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Controlled Release of Urea from Rosin-Coated Fertilizer Particles Byung-Su Ko,† Young-Sang Cho,‡ and Hyun-Ku Rhee*,† Department of Chemical Engineering, Seoul National University, Kwanak-ku, Seoul 151-742, Korea, and Division of Chemical Engineering, Korea Institute of Science and Technology, Sungbuk-ku, Seoul 136-791, Korea
A mathematical model describing the controlled release of urea from rosin-coated urea granules has been developed on the basis of the experimental results. The release mechanism was also elucidated from the experimental observation on the release from a single pellet. Two classes of release pattern were observed from single-pellet release experiments including (1) immediate and relatively rapid release through a few holes or many microscopic pores originally present in the coating and (2) little or no release for an extended period of time followed by a sudden, rapid release through the holes formed in the coating. It was found that individual granules could not give sustained release of urea by itself. Applying the mathematical model developed for a single pellet to real systems in which coating weight distribution is present, we obtained good agreements between model predictions and experimental results. Introduction Over the past score of years controlled release technology has undergone a rapid development in various fields and its products now span a variety of applications including medical, agricultural, and household uses. One approach that has received a great attention as a means of controlled release of active agents has been the incorporation of solutes in solid polymers (Paul, 1976; Langer, 1980). Many of the commercialized controlled release fertilizers have utilized this method that permits urea to be released at a controlled rate. Sulfur-coated urea (SCU), which uses sulfur as a coating material, is representative of commercialized controlled release fertilizers. Rosin-coated urea, which was investigated in this study, is also prepared by a rosin-coating process. A great deal of research effort has been devoted to the development of controlled release fertilizers by investigators around the world. Actually, many controlled release fertilizers such as urea-formaldehyde (UF) or SCU were commercialized. However, UF suffered from high production cost caused by chemical reaction (Lunt, 1971), while SCU required supplementary application of a sealant compound over the sulfur coating to seal minute fissures and pinholes in the coating (Davis, 1976). Moreover, acidification of soil owing to the use of sulfur can lead to serious problems afterward. In the development of controlled release fertilizers, economic evaluation of the products is essential because fertilizers are relatively cheap compared to other fine chemicals, so lower production cost should be available. The manufacture of controlled release fertilizer by a coating process is particularly useful in that a coating process is relatively simple and the release rate from this coated fertilizer may be designed as one wishes. In this case, the selection of coating material is important since it determines almost all the qualities of the controlled release fertilizer. Coating materials should not only be cheap but also have good coating properties. In addition, they should not contaminate the soil. * To whom correspondence should be addressed. E-mail address:
[email protected]. † Seoul National University. ‡ Korea Institute of Science and Technology.
0888-5885/96/2635-0250$12.00/0
When rosin is used as a coating material, there is no need for additional application of a sealant. Moreover, this material is very cheap and does not contaminate the soil, so it is being evaluated as a new coating material. In developing the rosin-coated fertilizers, it is aimed to reduce the number of fertilizer applications to one. Although split applications of nitrogen often enhance the crop yield, such a benefit can still be obtained by applying the controlled release fertilizers if their nitrogen release pattern is made similar to the case of split applications. In this sense, it is desirable to develop a mathematical model that can be utilized for the design of controlled release fertilizers showing specific release characteristics. There have been extensive studies on the mathematical modeling of controlled release (Baker and Lonsdale, 1974; Theeuwes, 1975; Peppas, 1984; Ritger and Peppas, 1987a,b; Fan and Singh, 1989; Polowinski et al., 1990). However, with regard to controlled release from coated fertilizer granules, none of those models seems to be adequate. Only the release mechanism from SCU has been reported so far (Davis, 1976). The purpose of this study is to develop a mathematical model that describes the controlled release of urea from rosin-coated urea granules and investigate the release mechanism. Experimental Section Physical Properties of Urea Granules. For the preparation of spherical particles for single-pellet experiments, urea granules were first reduced to powder and then repelletized by a presser. Then, these pellets (initially in the shape of cylinder) were cut manually to give almost perfect spheres. The diameter of these particles ranged from 5.58 to 6.03 mm, and the density was measured at 1.264 g/cm3. Meanwhile, sample granules ranging from 2.36 to 2.83 mm in diameter were sieved and used for further release experiments. Even though these granules were not perfect spheres, they were pretty much in the shape of spheres. During the sieving, we tried to remove such granules that had defects on the surface. The density of these sample granules was measured to give 1.375 g/cm3. The solubility of urea granules in 30 °C water was found to be 0.652 g/cm3, the value of which was obtained by weighing the amount of urea dissolved until saturation was reached. © 1996 American Chemical Society
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Figure 2. Typical examples showing release characteristics in single-pellet experiments.
