solid
`
`nanoparticles.
`
`9.
`
`In Shively's emulsions, the paclitaxel is dissolved in oil droplets suspended in an
`
`aqueous solution rather than in a solid core of albumin-coated nanoparticles. Such oil droplets are
`
`different from the solid nanoparticles in terms of composition, density, and buoyancy, and involve
`
`different stability considerations. For example, oil droplets, which have a lower density than that of
`
`water, would tend to float rather than precipitate. Thus, Shively's teaching of 5 mg/ml paclitaxel in
`
`an oil-in-water emulsion formulation provides no suggestion that a nanoparticle formulation having
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`a solid core of paclitaxel and an albumin coating would be stable at paclitaxel concentration of
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`between 5 mg/ml and 15 mg/ml.
`
`10.
`
`The Examiner relies on Example 5 ofDesai as teaching that "the composition of
`
`Desai is stable for 27 days at temperatures of 4°C, 25°C, and 38°C (see Example 5)." Page 4 of the
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`Office Action. Example 5 ofDesai refers to the stability of polymeric shells containing buoyant
`
`soybean oil. No drug was present within the polymeric shell. Like the oil droplets in Shively, the
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`Application No.: 111520,479
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`Docket No.: 638772000109
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`oil-containing polymeric shells in Example 5 of Desai are different from the solid nanoparticles in
`
`terms of composition, density, and buoyancy. The stability of the oil-containing polymeric shells
`
`discussed in Example 5 of Desai thus provides no suggestion that a nanoparticle formulation
`
`comprising a solid core of paclitaxel and an albumin coating would be stable at paclitaxel
`
`concentration of between 5 mg/ml and 15 mg/ml.
`
`11.
`
`The Examiner also points to Example 4 ofDesai as teaching that "a higher loading of
`
`taxol can be achieved by utilizing an additional solvent such as ethyl acetate, which is removed."
`
`Pages 3-4 of the Office Action. As the Examiner himself acknowledges, "[ s ]aid taxol suspension is
`
`taught as being protein walled polymeric shells enclosing an oil/taxol solution." Page 4 of the
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`Office Action. As discussed above, such oil-containing polymeric shells are different from the solid
`
`nanoparticles in terms of composition, density, and buoyancy, and involve different stability
`
`considerations. Furthermore, an increase in loading of paclitaxel (taxol) within the polymeric shells
`
`as taught in Desai would be expected to increase the particle size and/or the density of the particles,
`
`which in tum could increase the tendency of the particles to precipitate.
`
`12.
`
`The Examiner also states that "the preferred particle radii for the invention of Desai
`
`are 0.1 to 5 microns (100-5000 nm), which overlaps the instant [particle size] range." Page 5 ofthe
`
`Office Action. Example 2 of Desai shows that the protein shells have a size range of 1.35±0.73
`
`microns (1350±730 nm). The wide size range taught in Desai would be expected to lead to further
`
`instability. Specifically, according to the well-known phenomenon of Ostwald ripening, when both
`
`small and large particles are present (such as the compositions disclosed in Desai), smaller particles
`
`tend to dissolve, and the larger particles tend to increase in size. See Y ao et al., Theory and
`
`Simulation of Ostwald Ripening, Physical Review B, 1993)(Exhibit 2). Given the wide size range
`
`of the particles in Desai and the increased tendency of the particles to precipitate as the paclitaxel
`
`concentration increases, one would not reasonably have expected that the nanoparticle formulation
`
`of paclitaxel disclosed in Desai could be obtained at a concentration between 5 mg/ml to 15 mg/ml,
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`without causing precipitation and compromising the stability of the composition.
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`Application No.: 111520,479
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`Docket No.: 638772000109
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`13.
`
`Thus, neither Shively nor Desai suggests that a nanoparticle formulation comprising
`
`a solid core of paclitaxel and an albumin coating would be stable at a paclitaxel concentration of
`
`between 5mg/ml and 15 mg/ml.
`
`The advantageous stability of the nanoparticle formulations recited in the amended claims
`
`14.
