controlled drug release of oral dosage forms

CONTROLLED DRUG RELEASE OF ORAL DOSAGE FORMS

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CONTROLLED DRUG RELEASE OF ORAL DOSAGE FORMS

Professor JEAN-MAURICE VERGNAUD Faculty of Sciences, University of Saint-Etienne, France

ELLIS HORWOOD NEWYORK LONDON TORONTO SYDNEY TOKYO SINGAPORE

First published in 1993 by Ellis Horwood Limited Market Cross House, Cooper Street, Chichester, West Sussex, PO19 IEB, England A division of Simon & Schuster International Group

0 Ellis Horwood Limited, 1993

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form, or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission, in writing, from the publisher

Printed and bound in Great Britain by Bookcraft, Midsomer Norton

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A catalogue record for this book is available from the British Library

ISBN O-13-1749564

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Available from the publisher

Table of contents

PREFACE . . . . . . . . . . . . . . . . . 0 . . . . . . . . . . . . m . . . . . . . . . . . . m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 THE DIFFUSION EQUATIONS AND BASIC CONSIDERATIONS ............... 1.1 Introduction.. ................................................................................

1 .I .I Process of diffusion.. ....................................................... 1 .I .2 Diffusion of a substance through a polymer.. ......................

Polymer in the rubbery state (Case I). ................................ Polymer in the glassy state (Case II). ................................. Absorption of liquid in Case Ill.. ........................................

1 .l .3 Steady and non-steady conditions.. ................................... 1 .I .4 Initial conditions.. ............................................................ 1 .I .5 Boundary conditions.. ...................................................... 1 .I .6 Volume of the surrounding atmosphere and partition factor . .

i.2 Equations of diffusion for various shapes .......................................... 1.2.1 Equations of diffusion for a thin sheet.. .............................. 1.2.2 Equations of diffusion for a rectangular parallelepiped ..........

Isotropic materials.. ......................................................... Anisotropic materials.. .....................................................

1.2.3 Cylinder of infinite and finite length.. ................................. Cylinder of infinite length.. ............................................... Cylinder of finite length.. ..................................................

1.2.4 Radial diffusion in a sphere.. ............................................. 1.3 Methods of solution when the diffusivity is constant.. .......................

1.3.1 Kinds of solution ............................................................. 1 .3.2 Method of separation of variables.. .................................... 1.3.3 Method for the Laplace transform.. .................................... 1 .3.4 Method of reflection and superposition.. ............................

Plane source ................................................................... Reflection at a boundary.. ................................................ Extended initial distribution of the substance.. ....................

2 MATHEMATICAL TREATMENT OF DIFFUSION IN A PLANE SHEET ......... 21 2.1 Introduction.. ................................................................................ 21

. . . XIII

1 1 2 2 2 3 3 3 4 4 5 6 6 7 7 8 9 9 9

10 10 10 11 15 17 17 19 19

ii Table of contents

2.2 Non-steady state with a high coefficient of matter transfer on the surface and an infinite volume of the surrounding.. ............................

2.2.1 Uniform initial distribution in the sheet.. ............................. 2.2.2 Initial distribution f(x) in the sheet of thickness L.. ..............

Two different media.. ...................................................... The two media are identical.. ............................................

2.2.3 Non-steady state with a membrane with a uniform initial distribution and surface concentration different.. .................

2.3 Non-steady state with a finite coefficient of matter transfer on the surface.. .......................................................................................

2.4 Non-steady state diffusion in a stirred solution of limited volume ......... 2.4.1 Absorption of diffusing substance by the sheet.. ................. 2.4.2 Desorption of diffusing substance from the sheet in the

solution .......................................................................... 2.5 Steady-state with a membrane.. ......................................................

2.5.1 High value of the coefficient of matter transfer on the surf ace ...........................................................................

2.5.2 Case of finite value of the coefficient of matter transfer ....... 2.5.3 Composite membrane ...................................................... 2.5.4 Membrane separating gases or vapour.. .............................

3 MATHEMATICAL TREATMENT OF DIFFUSION IN AN ISOTROPIC RECTANGULAR PARALLELEPIPED .......................................................

3.1 Introduction.. ................................................................................ 3.2 Isotropic rectangular parallelepiped with a constant diffusivity ...........

3.2.1 Infinite coefficient of matter transfer on the surface. Constant concentration on the surface.. ............................. Use of trigonometrical series.. ........................................... Use of error function.. ......................................................

3.2.2 Finite coefficient of matter transfer on the surface.. ............ 3.3 Effect of the thickness of the sheet.. ...............................................

4 MATHEMATICAL TREATMENT OF RADIAL DIFFUSION IN A SPHERE..... 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Solid sphere in non-steady state with constant diffusivity . . . . . . . . . . . . . . . . . .

4.2.1 Infinite coefficient of matter transfer on the surface . . . . . . . . . . . . 4.2.2 Finite coefficient of matter transfer on the surface . . . . . . . . . . . . . . 4.2.3 Diffusion between a sphere and a well-stirred surrounding

atmosphere of finite volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Hollow sphere in non-steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Hollow sphere in steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.1 Hollow sphere with a constant concentration on each surface.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4.2 Hollow sphere with a constant concentration of the internal surface and a finite coefficient of matter transfer on the external surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 MATHEMATICAL TREATMENT OF DIFFUSION IN CYLINDERS . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Non-steady state with a solid cylinder of infinite length . . . . . . . . . . . . . . . . . . . . . .

5.2.1 Constant concentration on the surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Finite coefficient of matter transfer on the surface . . . . . . . . . . . . . . 5.2.3 Solid cylinder in a well-stirred surrounding of finite volume...

22 22 31 31 32

34

37 40 41

43 45

45 46 47 48

49 49 50

50 51 52 55 58

59 59 60 60 62

66 69 71

72

73

75 75 77 77 81 86

Table of contents

5.3 Non-steady state with a solid cylinder of finite length.. ...................... 5.3.1 Constant concentration on the surface ............................... 5.3.2 Finite coefficient of matter transfer on each surface ............

5.4 Non-steady state with a hollow cylinder of infinite length.. ................. 5.4.1 Surface concentration constant and equal on each surface ... 5.4.2 Constant concentrations on each surface.. .........................

5.5 Steady state with a hollow cylinder of infinite length.. ....................... 5.5.1 Constant concentrations on each surface.. ......................... 5.5.2 Constant concentration on the internal surface and a finite

coefficient of matter transfer on the external surface .......... 5.5.3 Composite hollow cylinder ................................................

5.6 Conclusions ..................................................................................

6 NUMERICAL ANALYSIS WITH ONE-DIMENSIONAL DIFFUSION THROUGH A PLANE SHEET ................................................................

6.1 Introduction.. ................................................................................. 6.2 Diffusion through a sheet with constant diffusivity.. ..........................

6.2.1 Infinite coefficient of matter transfer on the surface ............. 6.2.2 Finite coefficient of matter transfer on the surface ..............

6.3 Diffusion through a sheet with concentration-dependent diffusivity ..... 6.3.1 Infinite coefficient of matter transfer on the surface.. .......... 6.3.2 Finite coefficient of matter transfer on the surface.. .............

6.4 Membrane separating two different media.. ...................................... 6.4.1 Infinite coefficient of matter transfer on each surface .......... 6.4.2 Finite coefficient of matter transfer on each surface .............

6.5 Diffusion between two different sheets ........................................... 6.5.1 Constant diffusivities.. ..................................................... 6.5.2 Concentration-dependent diffusivities.. ..............................

6.6 Transfer with special conditions.. .................................................... 6.6.1 Programmation of temperature.. ........................................ 6.6.2 Programmation of the concentration in the surrounding ........

7 NUMERICAL ANALYSIS WITH A RECTANGULAR PARALLELEPIPED, AND A THREE-DIMENSIONAL TRANSFER.. ..........................................

7.1 Introduction.. ................................................................................ 7.2 Transfer through a rectangular parallelepiped with a constant

concentration on the surface.. ......................................................... 7.2.1 Constant diffusivity.. ........................................................ 7.2.2 Concentration-dependent diffusivity.. .................................

7.3 Transfer through a rectangular parallelepiped with a finite coefficient of matter transfer on the surface.. ...................................................

