There is evidently therefore a transfer of iodine molecules from the lower to the upper part of the vessel taking place in the absence of convection currents. The iodine is said to have diffused into the water. If it were possible to watch individual molecules of iodine, and this can be done effectively by replacing them by particles small enough to share the molecular motions but just large enough to be visible under the microscope, it would be found that the motion of each molecule is a random one.
In a dilute solution each molecule of iodine behaves independently of the others, which it seldom meets, and each is constantly undergoing collision with sol- vent molecules, as a result of which collisions it moves sometimes towards a region of higher, sometimes of lower, concentration, having no preferred direction of motion towards one or the other.
The motion of a single molecule can be described in terms of the familiar 'random walk' picture, and whilst it is possible to calculate the mean-square distance travelled in a given interval of time it is not possible to say in what direction a given molecule will move in that time. This picture of random molecular motions, in which no molecule has a preferred direction of motion, has to be reconciled with the fact that a transfer of iodine molecules from the region of higher to that of lower concentration is nevertheless observed.
Consider any horizontal section in the solution and two thin, equal, elements of volume one just below and one just above the section. Thus, simply because there are more iodine molecules in the lower element than in the upper one, there is a net transfer from the lower to the upper side of the section as a result of random molecular motions. Basic hypothesis of mathematical theory Transfer of heat by conduction is also due to random molecular motions, and there is an obvious analogy between the two processes.
This was recog- nized by Fick , who first put diffusion on a quantitative basis by adopting the mathematical equation of heat conduction derived some years earlier by Fourier The mathematical theory of diffusion in isotropic substances is therefore based on the hypothesis that the rate of transfer of diffusing substance through unit area of a section is proportional to the concentration gradient measured normal to the section, i.
In some cases, e. If F, the amount of material diffusing, and C, the concentration, are both expressed in terms of the same unit of quantity, e.
The negative sign in eqn 1. It must be emphasized that the statement expressed mathematically by 1. Because of this symmetry, the flow of diffusing sub- stance at any point is along the normal to the surface of constant concentration through the point. Differential equation of diffusion The fundamental differential equation of diffusion in an isotropic medium is derived from eqn 1.
Consider an element of volume in the form of a rectangular parallelepiped whose sides are parallel to the axes of coordinates and are of lengths 2 dx, 2 dy, 2 dz. Element of volume. Expressions 1. In many systems, e. In this case, and also when the medium is not homo- geneous so that D varies from point to point, eqn 1. If D depends on the time during which diffusion has been taking place but not on any of the other variables, i.
Diffusion in a cylinder and sphere Other forms of the above equations follow by transformation of co- ordinates, or by considering elements of volume of different shape. Anisotropic media Anisotropic media have different diffusion properties in different directions. Some common examples are crystals, textile fibres, and polymer films in which the molecules have a preferential direction of orientation.
This means that 1. Substituting from 1. The extension to non-constant Ds is obvious from 1. If we make the further transformation. Whether or not this can be done in a given case depends on the boundary conditions. Thus it can be shown that the square of the radius vector of the ellipsoid in any direction is inversely proportional to the diffusion coefficient normal to the surfaces of constant concentration at points where their normals are in that direction.
It means that the flow through a surface perpendicular to a principal axis of diffusion is proportional simply to the concentration gradient normal to the surface as is the case for isotropic media. Significance of measurements in anisotropic media Since in the majority of experiments designed to measure a diffusion coefficient the flow is arranged to be one-dimensional, it is worth while to see how such measurements are affected by anisotropy.
If the diffusion is one-dimensional in the sense that a concentration gradient exists only along the direction of x, it is clear from 1. If the direction of diffusion is chosen to be that of a principal axis, then Dllis equal to one or other of the principal diffusion coefficients Dl9 D2, or D 3. Similar remarks apply to a high polymer sheet in which there is both uniplanar and undirectional orientation, i. The principal axes of diffusion of such a sheet will be normal to the plane sheet, and along and perpendicular to the preferred direction of orientation in that plane.