`
`Example 37 ofthe present application has shown, unexpectedly, that a
`
`pharmaceutical composition with nanoparticles having a size ofless than 400 nm and having a solid
`
`core of paclitaxel and an albumin coating can be reconstituted to a paclitaxel concentration between
`
`5 mg/ml and 15 mg/ml without compromising the stability of the composition. As shown in
`
`Example 37, the lyophilized composition was reconstituted to 5, 10, and 15 mg/ml and stored at
`
`room temperature and under refrigerated conditions. The suspensions were found to be
`
`homogeneous for at least three days under these conditions. Particle size measurements performed
`
`at several time points indicated no change in size distribution. No precipitation was observed under
`
`these conditions.
`
`15.
`
`The advantageous properties of the nanoparticles recited in the amended claims is
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`further demonstrated in a subsequent experiment which compared the physical stability of two
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`pharmaceutical compositions containing nanoparticles comprising a solid core of paclitaxel and an
`
`albumin coating. In Composition 1, there was no detectable percentage ofnanoparticles that have a
`
`size above 400 nm. In Composition 2, by contrast, at least 10% of the nanoparticles in the
`composition had a particle size that is above 400 nm. 1
`
`16.
`
`Exhibit 3 shows photographs of vials containing Composition 1 and two lots of
`
`Composition 2 stored at 40°C for 24 hours at the concentration of about 5 mg/ml. The vials were
`
`inverted at the end of the storage period to show the sedimentation of the particles at the bottom of
`the vials. As shown in Exhibit 3, upon storage at 40 oc for 24 hours,2 there was a distinctly visible
`
`1 Size determined by disc centrifugation method immediately after reconstitution of the compositions at about 5 mg/ml.
`2 Storage at 40 °C for 24 hours is equivalent to storage at room temperature for at least three days.
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`sedimented layer at the bottom ofthe vials containing Lot 1 and Lot 2 of Composition 2. Such
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`sedimentation was not observed in the vial containing Composition 1.
`
`17. Microscopic observation of the reconstituted suspensions for Composition 2 stored at
`40 oc for 24 hours at 400x magnification further revealed large particles, which were not observed
`in reconstituted Composition 1. Exhibit 4.
`
`18.
`
`Table 1 further shows the stability evaluation results of Composition 1 and
`
`Composition 2. As shown in Table 1, upon storage of the composition at 40°C for 24 hours, the
`
`weight mean diameter of the nanoparticles in Composition 1 remained unchanged. In Composition
`
`2, by contrast, the weight mean diameter of the nanoparticles increased significantly upon storage.
`
`Storage Condition
`
`Weight Mean Diameter, nmj
`
`136.9
`
`135.2
`
`244.5
`
`1159.5
`
`228.0
`
`561.5
`
`Table 1
`
`Sample
`
`Composition 1
`
`0 time
`
`24 hours at 40°C
`
`Composition 2, Lot 1
`
`0 time
`
`24 hours at 40°C
`
`Composition 2, Lot 2
`
`0 time
`
`24 hours at 40°C
`
`19.
`
`I hereby declare that all statements made herein of my own knowledge are true and
`
`that all statements made on information and belief are believed to be true; and further that these
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`statements were made with the knowledge that willful false statements and the like so made are
`
`publishable by fine or imprisonment, or both, under Section 1001 of Title 18 of the United States
`
`Code, and that such willful false statements may jeopardize the validity of the application, any
`
`patent issuing thereon, or any patent to which this verified statement is directed.
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`3 Size determined by disc centrifugation method.