Figure 1. Scanning electron micrograph for the cross section of rosin-coated granule.
Coating Process. Coating solution was prepared by dissolving the solid rosin (U.S. Color Grade WW, China) in methanol. This coating solution consisted of 1 part of rosin in 3 parts of methanol by weight. The specific gravity of this solution was measured at 0.830 g/cm3. Rosin is not a chemical species but a complex mixture of mutually soluble organic compounds. The physical and chemical properties of rosin may vary depending on its source and method of refining. We used distilled wood rosin, the softening point of which was 76 °C. Rosin is insoluble in water and freely soluble in alcohol. It does not undergo any reaction with water. Pneumatic spraying of this coating solution onto the urea granules was carried out in a rotary coating fan. Since the temperature inside the fan was maintained at 40 °C throughout the coating process, much of the solvent was removed during this coating process. The amount of coating, thus coating thickness, was regulated by the coating time. In particular, coated fertilizer particles which had various coating thicknesses were prepared for single-pellet experiments. To remove the remaining solvent in the coating, we dried the coated urea granules in the rotary coating fan also maintained at 40 °C for 20 min. Then, these coated granules were exposed to room temperature for more than 24 h for complete removal of solvent. Less than 300 ppm of methanol was detected after all these procedures. The uniformity of rosin coating was investigated by a scanning electron micrograph (SEM), and a typical result from the smaller sample granule is shown in Figure 1. In most cases (except for the pellets that had defects on their surface) the coating was found rather uniform all over the urea pellet surface. Release Experiments. Coated urea granules (0.88 g, about 70 granules) were placed in 200 mL of water maintained at 30 °C, and the amount of urea released in time was measured by a high performance liquid chromatograph (HPLC) with refractive index detector. Temperature control was performed within (1 °C error range by an incubator operated on a PID-control algorithm. Also, the calibration curve for the measurement of urea concentration was obtained. The morphology of the rosin coating was completely maintained during the release of urea. It was a spherical shell with visible hole(s) in the surface. Even after the complete release of urea, no visible loss or erosion of rosin was observed for quite a long time.
Indeed, we developed the rosin-coated fertilizer with special attention to application to rice crops. Since rice plants grow with their roots in a pool of water in which the fertilizer is spread, the experimental release environment was not very much different from that of the real one in the case of rice crops. The present simplified release procedure has significance in that there is consistent correspondence between our experimental results and actual ones. Release from Single Pellet The release rate of urea from rosin-coated fertilizers is generally controlled by the amount and type of coating and other process variables. Although urea granules are coated by identical coating solution under the same conditions, they differ from one another in the amount of coating to give a variety of release rates. Therefore, one may be able to investigate the mechanism and the rate of urea release by conducting release experiments from a single pellet. Experimental Results. Time release data for some individual pellets having different coating thicknesses are shown in Figure 2. Except for the coating time, these individual pellets were prepared under the same coating conditions. In Figure 2, roughly two classes of release pattern are observed. For urea particle having a coating thickness equal to 34.6 µm, immediate and relatively rapid release of urea occurs, while for a coating thickness equal to or greater than 69.9 µm, there is an extended period of time with little or no release before a sudden and dramatic change in release rate takes place. It seems, therefore, reasonable to think that there exists a critical coating thickness, δcr, which divides these two release patterns. For urea particles having a coating thickness greater than this critical value, it was observed from experiments that small hole(s), developed suddenly in the coating, substantially increased the release rate. Thus, the sudden and massive loss of urea is closely related to the formation of hole(s) in the coating. After the formation of hole(s) the release of urea was almost completed within 7 days. Thus, individual granules could not give sustained release of urea by itself. The time required for the formation of hole(s) is plotted against the coating thickness in Figure 3. For the particles with coating thickness less than 50 µm, almost no time is needed before a rapid release of urea, indicating that there exist a few holes or many microscopic pores originally in the coating. For the coating thickness greater than 50 µm, the time required for the
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Figure 3. Time required for hole formation as a function of the coating thickness obtained from single-pellet experiments at 30 °C.
initially in the coating. Hence, the release of urea starts immediately after the coated granule is placed in water. The release of urea in this case is achieved by means of osmotic pumping and diffusion through holes or many microscopic pores originally present in the coating. In contrast, the release of urea from a granule with coating thickness greater than the critical value is accomplished by the following steps. Step 1. Urea imbibes water osmotically. Step 2. Owing to the semipermeable property of rosin coating, hydrostatic pressure is developed inside the coating shell. Step 3. When the hydrostatic pressure difference reaches the maximum hydrostatically tolerated pressure difference of rosin-coating shell, deformation of coating shell (i.e., swelling) occurs and further development in hydrostatic pressure makes holes or cracks in the weakest site of the coating shell. Step 4. Urea is then released by means of osmotic pumping and diffusion through holes. Model Development. After the formation of holes in the coating, the delivery of urea is generally performed by means of three different routes of release.