7.3.1 Constant diffusivity.. ........................................................ 7.3.2 Concentration-dependent diffusivities, and finite coefficient

of matter transfer on the surface.. ..................................... 7.4 Transfer with special conditions.. .....................................................

7.4.1 Anisotropic material.. ....................................................... 7.4.2 Programmation of temperature.. ........................................ 7.4.3 Programmation of the concentration in the surrounding ........

8 NUMERICAL ANALYSIS WITH A RADIAL TRANSPORT WITHIN A SPHERE.. ..........................................................................................

. . . 111

90 90 93 95 97 98 99 99

101 101 102

105 105 106 107 110 113 113 114 118 118 119 122 122 125 126 126 127

129 129

131 132 135

137 137

142 146 146 146 147

149 8.1 Introduction.. . . . . . . . . . . . . . . . . . . . . . . . ,.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . 149

iv Table of contents

8.2 Radial diffusion through a sphere with constant diffusivity ..,.............. 8.2.1 Infinite coefficient of matter transfer on the surface . . . . . . . . . . . . 8.2.2 Finite coefficient of matter transfer on the surface . . . . . . . . . . . . . .

8.3 Radial diffusion through a sphere with concentration-dependent diffusivity . . . . . . . . . . . . . . . . . . e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.3.1 Infinite coefficient of matter transfer on the surface . . . . . . . . . . . . 8.3.2 Finite coefficient of matter transfer on the surface . . . . . . . . . . . . . .

8.4 Hollow sphere with constant concentration on the internal surface, and constant diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8.4.1 Infinite coefficient of matter transfer on the external surf ace.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . .

8.4.2 Finite coefficient of matter transfer on the external surface.. 8.5 Hollow sphere with constant concentration on the internal surface,

and concentration-dependent diffusivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Infinite coefficient of matter transfer on the external

surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Finite coefficient of matter transfer on the external surface..

8.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ,...... . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 NUMERICAL ANALYSIS WITH CYLINDERS.. ........................................ 9.1 Introduction.. ................................................................................ 9.2 Solid cylinder of infinite length.. ......................................................

9.2.1 Constant diffusivity ......................................................... 9.2.2 Concentration-dependent diffusivity ..................................

9.3 Hollow cylinder of infinite length.. ................................................... 9.3.1 Constant diffusivity.. ....................................................... 9.3.2 Concentration-dependent diffusivity.. ................................

9.4 Solid cylinder of finite length.. ......................................................... 9.4.1 Constant diffusivity.. ....................................................... 9.4.2 Concentration-dependent diffusivity.. ................................

9.5 Conclusions ..................................................................................

10 DRUG DELIVERY FROM DOSAGE FORMS CONSISTING OF A DRUG DISPERSED IN A NON-ERODIBLE POLYMER.. .......................................

IO. 1 Introduction and definitions.. ........................................................... 10.1 .I Problems of the drug passing through the body.. ................. 10.1.2 Pharmacokinetics, pharmacodynamics and

biopharmaceutics.. .......................................................... 10.1 .3 Conventional dosage forms .............................................. 10.1 .4 Oral therapeutic systems.. ................................................ IO. 1.5 Simple monolithic devices with a polymer matrix.. .............. 10.1.6 Processes with a double matter transfer.. ...........................

10.2 Drug-Eudragit sheet in gastric liquid.. ............................................... 10.2.1 Introduction.. .................................................................. 10.2.2 Theoretical aspects.. ........................................................ 10.2.3 Experimental.. ................................................................. 10.2.4 Results obtained with Eudragit as polymer matrix.. .............. 10.2.5 Conclusions with Eudragit as polymer matrix.. .....................

10.3 Drug-Carbopol sheet in gastric liquid.. .............................................. 10.3.1 Introduction.. .................................................................. 10.3.2 Theoretical aspects .......................................................... 10.3.3 Experimental.. ................................................................. 10.3.4 Results with Drug-Carbopol devices.. .................................

150 150 154

155 156 157

159

160 161

162

162 163 164

167 168 169 173 176 177 177 179 181 181 193 196

199 199 199

203 205 206 208 212 215 215 216 219 221 224 225 225 226 226 227

Table of contents V

10.3.5 Conclusions with Drug-Carbopol devices.. ........................... 10.4 Effect of pH on drug release.. ..........................................................

10.4.1 Introduction ..................................................................... 10.4.2 Theoretical aspects.. ........................................................ 10.4.3 Experimental.. .................................................................. 10.4.4 Results with a Drug-Eudragit sheets.. ................................ 10.4.5 Conclusions.. ...................................................................

10.5 Spherical Drug-Eudragit beads in gastric liquid.. ................................. 10.5.1 Introduction.. ................................................................... 10.5.2 Theoretical aspects .......................................................... 10.5.3 Experimental.. .................................................................. 10.5.4 Results with Drug-Eudragit spheres.. .................................. 10.5.5 Conclusions.. ...................................................................

11 DRYING OF DOSAGE FORMS MADE OF A DRUG DISPERSED IN A POLYMER MATRIX ............................................................................

1 1 .I Introduction.. ................................................................................ 11.2 Drying dosage forms in a surrounding atmosphere of infinite volume . .

1 1 .2.1 Theoretical.. ................................................................... 1 1 .2.2 Experimental ................................................................... 11 .2.3 Results with a constant temperature.. ............................... 1 1.2.4 Effect of temperature.. .....................................................

1 1 .3 Drying with a programmed temperature ......................................... 1 1 .3.1 Theoretical.. ................................................................... 1 1 .3.2 Experimental.. ................................................................. 1 1 .3.3 Results with a programmed temperature.. .........................

11.4 Drying in a surrounding atmosphere of finite volume.. ........................ 1 1.4.1 Theoretical of drying in a surrounding atmosphere of finite

volume.. ......................................................................... 1 1 .4.2 Experimental.. ................................................................. 11 .4.3 Results.. .........................................................................

1 1 .5 Drying with a controlled vapour pressure.. ........................................ 11.5.1 Theory of the process with controlled vapour pressure ........ 1 1 .5.2 Simulation of the process.. ...............................................

1 1 .6 Conclusions.. ................................................................................

12 DRUG DELIVERY FROM DOSAGE FORMS CONSISTING OF A DRUG DISPERSED IN AN ERODIBLE POLYMER.. .............................................

12.1 Introduction.. ................................................................................ 12.2 Theoretical aspects ........................................................................

1 2.2.1 Diffusional process.. ........................................................ 12.2.2 Polymer erosion is the driving force.. .................................

1 2.3 Experiments.. ................................................................................ 1 2.4 Results.. .......................................................................................

12.4.1 Results with the diffusion process.. ................................... 12.4.2 Results with the erosion process.. .....................................

12.5 Conclusions.. ................................................................................

13 DOSAGE FORMS MADE OF A CORE AND SHELL, WITH AN ERODIBLE SHELL. CONSTANT RATE OF DELIVERY ..............................................

13.1 Introduction.. ................................................................................ 13.2 Theoretical aspects.. ...................................................................... 13.3 Experimental.. ...............................................................................

230 231 231 232 233 234 240 242 242 242 246 246 251

261 261 263 263 267 268 274 277 279 280 280 286

286 291 291 300 301 302 308

313 313 314 314 317 318 320 320 322 327

329 329 330 333

vi Table of contents

13.4 Results.. ....................................................................................... 13.4.1 Results with sodium salicycate.. ....................................... 13.4.2 Results with sulfanilamide.. ..............................................

13.5 Conclusions.. ................................................................................

14 DOSAGE FORMS MADE OF CORE AND SHELL, WITH A NON-ERODIBLE POLYMER.. .......................................................................................

14.1 Introduction.. ................................................................................ 14.2 Theoretical.. .................................................................................. 14.3 Experimental.. ............................................................................... 14.4 Results.. .......................................................................................

14.4.1 Data.. ............................................................................ 14.4.2 Validity of the model.. ...................................................... 14.4.3 Effect of parameters.. ...................................................... 14.4.4 Profiles of concentration.. ................................................

14.5 Conclusions.. .................................................................................

15 CONTROLLED RATE OF DELIVERY WHEN THE SOLUBILITY OF THE DRUG IS LOW, BY USING A SWELLING POLYMER.. ..............................