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`Exhibit 1
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`Actavis - IPR2017-01100, Ex. 1021, p. 8 of 30
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`Actavis - IPR2017-01100, Ex. 1021, p. 8 of 30
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`BIOGRAPHY
`NEIL P DESAI, PhD
`
`Vice President, Strategic Platforms
`Abraxis Bioscience, LLC
`A wholly owned subsidiary of Celgene Corp
`Los Angeles, CA
`
`Neil Desai is currently Vice President of Strategic Platforms at Abraxis Bioscience I
`Celgene. Prior to its acquisition by Celgene in Oct 2010, he was Sr. Vice President of
`Global Research and Development at Abraxis Bioscience, in Los Angeles, California,
`USA, where he was responsible for the development of the company's growing product
`pipeline and the development of the company's intellectual property portfolio. These
`responsibilities included the development of products from the early discovery phase
`through preclinical testing, late stage clinical studies and development for commercial
`manufacturing. Dr. Desai is an inventor of ABI's nanotechnology and nanoparticle(cid:173)
`albumin bound (nab ™) drug delivery platform, was primarily responsible for the
`development of its nanotechnology drug, Abraxane® and the discovery of the novel
`targeted biological pathway utilized by nab®-drugs. This platform has been clinically
`proven to enhance the efficacy and safety of cytotoxic drugs through a novel targeted
`biological pathway and is the first protein-based nanotechnology product to be approved
`by the FDA for the treatment of cancer. Abraxane is now approved in over 40 countries
`worldwide as a new class of nanotherapeutic for the treatment of metastatic breast cancer
`and is currently in phase III studies for lung cancer, pancreatic cancer and melanoma.
`Abraxis was recently acquired by Celgene for over $3B.
`
`Prior to joining ABI in 1999, Dr. Desai was Senior Director of Biopolymer Research at
`VivoRx, Inc and VivoRx Pharmaceuticals, Inc. (predecessor companies of ABI), where he
`worked on the early discovery and development of Abraxane, developed novel
`encapsulation systems for living cells and was part of the team that performed the world's
`first successful encapsulated islet cell transplant in a diabetic patient.
`
`Dr. Desai has more than 20 years of experience in the research and development of novel
`drug delivery systems and biocompatible polymers. He holds over 1 00 issued patents and
`peer-reviewed publications, has made over 150 presentations at scientific meetings and is
`also active in the research community having organized and chaired symposia in the areas
`of biocompatible polymers and nanotechnology-based delivery systems. He has served as
`reviewer for several scientific journals in the area of cancer therapeutics and drug delivery.
`He also is an active participant in the FDA Nanotechnology task force, FDA-Alliance for
`Nanohealth initiatives, FDA CDRH Nanotechnology Panel and the European Union
`Technology Program for Nanomedicine. Dr. Desai holds an M.S and Ph.D. in Chemical
`Engineering from the University of Texas at Austin, USA, and a B.S. in Chemical
`Engineering from the University Institute of Chemical Technology in Mumbai, India.
`
`DESAI, Page 1/15
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`Exhibit 2
`Exhibit 2
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`
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`Actavis - IPR2017-01100, Ex. 1021, p. 10 of 30
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`PHYSICAL REVffiW B
`
`VOLUME 47, NUMBER 21
`
`1 JUNE 1993-1
`
`Theory and simulation of Ostwald ripening
`
`Jian Hua Yao, * K. R. Elder, Hong Guo, and Martin Grant
`Centre for the Physics of Materials, Physics Department, Rutherford Building,
`McGill University, 9600 rue University, Montreal, Quebec, Canada H9A £T8
`(Received 17 August 1992; revised manuscript received 23 December 1992)
`
`A theoretical approach to the Ostwald ripening of droplets is presented in dimension D 2 2.
`A mean-field theory is constructed to incorporate screening effects in the competing many-droplet
`system. The mean-field equations are solved to infinite order in the volume fraction and provide
`analytic expressions for the coarsening rate, the time-dependent drople~tribution function, and
`the time evolution of the total number of droplets. These results are in good agreement with
`experiments in three dimensions and with a very large scale and extensive numerical study in both
`two and three dimensions presented in this paper. The numerical study also provides the time
`evolution of the structure factors, which scale with the only length scale, the average droplet radius.
`
`C(r)~r~R= 000(1 + i}.
`
`(2)
`
`where v is the capillary length, defined below, and the
`boundary condition far from all droplets:
`lim C(r) = C,
`(3)
`r-+oo
`where a· is the, mean concentration in the bull{. The
`
`I. INTRODUCTION
`
`When a binary mixture is cooled from a disordered
`phase into a two-phase metastable region (where the vol(cid:173)
`ume fraction¢> of the minority component is small), the
`minority component condenses into spherical droplets.