( )
dMt ) dt
( )
dMt + dt OP osmotic pumping
( )
dMt + dt DO diffusion through orifice
( )
dMt (1) dt DC diffusion through coating The contribution of molecular diffusion through coating to the total release rate would be very small because of the semipermeable property of rosin coating, so it can be neglected. In this case, eq 1 reduces to
( ) ( ) ( ) dMt dMt ) dt dt
Figure 4. Scanning electron micrographs of rosin-coated surface: (a) thickness ) 42 µm; (b) thickness ) 152 µm.
formation of holes increases linearly with the increase of coating thickness. The coating thickness of 50 µm, therefore, can be considered as a critical coating thickness. The microscopic structures of the rosin-coating surface which are taken by scanning electron micrographs (SEMs) are shown in Figure 4. Many microscopic pores are present initially in the coating for the coating thickness of 42 µm (Figure 4a), while there are no such pores when the coating thickness is greater than the critical value. (See Figure 4b.) Proposed Release Mechanisms. From the observations of the release patterns in single-pellet experiments, the following release mechanisms were proposed as a reasonable description of the release phenomenon. In the case of coating thickness less than the critical value, there are some holes or many microscopic pores
+
OP
dMt dt
DO
(2)
As long as there exists solid urea undissolved inside the coating shell, the zero-order release of urea is attained with the release rate given by (Katchalsky and Curran, 1967; Theeuwes, 1975)
( )
dMt Ao A ) kπsS + DS dt z δ δ
(3)
where A is the coating area, Ao the cross-sectional area of the orifice, k a constant, πs the osmotic pressure at saturation, S the solubility of urea, and δ the coating thickness. By using a lumped parameter D′ defined as DAo/A, eq 3 is reduced to
( )
dMt AS (kπs + D′) ) dt z δ
(4)
When the concentration of urea solution falls below the saturation value, nonzero-order delivery rate is given by the sum of osmotic pumping rate and the diffusional term:
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dMt Akπ AD′ ) C+ C dt δ δ Akπs 2 AD′ C + C ) δS δ
(5)
where C is the concentration of urea solution inside the coating. By substituting the relation dMt/dt ) -V dC/ dt for dMt/dt, the following differential equation is obtained for the concentration of urea inside the coating:
-
dC Akπs 2 AD′ ) C + C dt VδS Vδ
(6)
where V is the interior volume of coating assumed to be constant. Equation 6 can be integrated from time tz to t when the concentration changes from S to C. The concentration thus obtained is expressed as a function of time as follows:
C)
D′S AD′ (t - tz) -kπs (kπs + D′) exp Vδ
{
Figure 5. Plot of kπs + D′ versus coating thickness.
(7)
}
The release rate in this case is given by
dMt dC ) -V ) dt dt
(t - t )} {AD′ Vδ AD′ (t - t )} - kπ ] δ[(kπ + D′) exp{ Vδ ASD′2(kπs + D′) exp
z
2
s
z
(8)
s
Also, the amount of urea released can be determined by the equation
Mt ) M∞ - CV ) M∞ -
D′SV (9) AD′ (t - tz) - kπs (kπs + D′) exp Vδ
{
}
Determination of Model Parameters. The urea particles used in single-pellet experiments have the density of 1.246 g/cm3 and solubility of 0.652 g/cm3 in 30 °C water. Thus, the fraction of total mass delivered by the zero-order release can be calculated as (Theeuwes, 1975)
Mz S ) 1 - ) 0.484 M∞ Fp where Fp is the density of urea particles. This means that the concentration of the solution inside the coating shell remains constant at the saturation value until 48.4% of urea is released. The release rate during this period remains constant and is given by eq 4. The model parameter kπs + D′ can, therefore, be determined from the experimental zero-order release rate. This method of obtaining the model parameter was repeated over various coating thicknesses to give the model parameter as a function of the coating thickness. The results obtained are shown in Figure 5. For coating thicknesses greater than 38 µm, a relatively constant value of kπs + D′ is observed, while for coating thicknesses below 38 µm, a dramatic increase in the value of kπs + D′ occurs with the decrease of coating thickness. Actually, if there is not a sufficient amount of coating, relatively large holes are present
Figure 6. Fractional release of urea from an initially perforated rosin-coated urea particle at 30 °C.