15.1 Introduction.. ................................................................................ 15.2 Dosage form with a polymer matrix and a swelling polymer.. ..............

1 5.2.1 Theoretical.. ................................................................... 15.2.2 Experimental.. ................................................................. 15.2.3 Results.. .........................................................................

15.3 Dosage forms with gelucire and a swelling polymer.. ......................... 1 5.3.1 Theoretical.. ................................................................... 15.3.2 Experimental.. ................................................................. 15.3.3 Results and discussion.. ....................................................

15.4 Dosage form with a core (swelling Polymer-Drug-Eudragit) and an erodible polymer.. ..........................................................................

15.4.1 Theoretical.. ................................................................... 15.4.2 Experimental.. ................................................................. 15.4.3 Results.. .........................................................................

15.5 Conclusions ..................................................................................

16 DOSAGE FORMS WITH A DRUG ATTACHED TO A POLYMER DISPERSED IN A NON-ERODIBLE POLYMER MATRIX ............................

16.1 Introduction.. ................................................................................ 16.2 Theoretical.. .................................................................................. 16.3 Experimental.. ............................................................................... 16.4 Results.. ....................................................................................... 16.5 Conclusions ..................................................................................

334 334 338 341

345 345 346 352 352 353 354 354 357 358

363 363 364 364 365 366 373 373 373 376

380 380 382 383 390

393 393 394 395 397 404

INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..~........................................................... 409

PREFACE

Therapeutic systems represent a new route for drug administration: as the drug is delivered continuously at a controlled rate over a predetermined period of time, uniform and constant blood level is achieved, smaller amount of drug is needed reducing the side effects, and the therapy is improved. Various people beyond the patient are concerned with this therapy, physicians and pharmacists in various areas of specialization, of course, but also bio-engineers and even workers in chemical engineering. Various oral therapeutic systems consist of a polymer matrix through which the drug is dispersed, and thus good knowledge of the matter transfers through the polymer is necessary when they are in contact with the gastric or the intestine liquid.These matter transfers being controlled by transient diffusion, the mathematical treatment of diffusion must be known when it is feasible in simple cases and especially for constant diffusivity. Morever, in complex cases and when the diffusivity is concentration-dependent, numerical methods with finite differences must be used instead of the mathematical treatment. Finally, in order to accustom the users with these ways of calculation, various mathematical or numerical models are built and tested in the study of different oral dosage forms. Thus a new way of working is developed, coupling the experiments with the models of the process, these experiments being performed in so called in-vitro tests which simulate the conditions in the body as much as possible.

The drug, by using this term in the sense of a biologically active substance, is a chemical compound administered to the patients organism, with which it develops a reciprocal interaction for therapeutic purposes. Generally, for many reasons, the drug is not used in the pure state. The supply form of presentation of the drug, or dosage form, is the complete medication. Conventional dosage forms consist of the drug, the active agent,

. VIII Preface

and auxiliary substances biologically inert, the excipients. The role of excipients is essentially of binding the drug, filling the dosage form in order to ensure the consistency and volume necessary for the patient use. When in contact with the gastric liquid, the drug is released from the dosage form used for the administration. Two factors determine the release of the drug: the solubility and the rate of dissolution. When the drug is released from the dosage form, it must pass through several barriers, before reaching the site of action. The driving force responsible for the transport of the drug through these membranes is the concentration gradient across the membrane, the process being controlled by diffusion. The amount of unchanged drug that is absorbed by the organism in a certain time and that arrives at the target site through the circulatory system, or bioavailability, depends on the dosage form, and thus may be altered by this dosage form. The drug conveyed by the bloodstream, leaving the intravascular compartment is distributed between extracellular and intracellular compartment where it can reach the receptors for drugs lying in the tissues. Finally the drug is eliminated either by chemical alteration of the molecule with formation of metabolites or by excretion via various organs. The time required for elimination of half the plasma content of the drug by metabolism or excretion, the biological half-time, is of high interest for the dosage regimen prescribed by the physician. An exact dosage regimen is of high importance when the concentration of the drug must be maintained constant in the tissues over a long period of time. Morever, the concentration of the drug must be kept between the median lethal dose (causing the deaths of 50% of experimental animals) and the median effective dose (effective in 50% of cases). The therapeutic index, equal to the ratio of these above concentrations defines the safety margin.

All conventional dosage forms made of a drug dispersed in excipients, release the drug according to the following pattern. The drug is very rapidly dissolved from the dosage fonn and quickly builds up to a maximum high concentration, which then falls exponentially with tune because of the first order absorption. The result is an undulating concentration of the drug in the stomach or intestine, as well as in the blood and tissues, where high concentrations with overdosages alternate with low concentrations and underdosages. The limitations of conventional dosage forms made of drug and excipients appear then since they cause problems in maintaining therapeutic drug levels over only brief durations of tune :

(i) The fluctuating drug levels with conventional dosage forms lead to an insufficient efficacy of therapy provoking an excessive use of the drug.

Preface ix

(ii) Overdosage appearing after dissolution of the drug may be responsible for a high frequency of side effects, leading to iatrogenic damage. (iii) High frequency of administration of conventional dosage forms is limited by the reliability of the patient and the patient compliance (omission, wrong frequency) (iv) A potent drug may largely lose its therapeutic efficacy through improper formulation, and thus a pharmacologically active substance is not necessarily an effective drug.

Oral dosage systems able to release the drug at a constant rate for a given time period are thus of mterest. The result is then a constant uniform concentration of drug in blood and tissues over a given period of time, with the following advantages : (i) Significant smaller amounts of drug are generally prescribed with a therapeutic system of drug delivery. (ii) The reduced amount of drug administered reduces the problems of side effects, improving the safety of therapy. (iii) The patient compliance is usually better with these types of dosage forms, as the frequency of administration is considerably lower.

Simple oral dosage forms capable of controlling the release of the drug are often and easily obtained with monolithic devices where the drug is dispersed in a biocompatible polymer. This polymer which cau be either biodegradable or non degradable, plays the role of a polymer matrix. Not only the polymer brings the consistency to the dosage form, but also it controls the release of the drug. The process is generally as follows: the liquid (gastric liquid or intestine liquid) enters the polymer, dissolves the drug and enables the drug to leave out the dosage form through the liquid located in the dosage form. The matter transfers for the liquid and for the drug are controlled by transient diffusion, with concentration-dependent diffusivities, the diffusivity of the drug depending on the concentration of the liquid in the dosage form. The release of the drug being controlled by transient diffusion, exhibits a rather high rate at the beginning of the process which decreases with time in an exponential way. These dosage forms are very simple to prepare and rather inexpensive, but the process of release is controlled by diffusion, and the rate of release is far from being constant.

The drug delivery from the dosage form is studied by using in-vitro tests, these in- vitro tests being built up in such a way that they simulate as much as possible the story in the stomach or intestine of the patient. These in-vitro tests are very useful for many

X Preface

reasons, and the most obvious are only given : (i) The conditions of the in-vitro test are very well defined and standardized, enabling comparisons between various results. (ii) They are easy to perform, and the effect of each parameter can be analysed separately. (iii) In contrast with the in-vitro test, the in-vivo test is far more complex, as this latter is subject to a variety of influences that differ greatly among individuals.

There are several objectives in this book devoted to the study of the process of matter transfers in oral dosage forms with a polymer matrix able to control the release of the drug. As the driving force for the matter transfers of the liquid and the drug through the polymer is the gradient of concentration, the process is controlled by transient diffusion. Some emphasis is thus placed upon the mathematical treatment of diffusion in solids of various shapes, when the process is so simple that an analytical solution exists. As very often the process of matter transfers is rather complex, it must be studied by using numerical methods with finite differences. Finally, various examples are described by considering simple oral dosage forms with either a non-erodible or an erodible polymer matrix, and with more complex systems consisting of a core and shell. These studies are made by using the method coupling experiments with short tests and long real tests and modelling of the process.