`As time evolves, on average the droplets grow in radius
`R(t), while their number decreases: Large droplets grow
`by the condensation of material diffused through the ma(cid:173)
`trix from small evaporating droplets. This phenomenon
`is called Ostwald ripening. Figure 1 is a schematic pic(cid:173)
`ture of two-dimensional Ostwald ripening as time in(cid:173)
`creases. In this figure, the shaded circles stand for the
`droplets fixed in two-dimensional space, and time evolves
`from (a) to (d). Figure 1 clearly shows that the small
`droplets are shrinking, while the large ones are growing;
`i.e., the large droplets are swallowing small ones. As
`time t evolves, the total number of droplets decreases
`and the average droplet radius increases, but the volume
`fraction of droplets¢> (the shaded area) does not change
`with time. During the coarsening, the system tries to
`minimize its interfacial free energy by nonlocal diffusion.
`These are the main features of Ostwald ripening.
`The theory of Ostwald ripening determines how the
`droplets evolve with time. Important quantities of inter(cid:173)
`est·are the droplet-distribution function f(R, t), the aver(cid:173)
`age droplet radius R(t), and the total number of droplets,
`N(t). The classic Ostwald-ripening theory is attributed
`to Lifshitz and Slyozov,l and Wagner2 (LSW), who stud(cid:173)
`ied the case in which the volume fraction of the minority
`phase tends to zero, i.e., ¢>-+ 0, in dimension D = 3.
`The starting point of the LSW theory is the diffu(cid:173)
`sion equation for the concentration C in the steady-state
`limit:
`V 20(r) = 0 ,
`(1)
`where ac I at can be neglected. This determines the
`flow of material between droplets, subject to the Gibbs(cid:173)
`Thomson boundary condition at the surface of a droplet
`of radius R:
`
`lbl
`
`•
`
`•
`
`•
`
`,.,
`
`•
`
`lei
`
`•
`•••
`• • •
`•• •
`•• • •
`•
`•
`• • • •
`• • • • • • • •
`•• ldl
`•
`• • •
`.-
`•
`•
`•
`• • ••
`
`FIG. 1. This sketch shows the Ostwald-ripening phe(cid:173)
`nomenon in two dimensions. The shaded circles represent the
`droplets (the minority component) fixed in two-dimensional
`space. As time evolves from {a) to (d), the total number of
`droplets decreases and the average droplet radius increases,
`but the volume fraction of droplets 4> (the shaded area) is
`constant.
`
`47
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`THEORY AND SIMULATION OF OSTWALD RIPENING
`
`14lll
`
`capillary length is v = 2/VmCoo/(RgT), where 'Y is the
`surface tension, Vm the molar volume, Coo the solute
`concentration at a fiat interface, R9 the gas constant,
`and T the temperature. The mass balance
`!!:._(47rRa)=4 R2VdC(r)l
`dt
`dr
`3
`
`11"
`
`r=R'
`
`(4)
`
`where V
`is the diffusion constant, ensures that the
`changes in volume of the droplets, which are assumed to
`be spherical, are due to a change in concentration. The
`droplet-distribution function f(R, t) determines averages
`by
`
`with our convenient normalization
`
`-
`(·. ·) =
`
`fdRJ(R,t)(···)
`I dRf(R,t)
`J dRJ(R, t) = N(t),
`
`{5)
`
`(6)
`
`the total number of droplets. The distribution function
`obeys the continuity equation
`of(R,t) a .
`Bt + BR[Rf(R,t)] = 0;
`
`(7)
`
`thus there is no source of new droplets (nucleation has
`ceased), where the overdot denotes a time derivative, and
`the conservation law
`
`L\(t) + ~ lJO R3 f(R, t)dR = Q ,
`
`concentration, and V' is the volume of the system. We
`shall consider this equation in the limit in which the mi(cid:173)
`nority phase (within the droplets) and majority phase
`(outside the droplets) are at their equilibrium concentra(cid:173)
`tions, i.e., L\(t) = 0. These equations above follow for all
`small volume fractions, provided a. steady-state droplet
`picture is reasonable.