Conversion of heat flow to diffusion solutions Carslaw and Jaeger and other books contain a wealth of solutions of the heat-conduction equation. There is general awareness among scientists and engineers that the phenomena of heat flow and diffusion are basically the same. Nevertheless, many non-mathematicians experience difficulty in making the changes of notation needed to transcribe from one set of solutions to the other.
In this section we examine in detail the correspondence between the physical parameters, the variables, and the equations and boundary conditions which occur in heat-flow and diffusion problems. We take the one-dimensional case with constant properties as an illustration.
Diffusion theory is based on Fick's two equations 1. The space coordinate is x and t is time. This is a consequence of our having identified C with 0. The 'diffusing substance' in heat flow is heat not temperature. The factor cp is needed to convert temperature to the amount of heat per unit volume; but concentration is, by definition, the amount of diffusing substance per unit volume and so no conversion factor is needed, i.
There is no ana- logue to S in heat flow. Sometimes this is referred to as Newton's law of cooling. Thus, a boundary condition describing thermal contact with a perfect conductor also describes diffusion of solute from a well-stirred solution or vapour. In addition, 6S will be the temperature just within the surface but the two concentrations may be related by some isotherm equation.
Care is needed with algebraic signs as in the previous section. In the first case, the relevant feature is that latent heat is removed instan- taneously from the heat-conduction process, in which it takes no further part. The diffusion counterpart is the immobilizing of diffusing molecules on fixed sites or in holes. Useful collections of mathematical solutions of the diffusion equations are to be found in books by Barrer , Jost , and Jacobs Jacob's solutions are of particular interest to biologists and biophysicists.
Types of solution G E N E R A L solutions of the diffusion equation can be obtained for a variety of initial and boundary conditions provided the diffusion coefficient is constant. Such a solution usually has one of two standard forms. Either it is comprised of a series of error functions or related integrals, in wHich case it is most suit- able for numerical evaluation at small times, i.
When diffusion occurs in a cylinder the trigonometrical series is replaced by a series of Bessel functions. Of the three methods of solution described in this chapter, the first two illustrate the physical significance of the two standard types of solution.
The third, em- ploying the Laplace transform, is essentially an operator method by which both types of solution may be obtained. It is the most powerful of the three, particularly for more complicated problems. The methods are presented here as simply as possible. The fuller treatments necessary to make the discussion mathematically rigorous are to be found in works on heat conduction, e.
Carslaw and Jaeger Method of reflection and superposition 2. The expression 2. Thus, on substituting for A from 2. Concentration-distance curves for an instantaneous plane source. Numbers on curves are values of Dr. Reflection at a boundary Expression 2. Since this equation is linear the sum of the two solutions is itself a solu- tion, and we see that 2.
Extended initial distributions So far we have considered only cases in which all the diffusing substance is concentrated initially in a plane.
The solution to such a problem is readily deduced by considering the extended distribution to be composed of an infinite number of line sources and by superposing the corresponding infinite number of elementary solutions. With reference to Fig. Then, from 2. Extended initial distribution. Table 2. The form of the concentra- tion distribution is shown in Fig. It is clear from 2. The error function therefore enters into the solution of a diffusion problem as a consequence of summing the effect of a series of line sources, each yielding an exponential type of distribution.
Concentration-distance curve for an extended source of infinite extent. Concentration-distance curves for an extended source of limited extent. Therefore expression 2. Such a system is realized in practice in the classical experiment in which a cylinder contains a layer of solution having on top of it an infinitely-long column of water, initially clear. In practice, this means that concentration changes do not reach the top of the column during the time of the experiment.
Such solutions are most useful for calculating the concentration distribution in the early stages of diffusion, for then the series converges rapidly and two or three terms give sufficient accuracy for most practical purposes. In all cases the successive terms in the series can be regarded as arising from successive reflections at the boundaries. The nature of the reflection depends on the condition to be satisfied.