The book is divided in three parts with sixteen chapters : The first part presents an overview of the mathematical treatment of diffusion

through a polymer in the elastomeric state. Various shapes are considered for the solid : thin plane sheets, rectangular parallelepiped, cylinders and spheres. In order to help the readers understanding, some emphasis is placed upon the conditions in which the mathematical treatment is feasible constant diffusivity, uniform initial concentration, simple boundary conditions. For people wanting to improve their background knowledge of the mathematical treatment of diffusion, various examples are described in a didactic way in the first five chapters. Special consideration is given to the operational conditions : with a very high volume of the liquid in which the dosage form is immersed, or with a finite volume of this liquid ; with a very high coefficient of matter transfer on the surface leading to a constant concentration on the surface, or with a finite coefficient of matter transfer on the surface.

- In chapter 1, general equations of diffusion are given for various shapes of the dosage form. and basic considerations are described.

Preface xi

- In chapter 2, the mathematical treatment of diffusion is shown in various cases with a plane sheet and mono-directional diffusion. - In chapter 3, the mathematical treatment of diffusion is given with a rectangular parallelepiped and three-dimensional diffusion. - In chapter 4, radial diffusion through spheres is studied. - In chapter 5, cylinders of infinite and finite lengths are considered with radial diffusion in the first case and radial and longitudinal diffusion in the second case.

The second part is devoted to numerical treatment of diffusion, in order to accustom the readers to this new and powerful way of working. This method is very useful, as very often no analytical solution can be obtained from the mathematical treatment, because of the complexity of the process : double matter transfers of the liquid and drug, concentration-dependent diffusivity. Explicit numerical methods with finite differences are developed, because of their easy use with microcomputers. Four chapters enable one to consider various shapes. - In chapter 6, plane thin sheets are considered and classical examples of numerical analysis are developed in the following simple cases : the diffusivity is either constant or concentration-dependent, while various values of the coefficient of matter transfer on the surface are given. - In chapter 7, numerical analysis is developed with a rectangular parallelepiped and a three-dimensional transfer. - In chapter 8, numerical analysis for the radial transfer through a sphere is presented. - In chapter 9, the matter transfers, either radial with long cylinders or radial and longitudinal with cylinders of finite length, are studied with the help of numerical analysis.

The third part examines various approches to industrial problems with practical purposes. A new method coupling experiments and modelling of the process is widely used.

Experiments are used for the following reasons : - to get deep knowledge of the process - to obtain the values of parameters, such as the diffusivities by using short tests - to test the validity of the models.

Modelling of the process is widely used, either with the mathematical treatment when the problem is simple, or with the numerical treatment when the process is complex.

Each of these different cases are discussed in chapters 10 to 16, working through the difficulties encountered with experiments and calculation.

xii Preface

- Chapter 10 concentrates on the drug delivery from simple dosage forms consisting of a drug dispersed in a non-erodible polymer. Two matter transfers are considered with the liquid entering the polymer, dissolving the drug and enabling the drug to leave the dosage form through the liquid located in the polymer. These two transfers are connected with each other, and the diffusivity of the drug depends on the liquid concentration. - Chapter 11 shows the complexity of the process of drying of dosage forms with a polymer matrix, the process being controlled not only by evaporation but also by diffusion of the liquid through the polymer. Various examples are described and the effect of factors such as the temperature or programmation of temperature, the pressure of the vapour in the surrounding atmosphere, is evaluated. - Chapter 12 discusses the problem of drug delivery from dosage forms made of a drug dispersed in an erodible polyme matrix. - Chapter 13 focuses on the interest of preparing dosage forms with constant rate of delivery. Typical dosage forms are presented with a core containing the drug dispersed in a polymer and with an erodible shell surrounding the core. - Chapter 14 deals with dosage forms made of a core and shell, where the core contains the drug dispersed in a non-erodible polymer and the shell is a non-erodible polymer. The effect of the relative thickness of the shell is of high interest. - Chapter 15 is devoted to special dosage forms able to deliver the drug from the dosage form when the drug is poorly soluble in the liquid. A swelling polymer is thus added in the erodible polymer matrix which helps the disintegration of the dosage form and thus dissemination of the drug in the liquid. As some polymers swell differently in gastric and intestine liquid, they allow the dosage form to deliver the drug partly in the stomach and intestine. - Chapter 16 examines the problem of dosage forms where the drug is attached to a polymer, this branched polymer being dispersed in a polymer matrix.

ACKNOWLEDGEMENTS

A large part of this book covers various applications and industrial problems, as many people working in industrial firms have influenced this work through industrial contracts. I am glad to thank them for their interesting cooperation.

Many colleagues and students have supported my efforts and brought contributions worth noting.

Deep gratitude is extended to my colleagues M. Rollet who showed me round the world of galenic pharmacy as well as J. Bardon and C. Chaumat. I am grateful for the collaboration of my colleague J. L. Taverdet in the work concerned with the preparation and studies of dosage forms. I give my best thanks to my colleagues J. Bouzon for his participation in numerical analysis and modelling of the process, and J. P. Montheard who dealt with the polymerization problems in chapter 16. I appreciate the kind help of H. Liu and J. Paulet.

My best appreciation is given to my students : Y. Armand, D. Bidah, N. Chaffi, A. Droin, A. Eddine, N. Farah, M. Kolli, N. Laghoueg, F. Magnard, P. Magron, Y. Malley, E.M. Ouriemchi, M. Saber, who did their best for preparating their Theses. Many thanks to D. Berthet for his efficient help in calculation and for his drawings, and to C. Cervantes, N. Fauvet, D. Ianna and M. Novais Da Costa for their competent typing of the manuscript.

1

The diffusion equations and basic considerations

1 .I INTRODUCTION

1 .l.l PROCESS OF DIFFUSION Generally diffusion is the process through which matter is transferred from one

place to another, resulting from random molecular motions. Of course, on the average, the

matter is transferred by diffusion from the region of higher to that of lower concentration

of the matter. The example of diffusion of a drop of dye in motionless water is a good

example. Transfer of heat by conduction is also due to random molecular motions transferring

kinetic energy, and there is some analogy between these two processes of matter and heat

transfers. The mathematical equation of heat conduction was established by Fourier in

1822. A few decades later, Fick in 1855, recognizing this analogy, put diffusion on a

quantitative basis by adopting the same equation. In an isotropic substance the rate of

transfer of diffusing substance through unit area of a section is proportional to the gradient

of concentration measured normal to this section.

(1.1) F= -D K . ax

where g X

is the gradient of concentration C of the substance along the x-axis of diffusion,

F is the rate of transfer per unit area of the section perpendicular to the x-axis,

and the coefficient D is called the diffusion coefficient of diffusivity.

The diffusion equations and basic considerations [Ch. 1

The term diffusivity will be used in the book. The negative sign arises because the substance is transported from higher to lower concentration of the substance.

If the rate of transfer of substance per unit area (or the flux) F, and the concentration of substance are expressed in terms of the same unit of quantity, e.g. gram, it is clear that the diffusivity D is independent of this unit and has dimensions :

(1.2) (length)2. (time) or cm2/s as the flux F is expressed by g/cm2.s, and the concentration C is g/cm.

1.1.2 DIFFUSION OF A SUBSTANCE THROUGH A POLYMER When a dosage form is made of a drug dispersed in a polymer, the polymer playing

the role of a matrix, the whole process of diffusion is as follows. When the dosage form is in contact with a liquid, e.g. the gastric liquid, this liquid enters the polymer, dissolves the drug, and then enables the drug to diffuse out of the dosage form through the liquid located in the dosage form. Both these transfers are controlled by diffusion.

It is thus of interest to have a knowledge on the diffusion behaviour of various kinds of polymers.

Generally a polymer is in the glassy or in the rubbery state, depending on the temperature. Below the temperature of glassy transition Tg, the polymer is in the glassy state, and above this temperature it is in the rubbery state. Diffusion of a liquid differs notably when it goes through a glassy or a rubbery polymer. Segments of the polymer chain are continualy in motion, in the same way as Brownian motion for gases, creating voids. As the volume of these voids is of the same magnitude as the volume of a molecule of liquid, these motions enable the molecule of liquid to go through the polymer. Polymers have a wide spectrum of relaxation times associated with these structural changes. An increase in temperature or in concentration of liquid generally enhances the motion of the polymer segments and decreases the relaxation time.