`In the limit of¢-+ 0, the solution of the steady-state
`diffusion equation is
`C(r) = C- (C- C(R)]Rjr,
`since one need only consider one droplet. With the equa(cid:173)
`tion for mass balance, this gives the growth law in that
`limit:
`dR = V (~ _ !!..)
`dt
`R
`R
`
`(9)
`
`(10)
`
`'
`
`where, evidently, droplets larger (smaller) than the time(cid:173)
`dependent critical radius vj~(t) grow (shrink). In an
`elegant calculation, LSW determined the asymptotic
`growth rate of the average droplet radius to be
`
`R(t) = 4~vt
`(
`)
`
`1/3
`,
`
`(11)
`
`where the prefactor 4/9 is the dimensionless coarsening
`rate, and the overbar denotes an average. In addition to
`this prediction, an analytic form for the droplet distribu(cid:173)
`tion function was obtained:
`
`(8)
`
`f(R, t) ex: g(R(R){Jt
`
`(12)
`
`where L\(t) = C- C00 , is the supersaturation of the so(cid:173)
`lution, which vanishes as t-+ oo; Q = ¢V' is the initial
`
`for. late times. The explicit form of the scaled normalized
`distribution function is
`
`(13)
`
`]
`
`if 0 < z < J,
`otherwise.
`
`34e/2513)z2 exp[-1/(1- iz)]/[(z + 3) 713(~- z) 1113
`
`0(
`
`g(z) = -{
`
`This important work revealed both power-law growth
`and dynamic scaling, which are now considered univer(cid:173)
`sal characteristics of the kinetics of a first-order phase
`transition. 3
`Nevertheless, it has proved difficult to test their the(cid:173)
`ory rigorously by experiment or numerical simulation.
`Experiments typically study volume fractions apprecia(cid:173)
`bly larger than zero, and large-scale numerical work has
`been limited by previous computer facilities. Earlier work
`on extending the theory of LSW to nonzero ¢ has been
`attempted by many groups,4- 13 using both analytic and
`numerical methods.
`For the most part, analytic extensions have been based
`either on ad hoc assumptions (the work of Ardell9 and
`Tsumuraya and Miyati10), or on perturbative expansions
`in ¢, typically taken to order .j(/) (the work of Mar(cid:173)
`qusee and Ross4 (MR) and Tokuyama, Kawasaki, and
`Enomoto5 (TKE)]. In addition, an ambitious theory was
`developed by Mardar12 in which two-particle correlations
`
`were included for three-dimensional Ostwald ripening.
`All these approaches lead to the following growth law:
`
`R(t) = [F(O) +K{¢)t] 113 ,
`
`(14)
`
`where the coarsening rate K(¢) is a monotonically in(cid:173)
`creasing function of ¢. The droplet-distribution function
`satisfies
`f(R, t) rx g(z, ¢) ;R(D+l) ,
`
`(15)
`
`where z =. R/ R. The theories predict a broadening of
`g(z,¢) as the volume fraction is increased. Unfortu(cid:173)
`nately, the perturbative theories can neither go beyond
`0( ..fifJ) nor be applied to two-dimensional systems, and
`the ad hoc approaches contain uncontrolled approxima(cid:173)
`tions. Indeed, in many cases the theories for D = 3 give
`rather different results, as we shall show below. Experi(cid:173)
`ments are not of sufficient quality to distinguish them.
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`14112
`
`JIAN HUA YAO, K. R. ELDER, HONG GUO, AND MARTIN GRANT
`
`To our knowledge, two numerical studies have been
`conducted in three dimensions. In 1984, Voorhees and
`Glicksman7 (VG) carried out a numerical simulation, by
`a very interesting novel approach based on Ewald-sum
`techniques, reviewed below. Unfortunately that work
`was hampered by the computing facilities available at
`that time, and the number of droplets included was too
`few to give conclusive results. More recently Beenakker8
`attempted to extend their work, but he only included a
`rather small number ( ~ 20) of nearest-neighbor interac(cid:173)
`tions. Thus, in his simulation, the volume fraction was
`not fixed, and drifted 10-20 %. Thus neither work pro(cid:173)
`vided a good test for theory.