initially in the coating, so these holes take the action of increasing the lumped parameter D′ defined as DAo/A, while kπs remains relatively constant. For the separate determination of individual parameters, kπs and D′, a single pinhole through which negligible amount of urea could diffuse was perforated in the coating so that the delivery of urea mainly occurred by osmotic pumping (Theeuwes, 1975). The time release data from an initially perforated urea particle having a coating thickness of 272.9 µm are shown in Figure 6. In this figure, it is seen that the urea was released slowly at a relatively constant rate over a period of several weeks and then massive loss of urea followed after 45 days. Such a massive loss of urea was caused by the enlargement of hole(s) followed by shell rupture. As the hole was not large enough to allow diffusion through it, the delivery of urea was accomplished almost solely by osmotic pumping. As indicated by Theeuwes (1975), the release rate would be maintained at a relatively constant value if the size of the orifice is sufficiently small to minimize the contribution of urea diffusion through the orifice to the delivery rate, and if it is large enough to minimize the hydrostatic pressure developed inside the coating shell. For these reasons, the value of kπs was estimated at 6.13 × 10-10 cm2/s by using eq 3 with the second term on the right hand side neglected, which indicates that 13% of urea release is caused by the osmotic pumping mechanism. Model Application to Real Systems From the single-pellet experiments, we developed a model that can be used in predicting the released amount in time once the coating thickness is given. We
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Figure 7. Weight distribution curves of uncoated and coated urea granules.
are now going to apply this model to the release from a large number of coated granules for which coating weight is not uniform between granules. For this, it is essential to investigate the nature of nonuniformity of coating weight. Coating Weight Distribution. Standard control tests for release of urea are usually run with a large number of samples in which a coating weight distribution for individual granules is present. The distribution of coating weight for individual granules is difficult to determine experimentally because there already exists a distribution in the weight of uncoated urea granules. Average coating weight, however, can be obtained by simply weighing the total mass of coated sample granules. Generally, this average coating weight increases with the coating time under the same coating conditions. The coating weight distribution was, therefore, determined by the following circuitous procedures. Weight distribution curves of uncoated and coated urea granules were drawn in Figure 7 for the average coating weight of 2.58 mg (20.5% coating) per granule. In both cases, Gaussian probability distribution describes well those distributions. Thus, the following statistical fact (Brown and Hwang, 1992) can be used for the determination of the variance of coating weight distribution. Let X and Y be independent normal random variables with variance σX2 and σY2, respectively. Then, the probability density function for Z ) X + Y is also normal in form, and its variance is given by σZ2 ) σX2 + σY2. If we put X as a random variable representing the weight of uncoated urea granules and also put Y as a random variable for the weight of coating, then Z is a random variable that represents the weight of coated urea granules. The above statistical fact, however, indicates that coating weight distribution follows a Gaussian distribution for which the variance is given by subtracting the variance of X from the variance of Z. With this fact, the variance of the coating weight distribution was found to be 0.835 mg for the average coating weight of 2.58 mg. Method of Calculation. The procedures to calculate the amount of released urea and the release rate are considered here for the appropriate application of the mathematical model developed for a single pellet to real systems in which coating weight distribution is present. Although the size distribution of urea granules also exists in real systems, this effect was not considered here for mathematical simplification. Indeed, the effect of size distribution of urea granules on the average
release rate is very small and it can be minimized by using an average size of urea granules in the calculations. It is also to be noted here that we neglected the size effect in applying the mathematical model developed for a larger single pellet to real systems of smaller ones. Referring to Figures 3 and 5, we can say that the two important results of single-pellet experiments (hole formation time and model parameter, kπs + D′ are independent of the pellet diameter. Although the diameter range of single-pellet experiments was rather small compared to the whole range of interests, it can be speculated from Figures 3 and 5 that, at least, there would be no significant errors in applying the results from larger pellets to smaller ones. The coating weight distribution affects the amount of released urea in the following way:
(
)
mean value for amount ) of released urea, M ht
∑δ
(
)
amount of released urea from particles with coating thickness × between δ and δ + dδ, Mt(δ) fraction of particles which have coating thickness between δ and δ + dδ, f(δ) dδ
(
)
(10)
Equation 10 can be recasted to integral form as follows:
M ht)
∫0∞Mt(δ) f(δ) dδ
(11)
The integration on the right hand side of eq 11 can be approximated with a finite sum of the integrand,
M ht=
∑i Mt(δi) f(δi) ∆δi
(12)
where f(δ) is the probability density function for the coating thickness. A similar procedure can be used to find the effect of coating weight distribution on the release rate.