POLYMER IN THE RUBBERY STATE (CASE I)

A polymer in the rubbery state responds rapidly to changes in its condition. The polymer chains adjust very quickly to the presence of the molecule of liquid. The rate of diffusion of the liquid is much less than that of relaxation of the segments of the polymer. In this case, the diffusion is Fickian (case 1). The amount of diffusing substance absorbed (or desorbed) at time t can be expressed in terms of time by the following relation :

(1.3) M,=k. fi

Sec. 1.11 Introduction 3

when Mt is much lower than the corresponding amount after infinite time a. where k is a constant depending on &, the shape of the polymer and the diffusivity.

POLYMER IN THE GLASSY STATE (CASE II)

In a polymer in the glassy state, the stress may be slow to decay after this polymer has been stretched. Thus, the relaxation process is very slow compared with the rate of diffusion. In this case, called case II, the liquid diffuses through the polymer with a constant velocity showing an advancing front which marks the penetration limit of the liquid. Behind this advancing front of the liquid, the polymer may turn into swollen gel or rubber polymer, while ahead of this front, the polymer free of liquid is in the glassy state. The amount of liquid absorbed at time t is expressed in terms of time t by the following :

(1.4) M,=k.t

ABSORPTION OF LIQUID IN CASE III

When the rates of diffusion of the liquid and of the relaxation of the polymer are of the same order of magnitude, anomalous or non-Fickian diffusion is observed. This system lies between case I and case II and the amount of liquid absorbed at time t is given in terms of time by the expression

(1.5) Mt=k.tn

where n is between h and 1

The case I system is characterized by n = 0.5 and the case II system by n = 1, by considering eqn (1.5).

1 .I .3 STEADY AND NON-STEADY CONDITIONS Steady or stationary conditions are reached when the concentration of diffusing

substance within the solid does not depend on time. The mathematical condition is :

(1.6) g=O

The concentration of substance depends only on position, and the concentration distribution of the substance through the solid is constant.

Non-steady or transient conditions are obtained when the concentration of diffusing substance within the solid depends on position and time. The mathematical condition is :

(1.7.) g,o


Steady conditions are reached only in a few cases, when the solid through which the

liquid diffuses is considered as a membrane. Two cases are of high interest : the plane

membrane, the spherical membrane. Steady conditions can be obtained when constant

concentrations Ci and Co are maintained on each surface. At the beginning of the process,

the non-steady diffusion of the liquid takes place. After a given time, a steady state is reached in which the concentration of diffusing substance remains constant at all points

within the membrane.

1 .1.4 INITIAL CONDITIONS The initial conditions represent the concentration distribution of diffusing substance

within the solid and in the surrounding, at time t = 0, before the process starts.

For a simple dosage form, the concentration distribution of the drug is generally

uniform. In special dosage forms made of a core and shell, the drug concentration may be

uniform throughout the core and zero in the shell.

1 .1.5 BOUNDARY CONDITIONS The boundary conditions express the concentration of diffusing substance on the

surface of the solid.

When the drug does not diffuse out of the dosage form, there is no transfer through

the external surface of the dosage form. This condition is mathematical by expressed by

writing that the rate of transfer through the surface is zero. The following is thus obtained:

(1.8) -D.g=O orsimply g=O onthesurface

When the substance (liquid, drug) is transferred through the external surface of the

dosage form, the above equation becomes :

(1.9) %o 2X

on the surface

Two cases are of interest :

(i) When there is a finite coefficient of matter transfer through the external surface of

the solid, h. The rate at which the diffusing substance is transferred per unit area of

the external surface is thus expressed by :

(1.10) t>O - J$js=h(Cs-C.x~ on the surface

where C, and C,,, are the concentration on the surface and in the surrounding,

Sec. 1.11 Introduction 5

respectively.

is the gradient of concentration next to the surface

and D is the diffusivity

Of course, the diffusing substance enters or leaves the dosage form, depending on the respective value of C, and C,xt.

(1.11) cs > Gxt the substance leaves the dosage form

(1.11) c, < text the substance enters the dosage form

(ii) When the coefficient of matter transfer through the external surface of the solid is very high (compared with the diiusivity D), the concentration on the surface C, can be considered as constant during the whole process

(1.12) t>O C, = constant

REMARK - Instead of using C,, in eqn (l.lO), it is better to use the concentration on the surface which is at equilibrium with the surrounding atmosphere, C,. Eqn (1 .lO) expresses that the rate at which the substance goes through the external surface is constantly equal to the rate at which the substance is brought to this surface by internal diffusion. This rate is also proportional to the difference between the actual concentration C, and the concentration on the surface which is necessary to maintain equilibrium with the surrounding, C, or (C,,,).

1.1.6 VOLUME OF THE SURROUNDING ATMOSPHERE AND PARTITION FACTOR

It is of interest to consider the volume of the surrounding atmosphere as compared with the volume of the dosage form.

Moreover, there is often a partition factor, K, meaning that at equilibrium the concentration of diffusing substance is K times in the solid than in the surrounding atmosphere.

When the volume of the surrounding atmosphere is much higher than that of the dosage form, this volume can be considered as infinite. This means that the concentration of diffusing substance in the surrounding atmosphere does not vary. For instance, it

6 The diffusion equations and basic considerations [Ch. 1

remains zero (or negligible) for the drug leaving the dosage form, and constant for the liquid entering the dosage form.

In some cases, the volume of the surrounding atmosphere is not much higher than that of the dosage form. This means that the concentration of drug in the surrounding atmosphere increases.

The ratio of the volumes of the surrounding atmosphere and the dosage form is expressed by :

(1.13) a = Kvsu- * dos form

where K is the partition factor for the drug.

1.2 EQUATIONS OF DIFFUSION FOR VARIOUS SHAPES

Equations of diffusion are considered for various shapes of the dosage forms, and some emphasis is placed in the case of the thin sheet by developing calculation. The following shapes are examined successively : thin sheet, rectangular parallelepiped, cylinder of infinite and finite length, and sphere.

1.2.1 EQUATION OF DIFFUSION FOR A THIN SHEET The thin sheet of thickness dx perpendicular to the direction of diffusion is

considered (Fig. 1.1) with the area A through which the substance diffuses along the x- axis. The matter balance is evaluated during the time dt within this small volume Adx.

F x+dx X

Fig. 1.1. Diffusion of a substance through a thin sheet of thickness dx

Sec. 1.21 Equations of diffusion for various shapes 7

The rate at which the diffusing substance enters the sheet of area A is F,.A, and the change in the amount of substance during the time dt is A (F, - Fx+& dt. This change in the amount of substance is responsible for a change in the concentration within the sheet which can be written as follows :

(1.14) A. (Fx- Fx+,& . dt = A.dx.dC

The difference F, - Fx+& is given by :

(1.15) Fx- Fx+dx= -

and the matter balance becomes :

(1.16) aF, ac -ax=Ji

where the concentration C is a function of space and time.

As the flux F, of diffusing substance is expressed in terms of the diffusivity D and of the gradient of concentration eqn (l.l), the above equation can be rewritten in the final form :

(1.17) G=&(D. $j

When the difusivity is constant, this equation simplifies

(1.18) a% $=D.? with constant D

1.2.2 EQUATIONS OF DIFFUSION FOR A RECTANGULAR PARALLELEPIPED

ISOTROPIC MATERIALS

A rectangular parallelepiped whose sides are parallel to the axes of coordinates and of length dx, dy, dz is drawn (Fig. 1.2). The matter balance within this small parallelepiped can be evaluated in he same way as for the sheet by considering the diffusion along the three axes. Eqn (1.16) is thus rewritten as follows :


dx

Fig. 1.2. Diffusion of a substance through a small rectangular parallelepiped of

sides dx, dy, dz

(1.19) ac aF, aF, a&

-Ji=~+~+az

where Fx represents the flux of diffusing substance along the x-axis.

This equation becomes, by replacing each flux F by its value given in eqn (1 .l) :

l1.20) $=&ID. $$+$[D.$)+=$. $1

where the diffusivity D may be concentration-dependent. In the case of a constant diffusivity, the equation reduces to :

ANISOTROPIC MATERIALS

Anisotropic materials have different diffusion properties in different directions. Some examples are given with crystals, wood, and polymer films in which the molecules have been oriented. In this case, the direction of flow of diffusing substance at any point is not normal to the surface of constant concentration through this point. Eqn (1.1) must be replaced by the assumption :

Sec. 1.21 Equations of diffusion for various shapes 9

(1.22) -F,=D,,. E+D,*. $+DIS.$

showing that F x depends not only on the gradient but also on $ and g.