A further example of the use of this method is given by Jost For more complicated problems, however, the reflection and superposition method soon becomes unwieldy and results are more readily obtained by other methods. Method of separation of variables A standard method of obtaining a solution of a partial differential equation is to assume that the variables are separable.
Thus we may attempt to find a solution of 2. Substitution in 2. Both sides therefore must be equal to the same constant which, for the sake of the subsequent algebra, is conveniently taken as — X2D. Since 2. This reveals the physical significance of the trigonometrical series in 2. Method of the Laplace transform The Laplace transformation is a mathematical device which is useful for the solution of various problems in mathematical physics.
This is then interpreted, according to certain rules, to give an expression for the concentration in terms of x, y, z and time, satisfying the initial and boundary condition.
Historically the method may be regarded as derived from the operational methods introduced by Heaviside. Full accounts of the Laplace transform and its application have been given by Carslaw and Jaeger , Churchill , and others. Shorter accounts by Jaeger and Tranter are also available.
Here we shall deal only with its application to the diffusion equation, the aim being to describe rather than to justify the procedure. The solution of many problems in diffusion by this method calls for no mathematics beyond ordinary calculus. No attempt is made here to explain its application to the more difficult problems for which the theory of func- tions of a complex variable must be used, though solutions to problems of this kind are quoted in later chapters.
The fuller accounts should be consulted for the derivation of such solutions. Definition of the Laplace transform Suppose f t to be a known function of rfor positive values of t. It may be a complex number whose real part is sufficiently large, but in the present discussion it suffices to think of it simply as a real positive number. Laplace transforms of common functions are readily constructed by carry- ing out the integration in 2. A short table of transforms occurring fre- quently in diffusion problems is reproduced from Carslaw and Jaeger's book in Table 2.
Thus 2. The solution of 2. Reference to Table 2. Plane sheet In the problem just considered the transform solution could be interpreted immediately by reference to the table of transforms. We shall first obtain a solution useful for small values of the time.
We express the hyperbolic functions in 2. Thus we obtain from 2. A proof of this by Jaeger is reproduced in the Appendix to this chapter. It is derived by expressing 2. Since the hyperbolic functions cosh z and sinh z can be represented by the following infinite products see, e.
Carslaw , p. The a1,a2,— are the zeros of g p 9 i. The justification of this assumption involves the theory of functions of a complex variable in order to carry out a contour integration and is to be found in the fuller accounts of the subject. There is, in fact, a rigorous mathematical argument by which the use of 2. It must not be applied to 2.
The above refers to al9 a 2 , The extension of 2. Its application to an infinite number of factors is still justifiable. We may now consider the application of 2. First the zeros of the denominator must be found. For the other zeros, given by 2. The series converges rapidly for large values off. Solutions in two and three dimensions 2. The following proof is given by Carslaw and Jaeger , p. Clearly the initial and boundary conditions 2. An essential condition is 2. Carslaw and Jaeger give solutions for a rectangular corner, rectangles, parallelepipeds, cylinders and some examples of isotherms are shown graphically.
A general relationship Goldenberg derived a much more general relationship between the transient solutions of two-dimensional problems for an infinite cylinder of arbitrary cross-section, and the transient solutions of the corresponding three-dimensional problems in finite cylinders. The same relationship is valid for a hollow cylinder of arbitrary cross- section and for the region external to a cylinder of arbitrary cross-section. Similar relationships hold in other situations discussed by Goldenberg.
Other solutions Langford obtained new solutions of the one-dimensional heat equation for temperature and heat flux both prescribed at the same fixed boundary. They take the form of series of polynomial and quasi-polynomial solutions for plane sheets, cylinders, and spheres.