`For two-dimensional systems, theory has been ham(cid:173)
`pered by the logarithmic divergence present in the
`steady-state limit of the two-dimensional diffusion equa(cid:173)
`tion. Marqusee15 proposed a self-consistent theory, which
`Zheng and Gunton16 extended by including two-particle
`correlations. However, Marqusee's two-dimensional the(cid:173)
`ory is very different from his three-dimensional theory
`with Ross,4 while Zheng and Gunton's theory involved
`numerous approximations following Mardar's theory. Re(cid:173)
`cently Ardell9 published an extension to two dimensions
`of his phenomenological theory for three-dimensional
`coarsening. His theory, however, involves an ad hoc,
`although physically motivated, free parameter. These
`groups obtain the growth law and scaling function as for
`three dimensions (Eqs. (14) and (15) above). A different
`result was obtained in the non-steady-state calculation
`for¢ -4 0 by Rogers and Desai17 (RD). They found scal(cid:173)
`ing with R "' (t/ In t) 113 , for that limit in D = 2. Up to
`now, there have been no experiments to test these the(cid:173)
`ories, which have rather different predictions, although
`numerical work on a nonlinear Langevin equation has
`been done by Toral, Chakrabarti, and Gunton. 18
`In summary, although a great deal of progress has
`been made in understanding Ostwald ripening, a fully
`satisfactory approach has not yet been found, and it
`has remained a vexing problem in the field. Thus,
`we felt it worthwhile to reinvestigate this fundamental
`phenomenon. 13 The goal of this article is to present a sys(cid:173)
`tematic method to study Ostwald ripening in D ~ 2 at .
`nonzero volume fractions {for simplicity,¢ will be called
`the volume fraction, although, for example, in D = 2 it is
`an area fraction). To do so, we have introduced a mean(cid:173)
`field model that we solve exactly in arbitrary dimension.
`We then test our results by comparison to an experiment
`in three dimensions, and to a large-scale simulation we
`have done in two and three dimensions. This numeri(cid:173)
`cal simulation also provides an estimate of the dynamic
`structure factor, which we find obeys a dynamic scaling
`relationship.
`The organization of the paper is as follows: In Sec. II
`we introduce a Thomas-Fermi-type approximation tore(cid:173)
`duce the steady-state many-body diffusion equation to a
`set of one-body diffusion equations, where the conserva(cid:173)
`tion law plays the role of charge neutrality. Applying the
`Gibbs-Thomson condition to these solutions and using
`the conservation law, we obtain the basic equations of
`our mean-field theory. Section III presents the solution
`of the basic equations. In Sec. IV formulas for both two-
`
`and three-dimensional numerical simulations are derived
`by means of the Ewald techrlique. 19 Section V presents
`the simulation results and compares them with those of
`our mean-field theory, previous theories, and an exper(cid:173)
`iment in three dimensions. A short conclusion to this
`article is given in Sec. VI.
`
`II. MEAN~FIELD THEORY
`
`Our study makes use of dimensionless variables. Units
`of length and time are given in terms of a characteristic
`length lc = (D -1)··rVm/R9T and a characteristic time
`t* = l~/(VC00 Vm)· It is also convenient to introduce a di(cid:173)
`mensionless concentration field B(r) = [C(r)- C00]/C00 •
`All the quantities involved have been defined in Sec. I.
`The many-droplet diffusion problem is intractable
`without approximation. In the steady-state limit, the
`fundamental equation is7
`
`N
`'V2B{r) =a EB.:8(r- r,),
`
`(16)
`
`where N is the number of the droplets in the system,
`a = 2TrD/2jr(D/2), r,; giv~ the location of the ith
`droplet, and B,; is the strength of the source or sink of cur(cid:173)
`rent for diffusion. This is the multidroplet diffusion equa(cid:173)
`tion in the quasistationary approximation, where BB/8t
`is neglected because the growth rate of droplets is much
`slower than the relaxation time of concentration field in
`the matrix. The o functions on the right-hand side of Eq.