They are three principal axes of diffusion, and three principal diffusivities Dx, Dy, Dz, in the case where the principal axes of diffusion are the same as the x, y, z axes.

The equation of diffusion can thus be written :

or more simply when each principal diffusivity is not concentration-dependent.

(1.24)

1.2.3 CYLINDER OF INFINITE AND FINITE LENGTH

CYLINDER OF INFINITE LENGTH

The case of a cylinder of infinite length is easy to study, as the transfer of substance is radial only.

When the diffusivity is concentration-dependent, the equation of diffusion is as follows :

(1.25) -g- C=f.#.D.$)

and when the diffusivity is constant it becomes :

(1.26)

CYLINDER OF FINITE LENGTH

The diffusion is radial and longitudinal along the z-axis. When the diffusivity is concentration-dependent, the equation of diffusion results

from the superposition of the radial and longitudinal diffusion.

10 The diffusion equations and basic considerations ICh. 1

(1.27) ?$;.;(r.D.~j+~(D.$)

When the diffusivity is constant, this equation reduces to :

(1.28) a2c +- az2 1

Of course, in these two cases, the material is isotropic.

1.2.4 RADIAL DIFFUSION IN A SPHERE

When the diffusion is radial and the material is isotropic, the equation of diffusion is given by :

with a concentration-dependent diffusivity :

(1.29) G=$.g[D.r.$]

with a constant diffusivity :

(1.30) a2c 2 ac $=D. ar2+- [ 1 r?F 1.3 METHODS OF SOLUTION WHEN THE DIFFUSIVITY IS CONSTANT

1.3.1 KINDS OF SOLUTION

When the diffusivity is constant, general solutions of the equation of diffusion can be obtained for various initial and boundary conditions, when these latter conditions are not too complex.

Two standard forms can generally be obtained. The one, when the concentration of diffusing substance and the amount of substance transferred are expressed in terms of error functions and of related integrals. Because they converge rather quickly for small times, they are of great interest for evaluation during the early stages of diffusion, when the amount of substance transferred is low. The other, when the concentration of susbstance and kinetics of substance transferred are given in the form of trigonometrical

Sec. 1.31 Methods of solution when the diffusivity is constant 11

series. These series can be used for large values of time, when the amount of matter transferred as a fraction of the corresponding amount after infinite time is high, because these types of series converge very quickly in these conditions.

Three methods of solution of the diffusion equation can be used. (i) The method of separation of variables, which is widely used. It gives solutions

expressed in terms of trigonometrical series. (ii)The Laplace transform which is an operator method through which the partial

differential equations are transferred in ordinary equations. The two kinds of solutions can be obtained.

(iii)The method of superposition and reflection. Solutions are obtained in terms of error functions. In the case of a cylinder with radial transfer, series of Bessel functions are obtained instead of trigonometrical series.

1.3.2 METHOD OF SEPARATION OF VARIABLES By making the assumption that the variable x and t are separable, an attempt can be

made to find a solution for the partial differential equations of diffusion. For instance, in the case of a one-dimensional difusion through a sheet, upon

putting :

(1.31) C&t= c,. Ct

where C, and Ct are functions of x and t, respectively, the general equation of diffusion (1.18) becomes :

(1.32) 2. C,=D. Ct. s dx2

This equation can be rewritten after separation of variables :

(1.33) 1 dc, D d2C, -- q. z= c, * dx2

where the left-hand side depends on time only, and the right-hand side depends on x only.

The two ordinary equations are thus obtained


(1.34) &. ?=-A. D

(1.35) $ . KS + X2. D=O x dx2

the solutions of which are :

(1.36) Ct = Constant x exp ( - h2 Dt)

and

(1.37) C, = A. sin hx + B . cos hx

The solution (1.3 1) of eqn (1.18) is thus

(1.38) C&= C,. Ct = Constant (A . sin hx + B . cos hx) . exp ( - A2 Dt)

where A and B are constants.

Eqn (1.38) being a linear equation, a general solution is obtained by summing these solutions as follows :

(1.39) C,,,= %[A. sh-~h,x+B,. cosh,,x). exp(-(Dt) n=O

where A,,, B, and h, are constants.

The constants A,.,, B, and ?+, are determined by the boundary and initial conditions for each problem.

The simple problem of a sheet of thickness L with 0 < x < L is considered (Fig. 1.3). The diffusing substance is initially uniform throughout the sheet, and the coefficient of matter transfer on both surfaces x = 0 and x = L is so high that the concentration on these surfaces is zero as soon as the process starts.

The initial and boundary conditions are as follows :

(1.40) t=O O


in

X

0 L

Fig. 1.3. Diffusion of a substance out of a sheet 0 c x c L with a uniform initial concentration and zero concentration on the surfaces

(1.41) t>o x=0 c=o boundary conditions

andx=L

The boundary condition C = 0 for x = 0 necessitates

(1.42) B, = 0

since cos 0 = 1, while the other boundary condition C = 0 for x = L is satisfied by :

(1.43) sin LL = 0 = sin nx

andby:

(1.44) h,= r-r:

The initial condition can thus be written :

(1.45) Cx,o = gA,. sinn? for 0 c x L 1

By multiplying both sides of this equation by sin n y and integrating from 0 to L it

becomes :


J L o sinn~dx=Ar. J L (1.46) C,., ,,sin~. sin.Fdx+ . . . . A,, J L 0 sin2~dx+ J L +A, . prcx . nrcx 0 sm-. slnLdx+ . . . . L

As

sinp. sinq=;[cos(p-q)-cos{p+q)l

sin2p=; 1 l-cos2 p I

it becomes obvious that :

J L . pxx . n7tx 0 sm-. SlnLdx = 0 L forp#n J

L 2n7cx

0 hlpX=; forp = n

Moreover,

J L 0 sin~clx = -&[l-cosnrr] The even terms of n make A,, vanish, and only the odd terms of n are considered : n

= 1, 3, 5 The constant A can thus be expressed in this way with odd values of n, or 2n + 1 :

4 cx.0 4 tin 147 A=(2n+ 1)rr = (2n+ 1)7t

The final solution for the concentration of substance within the sheet 0 c x < L is thus given by the trigonometrical series :

(1.48) 4Ci, O 1 . (2n+l)rrx

Cx,t= x c -. sm +p+ l L !


1.3.3 METHOD OF THE LAPLACE TRANSFORM

The Laplace transform is a mathematical treatment through which a partial differential equation expressed in terms of the variable x and t is transformed in an ordinary equation expressed in terms of the variable x only. Application of the Laplace transform to the diffusion equation removes the time variable, leading to an ordinary equation with space variables.

The Laplace transform f(p) of a function of time f(t) for positive values of time is given by :

ca

(1.49) T(p) = o f(t) . exp (- pt) . dt

where p is a number which is chosen high enough to make the integral converge.

By integrating (1.49), it is thus possible to obtain the Laplace transform of a function. A few simple examples are given :

f(t) = 1 f(p) = o exp I b-P4 f(P) = ;

ea f(t)=exP(-at) f(p)= uexp-(p+a)t.dt f(p)=---&

Tables of Laplaces transfonns are available in some books of mathematics Consider a plane sheet of thickness 2 L, with - L < x < L, initially free from liquid,

with a constant concentration on each surface. The initial and boundary conditions are :

(1.50) t= 0 -L


by exp ( - pt) and integrating with respect to t from 0 to 00, there is :

-aC O 2 oxexp(-pt)dt = D. ac exp ( - pt) dt

O ax2

Integrating by parts the left-hand side

-ac 00

ox.exp(-pt)dt=[C.exp(-pt)]i+p oC.exp(-pt)dt=p.C

as the term in bracket vanishes at t = 0 because of the initial conditions and at t = 00 because of the exponential term.

The right-hand side is written as follows :

by interchanging the orders of integration and differentation.

The equation for the Laplace transform c is :

(1.53) e= q2. C dx2

(1.54) q2+

It is more convenient to use the condition at the midplane with eqn.(l.S2) and to consider only half the thickness of the sheet, 0 I x I L, instead of the condition x = L, C = C-J for t > 0.