They include as special cases some of the old or classical solutions. They also have applications to phase change problems with boundaries moving at a constant velocity. Thus 10 implies that to each linear factor p — ar of the denominator of y p there corresponds a term in the solution. The generalization is that, to each squared factor p — b 2 of the denominator of y p there corresponds a term i in the solution. Introduction IN this and the following three chapters solutions of the diffusion equation are presented for different initial and boundary conditions.
In nearly all cases the diffusion coefficient is taken as constant. In many cases the solutions are readily evaluated numerically with the help of tables of standard mathe- matical functions. Where this is not so, and where numerical evaluation is tedious, as many graphical and tabulated solutions as space permits are given. For example, it may be located initially at a point, or in a plane, or within a sphere, when we have an instantaneous point, plane, or spherical source as the case may be.
The solution for an instantaneous plane source in an infinite medium has already been given in Chapter 2, eqn 2. This is the same as expression 2. The corresponding result for a line source of strength M per unit length in an infinite volume, obtained by integrating 3.
Results for a variety of sources are derived by Carslaw and Jaeger , p. The spherical and cylindrical sources are likely to be of practical interest. Expression 3. The integral in 3. Curves showing the con- centration distribution at successive times are given in Figs. Concentration distributions for a spherical source. Concentration distributions for a cylindrical source.
The analysis holds for infinite and semi- infinite media, and physical applications are discussed. Continuous sources A solution for a continuous source, from which diffusing substance is liberated continuously at a certain rate, is deduced from the solution for the corresponding instantaneous source by integrating with respect to time t.
The problem of the semi-infinite medium whose surface is maintained at a constant concentration C o , and throughout which the concentration is initially zero, was handled by the method of the Laplace transform in Chapter 2, p.
Other results of practical importance which may be obtained in the same way are given below. The rate of loss of diffusing substance from the semi-infinite medium when the surface concentration is zero, is given by - Jw so that the total amount Mt of diffusing substance which has left the medium at time t is given by integrating 3.
Cases of practical interest are given below. Here Mt is used through- out to denote the total amount of diffusing substance which has entered the medium at time t. The effect of an increasing surface concentration is shown in Fig. Sorption curves for variable and constant surface concentrations in a semi-infinite medium.
In this case M is directly proportional to t and so the rate of uptake of diffusing substance is constant. In such a case the complete expression for the concentration at any point is the sum of a number of terms of type 3.
Consider, as an example, the problem of desorption from a semi-infinite medium having a uniform initial concentra- tion C o , and a surface concentration decreasing according to 3. Surface evaporation condition In some cases the boundary condition relates to the rate of transfer of diffusing substance across the surface of the medium.
Thus, if a stream of dry air passes over the surface of a solid containing moisture, loss of moisture occurs by surface evaporation. In each case the rate of exchange of moisture at any instant depends on the relative humidity of the air and the moisture concentration in the surface of the solid.
The simplest reasonable assumption is that the rate of exchange is directly proportional to the difference between the actual concentration Cs in the surface at any time and the concentration C o which would be in equilibrium with the vapour pressure in the atmosphere remote from the surface.
If the concentration in a semi-infinite medium is initially C 2 throughout, and the surface exchange is determined by 3. The rate at which the total amount Mt of diffusing substance in the semi-infinite medium per unit cross-sectional area changes is given by -CX 3.
The ex- pression 3. Concentration distribution for a surface evaporation condition in a semi-infinite medium. The evaluation for large hyJ Dt is made easier by using the asymp- totic formula 1 1 1.
Square-root relationship Expression 2. Sorption curve for a surface evaporation condition in a semi-infinite medium. These fundamental properties hold in general in semi-infinite media, pro- vided the initial concentration is uniform and the surface concentration remains constant. The infinite composite medium Here we consider diffusion in systems in which two media are present.
The condition 3. A solution to this problem is easily obtained by combining solutions for the semi-infinite medium so as to satisfy the initial and boundary conditions. By choosing the constants Al9 Bl9 A2, B2 to satisfy the initial conditions and 3. Graphs for other cases are shown by Jost and by Barrer Concentration distribution in a composite medium. Interface resistance If we have the same problem as on p. This, and the general distribution at successive times, is illustrated in Fig.