(
)
mean value for
) release rate of urea, (dMt/dt) release rate of urea from particles with coating thickness × δ between δ and δ + dδ, dM (δ)/dt t fraction of particles which have coating thickness between δ and δ + dδ, f(δ) dδ
∑
(
)
(
Equation 13 then gives
dMt ) dt
∫0∞
dMt(δ) f(δ) dδ dt
)
(13)
(14)
Again, the finite sum approximation can be used for the evaluation of integral term in eq 14:
dMt = dt
∑i
dMt(δi) dt
f(δi) ∆δi
(15)
Results and Discussion In the modeling of urea release from a single pellet, it is crucial to predict accurately the time required for
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Figure 8. Comparison between calculated results and experimental data for various coating thicknesses. (Release rates are calculated from the model equation.)
the formation of holes. Experimental results in Figure 3, however, show that the hole formation time of the coated urea particles having a coating thickness greater than 50 µm increases linearly with the coating thickness. Once the holes are formed in the coating, urea is then rapidly released by means of osmotic pumping and diffusion through holes. The release rate in this case can be obtained by using eqs 4 and 8. The comparisons between calculated results and experimental data are shown in Figure 8 for various coating thicknesses. For most of the coating thicknesses, the model predictions are in good agreement with the experimental results. For the application of the single-pellet release model to real systems in which coating weight distribution exists, it is necessary to find the coating weight distribution. As mentioned before, coating weight distribution follows a Gaussian distribution which is described by two characteristic values: mean and variance. The determination of the variance of coating weight distribution is a time-consuming task, whereas the average coating weight can be obtained easily. Thus, back calculations of the variance from the experimental data were performed to investigate the relation between the variance of coating weight distribution and the average coating weight. Model predictions based on various variances of coating weight distribution are compared with the experimental results in Figure 9. From this, it is seen that the experimental results of urea release are best described when a particular value of variance is used in each case. This best fitting variance was obtained for every coating weight by calculating the one with which model calculation best fits the experimental data in the sense of least sum of error squares. The corresponding plot of best fitting variance versus average coating weight revealed that the variance of coating
Figure 9. Comparison of model predictions based on various variances of coating weight distribution with the experimental data.
weight distribution is linearly dependent upon the amount of coating as shown in Figure 10, which is valid under the same coating conditions. This means that one
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Figure 10. Comparison of experimentally determined variance with the variance determined by the best fitting method.
will obtain a broader distribution of coating weight with the increase of average coating weight. The validity of these variances that are obtained by back calculations was supported by comparison of these values with the one obtained experimentally. The comparison of the results that were obtained by these two methods is also given in Figure 10. Here, the experimental value not only proves the tendency of linear dependence of variance on the amount of coating, it also provides a basis for the justification of back calculations. On the basis of the fact that the variance of the coating weight distribution is linearly dependent on the amount of coating, model predictions were carried out for various average coating weights. Figure 11 shows that model predictions are performed with satisfaction. The delivery rates of urea are also presented in Figure 11. For an initial short period of time, very rapid release of urea occurred and then urea was released slowly with a “sigmoid” release pattern. Very rapid release of urea in an early stage is thought to be caused by the massive loss of urea from granules having insufficient coating weight to cover all the surface completely. Incidentally, the rosin-coated fertilizer has been tested in a rice field by the Korea Rural Development Administration over the past 3 years. Their results were found to be consistent with the experimental results of this study. In other words, a sample controlled release fertilizer whose effective release time was longer than 45 days in our experimental environment did not show any nitrogen deficiency during the whole crop growth period whereas those with a shorter effective release time resulted in the shortage of nitrogen for crop growth later in the season. According to their comparative study between the conventional fertilizer and the rosin-coated fertilizer, a 5% (or 2%) increase in the rice crop yield was achieved when the four separate applications of the conventional fertilizer were replaced by a single application of the same amount (or 80%) of the rosin-coated fertilizer.
Figure 11. Comparison of calculated fractional release of urea with experimental data. (Release rates are calculated from the model equation.)