`(16) result from the assumption that the droplet locations
`remain fixed in space and the distances between droplets
`are much larger than the average droplet radius. This
`is a very good description for systems with small vol(cid:173)
`ume fractions. The necessary boundary conditions are
`the Gibbs-Thomson condition for the concentration field
`at the curved surface of each droplet and the imposed
`supersaturation far from all droplets:
`
`lim B(r) = Ba.v
`r-+oo
`
`(17)
`
`for i = 1, ... , N, where Ba.v is the average concentration
`outside the droplets. The conservation law is
`
`N
`
`LB,;=O,
`
`(18)
`
`i=l
`which implies that we shall consider the limit in which
`the minority phase (within the droplets) and major(cid:173)
`ity phase (outside the droplets) are at their equilibrium
`concentrations, 7 and the growth law satisfies
`
`d(vRf) =-1 J·ndu,
`where s,; is surface of the ith droplet, n is the unit vector
`normal to the droplet surface, and v = ~/2/r(D/2+1).
`Substituting the Fourier-Fick law J = - V (} into Eq. (19)
`and transforming the surface integral over the ith droplet
`into a volume integral gives
`
`dt
`
`••
`
`{19)
`
`Actavis - IPR2017-01100, Ex. 1021, p. 13 of 30
`
`

`

`THEQRY AND SIMULATION OF OSTWALD RIPENING_
`
`(20)
`
`80
`fJ + S- aB,o(r- ri)·
`-2
`2
`8t =-V fJ- e
`
`14113
`
`(26)
`
`Here we have approximated the contribution from other
`droplets in Eq. (16) through the introduction of a screen(cid:173)
`ing length e and a source or background field se. We
`
`shall now self-consistently rel~te thes_e quantities to I(R)
`by integrating the equation above and comparing it with
`Eq. (25), i.e.,
`
`Equations (23) and (26) completely specify our mean(cid:173)
`field approximation; indeed, they are the only approxi(cid:173)
`mations needed to solve the equations in the steady-state
`limit. Their form implies we consider a one-body problem
`without correlations. A systematic derivation of these
`equations from first principles would be valuable, since
`corrections to our equations, involving correlations, could
`be calculated. However, we have not been able to obtain
`such a derivation, although, as we have indicated above,
`a coarse graining of the microscopic equations, with the
`requirement that only a one-body distribution function
`is involved, will lead to our self-consistent starting point.
`In the steady-state limit, the concentration field obeys
`\120 - e-20 + e-20av = aB,o(r- ri)
`(29)
`in the vicinity of the ith droplet. 13•14 The solution of Eq.
`(29) at the boundary (lr - r,l = R.) is then
`1 Rs = Bav- Bi V(R./e, R.)
`fori= 1, ... , N, where V(Rjt;, R) is the Green's function
`of Eq. (29). In D = 3, V(Rjt;,R) = exp(-R/e)/R; in
`D = 2, V(Rjt;,R) = K 0(R/e), where K 0 is the zeroth(cid:173)
`order modified Bessel function. Equations (30) and (18)
`can then be used to solve for Bi and 8av· Substituting
`these solutions into Eq. (21) gives
`
`(30)
`
`R1-D
`{ [RV(R/1;, R)]- 1
`dR
`dt = V(Rfe,R)
`[V(R/e,R)] 1
`
`1 }
`- R
`
`'
`
`31
`
`(
`
`)
`
`where the overbar is defined as
`A= fooo Af(R,t)dR/ 1oo f(R,t)dR.
`Comparing Eq. (31) with Eq. {23) gives I(R) =
`afV(Rfe, R), so that
`
`(32)
`
`f(R, t)/V(R/1;, R)dR.
`
`(33)
`
`The conservation law Eq. (18) can be rewritten as
`
`RD f(R, t)dR = ¢.
`
`(34)
`
`Equations