Concentration distribution in a composite medium with a resistance at the interface. The semi-infinite composite medium This is the case of a semi-infinite medium which has a skin or surface layer having diffusion properties different from those of the rest of the medium.
This is true also of the error-function complements in 3. Whipple has given formulae for the concentration in a semi-infinite region of low diffusion coefficient bisected by a thin well-diffusing slab, at different times after the boundary of the semi-infinite region has been raised suddenly from zero to unit concentration. This is of interest in grain- boundary diffusion.
Sorption curves for a composite semi-infinite medium. Numbers on curves are values of DJD2. Weber's disc This is a classical problem of the field due to an electrified disc. More recent interest relates to the diffusion current at a circular electrode. Several authors Tranter ; Grigull ; Saito have developed the same solution in different ways.
When the diameter of the electrode is 2a we require solutions of cr r or cz where C is the concentration say of oxygen in the solution, the axis of z passes perpendicularly through the centre of the disc and r is the radial distance from the z-axis.
Introduction I N this chapter we consider various cases of one-dimensional diffusion in a medium bounded by two parallel planes, e.
These will apply in practice to diffusion into a plane sheet of material so thin that effectively all the diffusing substance enters through the plane faces and a negligible amount through the edges. After a time, a steady state is reached in which the concentration remains constant at all points of the sheet. Experimental ar- rangements for measuring D in this and other ways have been reviewed by Newns Permeability constant In some practical systems, the surface concentrations C1,C2 may not be known but only the gas or vapour pressures px,p2 on the two sides of the membrane.
The rate of transfer in the steady state is then sometimes written and the constant P is referred to as the permeability constant. Here P is expressed, for example, as cm 3 gas at some standard temperature and pres- sure passing per second through 1 cm 2 of the surface of a membrane 1 cm thick when the pressure difference across the membrane is 1 cm of mercury.
If the diffusion coefficient is constant, and if the sorption isotherm is linear, i. Since Cx, px and C 2 , p2 in 4. Concentration-dependent diffusion coefficient If the diffusion coefficient varies with concentration it is clear that the simple value of D deduced from a measurement of the steady rate of flow is some kind of mean value over the range of concentration involved.
Thus, if D is a function of C 4. It follows from 4. Concentration distributions for D depending on C in a number of ways are given in Chapter 9. Thus the resistance to diffusion of the whole membrane is simply the sum of the resistances of the separate layers, assuming that there are no barriers to diffusion between them.
Many of the results are quoted by Barrer , Carslaw and Jaeger , Jacobs , Jost and others. The emphasis here is on numeri- cal evaluation. Surface concentrations constant. Uniform initial distribution. Surface concentrations equal This is the case of sorption and desorption by a membrane.
Jason and Peters analyse the bimodal diffusion of water in fish muscle by combining two expressions of the type 4. Eqn 4. The solution 4. Talbot and Kitchener also obtained a solution for a slightly tapering tube. For small ju, e. It is clear that expressions 4. These are re- produced with change of nomenclature from Carslaw and Jaeger's book , p.
The curve labelled zero fractional uptake in Fig. Surface concentrations different This is the case of flow through a membrane. Barnes examined the errors introduced by assuming a linear gradient to exist across the membrane during the whole course of diffusion. The expression 4. By integrating then with respect to t, we obtain the total amount of diffusing substance Qt which has passed through the membrane in time t.
Thus from an observation of the intercept, L, D is deduced by 4. The intercept L is referred to as the 'time lag'. Approach to steady-state flow through a plane sheet. Rogers et al. They discuss the advantages of their method compared with the use of the time lag given by eqn 4. Jenkins, Nelson, and Spirer examined the more general problems of deducing both the diffusion coefficient and the solu- bility coefficient from experimental data and mathematical solutions when the outflow volume is finite so that the concentration varies with time at the outgoing face.