Two classes of release pattern were observed from single-pellet experiments. For urea particles having a coating thickness less than about 50 µm, immediate and relatively rapid release of urea through a few or many microscopic pores originally present in the coating occurred, while for the coating thickness greater than 50 µm, there was an extended period of time with little or no release before a sudden and rapid release through holes formed in the coating occurred. Also, a linear relationship between the hole formation time and the coating thickness was observed in case of coating thicknesses greater than 50 µm. After the formation of holes in the coating, the release of urea was nearly completed within 7 days indicating that individual granules cannot give sustained release of urea by themselves. The release phenomenon from a single pellet was described accurately by the mathematical model developed on the basis of experimental results. This model was then applied to real systems by considering the coating weight distribution for individual granules to obtain good agreement between model predictions and experimental results. Coating weight distribution in this case followed a Gaussian form, and the variance of this distribution was found to be linearly dependent upon the amount of coating.
Conclusions
Nomenclature
In this study, release experiments for rosin-coated urea granules were conducted in a water bath at 30 °C and the corresponding mathematical model has been developed on the basis of the experimental results. The release mechanism was also elucidated from singlepellet release experiments.
A ) coating area, cm2 Ao ) cross-sectional area of orifice, cm2 C ) urea concentration, mol/cm3 D ) diffusivity of urea, cm2/s D′ ) lumped parameter defined as DAo/A, cm2/s Mt ) quantity of released urea at time t, g
Ind. Eng. Chem. Res., Vol. 35, No. 1, 1996 257 Mz ) amount of urea delivered at zero order, g M∞ ) quantity of released urea at infinite time (initial loading), g m ) average coating weight, g S ) solubility of urea, g/cm3 t ) release time, s tz ) zero-order release time, s V ) interior volume of coating, cm3 Greek Letters δ ) coating thickness, cm πs ) osmotic pressure at saturation, Pa Fp ) density of urea particle, g/cm3 σ ) standard deviation of the coating weight distribution, g Subscripts cr ) critical o ) orifice s ) saturation z ) zero order
Literature Cited Baker, R. W.; Lonsdale, H. K. Controlled Release: Mechanisms and Rates. In Controlled Release of Biologically Active Agents; Tanquary, A. C., Lacey, R. E., Eds.; Plenum Press: New York, 1974; Vol. 47. Brown, R. G.; Hwang, P. Y. C. Introduction to Random Signals and Applied Kalman Filtering; John Wiley & Sons, Inc.: New York, 1992. Davis, C. H., Ed. Controlled-Release Fertilizers. TVA Bulletin Y-107; Oct 1976; pp 21-31.
Fan, L. T.; Singh, S. K. Controlled Release: A Quantitative Treatment; Springer-Verlag: Berlin, 1989. Katchalsky, A.; Curran, D. F. Nonequilibrium Thermodynamic in Biophysics; Harvard University Press: Cambridge, MA, 1967. Langer, R. Polymeric Delivery Systems for Controlled Drug Release. Chem. Eng. Commun. 1980, 6, 1-48. Lunt, O. R. Controlled-Release Fertilizers: Achievements and Potential. J. Agric. Food Chem. 1971, 19, 797-800. Paul, D. R. Polymers in Controlled Release Technology. In ACS Symp. Ser. 1976, 33, 1-14. Peppas, N. A. Mathematical Models for Controlled Release Kinetics. In Medical Applications of Controlled Release Technology; Langer, R. S., Wise, D., Eds.; CRC Press: Boca Raton, FL, 1984; Vol. 2. Polowinski, S.; Szosland, L.; Szumilewicz, J.; Polowinska, A.; Pierzchlewska, A. Controlled-Release Model Systems. Br. Polym. J. 1990, 23, 241-244. Ritger, P. L.; Peppas, N. A. A Simple Equation for Description of Solute Release: I. Fickian, and Non-Fickian Release from Nonswellable Devices in the Form of Slabs, Spheres, Cylinders or Discs. J. Controlled Release 1987a, 5, 23-36. Ritger, P. L.; Peppas, N. A. A Simple Equation for Description of Solute Release: II. Fickian and Anomalous Release from Swellable Devices. J. Controlled Release 1987b, 5, 37-42. Theeuwes, F. Elementary Osmotic Pump. J. Pharm. Sci. 1975, 64, 1987-1991.
Received for review March 7, 1995 Revised manuscript received August 10, 1995 Accepted September 20, 1995X IE950162H X Abstract published in Advance ACS Abstracts, December 1, 1995.