The boundary condition C = Cm for x = L and t > 0 gives the following in terms of Laplace transform :

ca

oC, .exp(-pt)&=$ x= L


The equation at the midplane becomes :

(1.56) $=O x= 0

The solution of eqn.(1.53) satisfying the conditions (1.55) and (1.56) is given by eqn. (1.1)

This hyperbolic function can be expressed in terms of negative exponentials and can thus be expanded is a series by using the binomial value.

CO Gp. explqxl+exp(-qxl

exp kd . [ 1 + exp (- 2 9 L)]

C=:[exp(-q(L-x)+exp(-q(L+x)]. t(- l)n. exp (- 2 n q L) n=O

(1.58) c=$o. g( - l)? n=O

exp(-q(2n+ljL-x)+:. g(-l)(-q(2n+l)L+x) n=O

As the Laplace transform of

erfc x . i I i-75 lS

exp( -9x) P

the concentration is thus given by :

(1.59) ~=~(-l)n.erfc(2n~~-Xj+~(-l)n.erfc/~2n~~+x) n=O n=O

1.3.4 METHOD OF REFLECTION AND SUPERPOSITION

PLANE SOURCE

The function with the constant A


(1.60) C = $exp

is a solution of the equation of diffusion in one dimension with the constant diffusivity

This function is symmetrical with respect to the plane x = 0, vanishes when it tends to infinity.

The amount of substance diffusing along the x-axis through a cross-section of unit area is

+- (1.61) M = -co

Upon putting

2 x2 v =4 dx=2fidx

this integral becomes

(1.62) M = 2AI% exp -v dv = 2Am ( 1 -

On substituting for the constant from the above relation, the function C expressed by eqn (1.60) becomes :

(1.63) C = &exp

This equation expresses the diffusion of the amount of substance M located in the plane x=Oofunitareaattimet=O.


REFLECTION AT A BOUNDARY

The concept of reflection of the diffusion substance can be used at an impermeable boundary.

For instance, the diffusion of the amount M of substance from the plane surface x = 0 through the cross-section of unit area is considered along positive x, with an impermeable boundary to prevent the substance diffusing along negative x.

The solution for the diffusion of the substance given by eqn (1.63) can be used. By using the principle of reflection at the boundary and superposition of the substance along the positive x, it is clear that the concentration along the positive x only is given by :

(1.64) C = -$& exp

As shown above, the condition at the impermeable boundary is given by :

?$ 0 x=0 along negative x

EXTENDED INITIAL DISTRIBUTION OF THE SUBSTANCE

Generally, the substance is not located in a plane, but it occupies a finite region. In the case where the substance initiaIly occupies the semi-infinite medium x = 0 the

initial conditions are :

(1.65) t = 0 xCO C=Ci and x>O C=O

By considering that the extended distribution of substance is decomposed in an infinite number of plane sources, the solution to this problem of transfer through the plane x = 0 is given by superposing the corresponding infinite number of solutions eqn.cl.63).

The amount of diffusing substance initially located within the element of thickness d&

is Ci, de diffuses along the x-axis. The concentration at a given point P, distance e from this element, resulting from this diffusion is given at time t by :

(1.66) ?!!!- 2m exp

The solution for the concentration at point P resulting from the initial distribution with the semi-infinite medium x < 0 full of substance eqn(1.65) is thus given by integrating the eqn.tl.66) over the limits x and -.


(1.67) C,,, = &. J~~FIp( -&)d&

2

Upon putting v2 = -&

C ci* m x,t = lr;; J x+2fiexP (-v)dv as the error function is given by :

2 y erfcy=- J v5 0 exp (- v2) dv the concentration at point P at position x and time t resulting from the initial uniform distribution of substance in the semi-infinite medium x < 0, is expressed by :

(1.68) C&=$erfc(&)

The error function has the following properties :

erf t-y) = - erf (y) erf(O)=O erf(=)= 1

and the error function complement is given by :

erfc (y) = 1 1 -a-f(y)

REFERENCES

1 J. Crank, The Mathematics of Diffusion, Clarendon Press, Oxford, 1975, p.22.

Mathematical treatment of diffusion in a plane sheet

2.1 INTRODUCTION

A plane sheet is a medium bounded by two parallel planes of infinite dimensions. In this case, a one-dimensional diffusion of substance is thus considered.

In fact, the plane sheet is so thin with respect to the other two dimensions that all the substance diffuses along the axis perpendicular to the plane faces and the amount transferred through the edges is negligible.

Various cases are of interest, depending on the value of the coefficient of matter transfer on the substance and on the volume of solution in which the plane sheet is placed.

Generally there is a non-steady diffusion. The steady state diffusion is only observed in the case of a membrane, e.g., when the concentrations on each surface are constant and different.

The diffusion must be constant for a mathematical treatment. In the case of a concentration-dependent diffusivity, a numerical method is necessary to resolve the problem.

The equation of one-dimensional diffusion with constant diffusivity is :

(2.1) $ = IS ax2

As shown already in Chapter 1, thene are two types of solutions, one expressed in

22 Mathematical treatment of diffusion in a plane sheet [Ch. 2

terms of trigonometrical series, and the other in terms of error functions. These two types of solution are given for the simple problems of diffusion. Three ways of calculation exist (the method of separation of variables, the Laplace transform, the method using superposition and reflexion in finite system), but the method of separation of variables is widely used, and full calculations are given for the simple problems of diffusion. In other more complex cases, the solutions are only given without calculation, but some emphasis is placed upon the conditions for which they can be used, e.g., initial and boundary conditions.

2.2 NON-STEADY STATE WITH A HIGH COEFFICIENT OF MATTER TRANSFER ON THE SURFACE AND AN INFINITE VOLUME OF THE SURROUNDING

This is the case where the volume of the surrounding atmosphere is so high with respect to the volume of the sheet that it can be considered as infinite. A value of the ratio of the volumes of the surrounding atmosphere and the sheet for which this case is obtained is determined in Section 2.4.

The concentration on the surfaces of the sheet can thus be considered as constant. They are equal to the concentration of the surrounding atmosphere, when the partition factor is 1 and to KC,,, when there is a partition factor K.

Three Subsections are of interest : - when the initial distribution is uniform in the sheet - when the initial distribution is f(x) in the sheet - when the concentrations on each surface of the sheet are different. This is the case of the plane membrane.

2.2.1 UNIFORM INITIAL DISTRIBUTION IN THE SHEET CASE OF A SHEET OF THICKNESS 2L, WITH - L < x < L

The sheet in the region - L < x < L is initially at the uniform concentration Ck, and the surfaces are at a constant concentration C,.

The initial and boundary conditions are :

(2.2) t=O -L

Sec. 2.21 Non-steady state with a high coefficient of matter transfer 23

As the midplane x = 0 of the sheet is a plane of symmetry, there is also the condition at any times :

(2.4) t 2 0 2 = 0 x= 0 midplane

The general solution for the equation of diffusion (2.1) is shown in Chapter 1, by using the method of separation of variables :

(2.5) c,,,- c, = %[A*. sin h,,x + B,. cos h,,x) . exp[ -krDt) n=O

The condition (2.4) written as follows :

n=O

necessitates that

(2.6) A,,=0

The condition (2.3), for example x = L and CXWt - C, = 0, for t > 0, is fulfilled when

(2.7) cos h,L=O=cos(2n+l)t and h,=12n2+L1z

The initial condition becomes

(2.8) Ci,- Cm = c B,.cosw for -L

24 Mathematical treatment of diffusion in a plane sheet Kh. 2

L

+B J P -LCOS (2n+ 1JKx 2L .cos (2p+ llxx dx + 2L . . . . . Ascosn. cosp = $ [ cos (n + p) + cos (n - p)]

cos2n = ;[l + cos 2n]

it is obvious that :

J L coJ2 2L . cos2p;;)~xdx=o n+ 1)xx -L J L Cos2(2n+ l)lcx -L 2L dx=L

Moreover

J L cos12n+ 1)x:x dx = 4L . (2n+ 1)n -L 2L (2n+ 1)7c sm 2 which can be also written in the form

4L (-1) (2n+ 1)X

because for even values of n sin% 1 2 = &I?!