The necessary solution is given by Carslaw and Jaeger Jenkins et al. They concluded that the time lag is underestimated by about 4 per cent by making this assumption. Paul and Dibenedetto also obtained solutions for finite outflow volumes.
Spacek and Kubin allowed the concentrations on both sides of the membrane to vary with time. In certain cases, however, where the surface concentration can be represented by a mathematical expression, the solution can be considerably simplified.
The sorption- time curve, i. Calculated sorption curves for surface concentration given by C0 l — e fit. At first the rate of uptake increases as sorption proceeds but later decreases as the final equilibrium is approached. Curves of this kind are often referred to as sigmoid sorption curves.
Concentration distributions in a plane sheet for surface concentration kt. Sorption curve for plane sheet with surface concentration kt. Diffusion from a stirred solution of limited volume If a plane sheet is suspended in a volume of solution so large that the amount of solute taken up by the sheet is a negligible fraction of the whole, and the solution is well stirred, then the concentration in the solution remains constant.
If, however, there is only a limited volume of solution, the concentration of solute in the solution falls as solute enters the sheet. If the solution is well stirred the concentration in the solution depends only on time, and is determined essentially by the condition that the total amount of solute in the solution and in the sheet remains constant as diffusion proceeds.
It is useful from an experimental point of view to have only a limited amount of solution since the rate of uptake of solute by the sheet can be deduced from observations of the uniform concentration in the solution. It is often simpler to do this than to observe directly the amount in the sheet. This has been stressed by Carman and Haul , who have written mathematical solutions in forms most appropriate for the measurement of diffusion coefficients by this method.
The general problem can be stated mathematically in terms of a solute diffusing from a well-stirred solution. The modifications necessary for cor- responding alternative problems, such as that of a sheet suspended in a vapour, are obvious. The concentration of the solute in the solution is always uniform and is initially C o , while initially the sheet is free from solute. This may not be so but there may be a partition factor K which is not unity, such that the concentration just within the sheet is K times that in the solution.
A solution of this problem by March and Weaver , based on the use of an integral equation, was cumbersome for numerical evaluation.
More convenient forms of solution have been obtained by Carslaw and Jaeger , p. The solution is most readily obtained by the use of the Laplace transform. Roots for other values of a are given by Carslaw and Jaeger , p. It is sometimes convenient to express a in terms of the fraction of total solute finally taken up by the sheet. When more than three or four terms are needed it is better to use an alternative form of solution. This has been suggested by Berthier as a method for measuring self-diffusion using radioactive isotopes.
For precision measurements it is advisable to check Berthier's values as in some instances not enough terms of the series solutions have been retained to obtain the accuracy quoted. There is the complementary problem in which all the solute is initially uniformly distributed through the sheet and subsequently diffuses out into a well-stirred solution.
Uptake by a plane sheet from a stirred solution of limited volume. Numbers on curves show the percentage of total solute finally taken up by the sheet. For the problem of desorption from the sheet we require a solution of 4. Instead we have that the fractional uptake of the solution is given by 1 4. Roots of 4. The terms in the series expres- sion for concentration very soon become cumbersome for numerical evalua- tion, however.
In practice, it is usually sufficient to use only the leading terms corresponding to the interval during which the sheet is effectively semi- infinite, when the concentration is given by expression 3. Carslaw and Jaeger , p. Sorption or desorption curves for the surface condition 4. All these equations and solutions have a practical application in the drying of porous solids.
Jaeger and Clarke have also given in graphical form the solutions of a number of problems with an evaporation type of boundary condition. The more complicated case in which the rate of transfer on the surface is proportional to some power of the surface concentration was discussed by Jaeger a.