2

oddvaluesofn sin- 3n = - 2

1= sin--- (2n+l) z 2

The coeffkient B, is thus equal to :

(2.10) B, = (2n+41,x(- 1)"


The profile of concentration of the diffusing substance within the sheet of thickness 2L, (- L < x c L), is thus expressed in terms of space x and tune t by the following

series :

C 2.11 X+t -CC4

- (- lIn (2n+ 1)7cx .exp -(2n+ 1t21r2Dt

tin- cca = ;. C~.COS

2n+ 1 2L n=O 1 4L2

The amount of substance which is transferred into or out of the sheet at time t, M,, is calculating by integrating the flux of substance through the surface with respect to time.

(2.13)

and

for x=fL

ac I I

2(cin-cJ m _ 1 n .&2n+ 1)Xx. exp 2-F= L C( 1. 2L

_ (2n+ 1)2n2Dt

n=O 4 L2

The amount of matter transferred Mt becomes :

or


(2.16) M, =

co

As the series c 1 iT2 =-

*So (2 n + Ii2 *

and the amount of substance which has entered of left the sheet after infinite time, a, is given by :

(2.17) M,= 2 (Ci,- C,( L

the total amount of diffusing substance which has entered (or left) the sheet of thickness 2L (- L c x < L), at time t, M,, is expressed as a fraction of the corresponding quantity after infinite time M.,,, by the following relation :

(2.18) x= exp -(2n+ 1)2n2Dt 00 i 4L2

Of course, when the diffusing substance enters the sheet :

(2.19) Gin < Co3

and when the diffusing substances leaves the sheet

(2.19) Gin > Ccu

The corresponding solutions are also obtained in terms of error functions by using either the Laplace transform or the method of superposition and reflection.

The profiles of concentration are expressed as a function of space x and time t, for the same sheet of thickness 2 L.

(2.20) :*t - tin = z(- l)n. erfc (2n+ l)L-x

co- tin 21cDi n=O


m

+ C( )

- 1 n. erfc (2n+ l)L+x 2l-D-t

n=O

where C, represents the constant concentration on the two surfaces of the sheet for t > 0. This is also the concentration obtained in the sheet after infinite time, when the sheet and the surrounding atmosphere are at equilibrium.

The kinetics of the diffusing substance which has entered or left the sheet is given by :

(2.21) $ ca

= 2fE.

I

$+2. z(- l)n. ierfcs n=l 1

REMARKI - Diffusion for long times with a sheet of thickness 2 L (- L < x c L) The equations (2.11) and (2.18) are of better use for long times of diffusion, as

the series does not converge fast for short values of time, and low values of the ratio $. OD

For very long times of diffusion or rather for high values of the ratio 2, the first

term of the series in eqn (2.18) is preponderent. The simple equation is thus obtained, for Mt M > 0.5 - 0.7 depending on the accuracy which is wanted.

0

(2.22) z Mt = +exp _ nZDt ca x 1 ! 4 L2

Expressed in the logarithmic form, a straight line is obtained by plotting log as a

function oft. The diffusivity D can this easily be calculated from the slope of this straigth line.

REMARK 2 - Diffusion for small times with a sheet of thickness 2 L (- L < x < L) The solutions expressed in terms of the error functions are of better use for small

tunes of diffusion, as the serie converge very fast for short tunes and small values of the M

ratio 2 M,


For very small times of diffusion, the series in eqn (2.21) tends to 0, and the Mt

very well known equation is obtained for M c 0.3 - 0.5, depending on the desired ca

accuracy.

(2.23) 2 = t E with thickness = 2L

This equation is of high interest for calculating the diffusivity D obtained by

plotting the ratio $ as a function of the square root of time, at the beginning of the process 00

when the coefficient of matter transfer is very high and the volume of the surrounding

atmosphere is much larger than that of the sheet.

REMARK 3 - Half-time of sorption (desorption) with a sheet of thickness 2 L (- L < x < L).

The half-time of the sorption (or desorption) process is sometimes used for w calculating the diffusivity. This is the time necessary for M = 1.

m 2

By considering only the first term of the trigonometrical series obtained for long

time, e.g., eqn (2.22) the following relation is obtained :

(2.24) D= 0.1958 for 2 = k ce

By considering the simple relation-ship obtained for small times, e.g. eqn (2,23),

the diffusivity is expressed in terms of $ by :

(2.25) D= 0.196

where L is half the thickness of the sheet.

REMARK 4 - Constant diffusivity

When the diffusivity obtained by using these three methods (long times, small

tunes, half-time of transfer), is the same, the diffusivity can be considered as constant.

Otherwise, the diffusivity is concentration-dependent.

sec. 2.21 Non-steady state with a high coefficient of matter transfer 29

REMARK 5 - Partition factor of the diffusion substance If there is a partition factor for the diffusing substance between the solid and

the surrounding atmosphere, it is expressed by K = $, ext

where C, is the constant concentration on the surface. This concentration is also the concentration required to maintain equilibrium with the surounding in which the constant concentration of diffusing substance is C,,.

REMARK 6 - Infinite value of the coefficient of matter transfer on the surface The case of a constant concentration on the surface is obtained when the coefficient

of matter transfer on the surfaces of the sheet is infinite, as shown in the following relation :

F=h(C,-C,)

where C, is the concentration on the surface, which is always equal to C,, h is the coefficient of matter transfer on the surface (cm/s), and F is the flux of matter through the surface.

In practical use, this case of a constant concentration on the surface is encountered when the coefficient of matter transfer on the surface is sufficiently high with regard to the diffusivity of the substance within the sheet of thickness 2 L, or rather when the dimensionless L$ is high enough.

For instance. when

(2.27) +$ > 20-50

dependency on the accuracy wanted, h can be considered as infinite.

REMARK 7 - Case of a sheet of thickness L, with 0 < x < L The solution of the equation of diffusion in this simple case is also obtained by

using the method of separation of variables. The initial and boundary conditions are :

(2.28) t=O O

30 Mathematical treatment of diffusion in a plane sheet

(2.29) t>o x= 0 c=c, surfaces

andx=L

The profiles of concentration CXt at any time is given by the relation :

(2.30) :*yz- = 4. &-&-. sb(2n+L1)Kx. exp -(2n+ 12A2Dt

in 00 n=O 1 L2

[Ch. 2

i

The amount of substance transferred through the surfaces of the sheet is calculated

by integrating the flux of substance through the surfaces with respect to time :

(2.31) M, = 2D for x=0 or x=L

The total amount of diffusing substance which has entered (or left) the plane sheet of thickness L, (0 < x < L), at time t, M,, is expressed in terms of time, as a fraction of the

corresponding quantity after infinite time M, :

(2.32) 2 = l-+.2 .exp -i2n+1)2rc2D1 Co x n=~Pn+ 1j2 j L2

REMARK a - Volume and nature of the surrounding atmosphere The case of a constant concentration of diffusing substance on the surface of the

sheet necessitates that the concentration of this substance in the surrounding atmosphere is

always constant and uniform. This fact can only be obtained when the surrounding

atmosphere is strongly stirred and of infinite volume. In practical use, this case can be encountered when the volume of the surrounding is

high enough with respect to the volume of the sheet. This is obtained when the dimensionless number 01 which is the ratio of the volumes of the surrounding atmosphere

and of the sheet is high enough.

The dimensionless number a can also be expressed in terms of the partition factor K,


when this partition factor is not 1. Generally, the equations for the concentration distribution and for the kinetics of

transfer of the diffising substance can be used when

(2.34) a> 20-50

depending on the accuracy which is desired.

2.2.2 INITIAL DISTRIBUTION f(x) IN THE SHEET OF THICKNESS L (0 < x < L)

The sheet with the initial concentration distribution of the diffusing substance f(x) is immersed in a surrounding atmosphere of infinite volume and an infinite coefficient of matter transfer on the surfaces.

Two cases of interest can be considered, when the plane sheet separates two media of different concentrations, and when the sheet is immersed in the same surrounding atmosphere.

TWO DIFFERENT MEDIA

In the first case, the initial and boundary conditions are :

(2.35) t=O OCXCL ($0 = f(x) sheet

(2.36) t>o x=0 x=L

CO*-

CL.-

surface 0 surface L

The solution of the equation of diffusion can be obtained by the method of separation of variables. The concentration distribution C,,t within the sheet is thus expressed by the trigonometrical se

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