Concentration-distance curves for various times are shown in Fig. An alternative form of solution suitable for small times is given by Macey Concentration distributions in a plane sheet for constant flux Fo at the surface. Impermeable surfaces An impermeable surface is one at which the concentration gradient is zero. This condition holds at the central plane of a sheet provided the initial and boundary conditions are symmetrical about that plane.
Numerical values based on 2. Another special case of 4. They consider the problem in which a sheet, initially at zero concentration throughout, has its surfaces maintained at a constant concentration Co for a time t0, after which they are rendered impermeable.
The subsequent change in concentration is described by 4. Composite sheet Various problems of diffusion into a composite sheet comprised of two layers for which the diffusion coefficients are different have been solved see, for example, Carslaw and Jaeger , p.
The solutions are similar in form to those presented in this chapter but obviously more complicated. In view of the additional number of parameters involved, no attempt is made to give numerical results here. Edge effects in membranes We have treated flow through a membrane as a one-dimensional pheno- menon. In the usual experimental arrangements an appreciable portion of the membrane is clamped between impermeable annular plates of outer radius b.
But inside the membrane the flow lines spread into the clamped region. This 'edge effect' means that the usual assumption of one-dimensional diffusion is not strictly correct.
Barrer, Barrie, and Rogers , using a solution developed by Jaeger and Beck , examined the importance of the edge effect. They replaced the usual experimental condition of constant concentrations maintained on the two faces of the membrane by conditions of uniform constant flux. The mathe- matical solution yields a mean concentration difference between the two faces of the membrane which can be equated to the prescribed uniform difference in the original experiment.
Barrer et al. From Barrer et al. Approximate two-dimensional solutions Problems in which the diffusion is predominately in one direction occur frequently, perhaps because of the boundary conditions or the shape of the medium.
Crank and Parker obtained approximate solutions to two- dimensional problems in biased heat flow. Previously it was widely accepted that for a thin sheet a good, one-dimensional approximation is obtained by assuming the temperature to be uniform across the thickness of the sheet.
It appears, however, that a 'quadratic profile' provides a better approximation than the usual 'constant profile'. On integrating 4. Crank and Parker discuss two ways of improving this crude approximation.
Then from 4. In order to satisfy 4. Then 4. A least-squares fitting of i?! An alternative approach starts again from 4. They are obtained by making w obey the conditions 4. Thus by using w for i; in 4. The constant k is arbitrary but must not differ from unity by more than O hb. They concluded that the 'constant-profile' solution which is usually used as an approximation for a thin sheet or rod is, in fact, poor for a thin sheet with a high surface evaporation. But it provides a good approximation for a thick sheet with low surface loss.
Both the quad- ratic profile' and the 'partial separation of variables' yield much better approximations over all. Concentration profiles across the sheet support these general conclusions. Crank and Parker show, however, that the approximate methods can readily be extended to cover some non-linear situations. Introduction W E consider a long circular cylinder in which diffusion is everywhere radial.
Concentration is then a function of radius r and time t only, and the diffusion equation 1. The concentration distribution defined by 5. Typical distributions are shown in Fig. Steady-state concentration distributions through cylinder wall. This is due to the two opposing changes associated with an increase in b. In certain circumstances, therefore, the rate of diffusion through the wall of a pipe may be increased by making the wall thicker Porter and Martin Page 71 4.
Approximate two-dimensional solutions Page 72 5. Page 76 5. Non-steady state: solid cylinder Page 78 5. The hollow cylinder Page 89 5. Page 94 6. Page 96 6. Page 97 6. Hollow sphere Page 6. Page 7. Time-dependent diffusion coefficients Concentration-dependent diffusion: infinite and semi-infinite media Methods of moments Approximation by orthogonal functions Page 8.
Numerical solutions Non-dimensional variables Physical derivation of a numerical solution Finite-difference solution: explicit method Crank-Nicolson implicit method Other boundary conditions Finite-difference formulae for the cylinder and sphere Composite media Two- and three-dimensional diffusion Singularities: local solutions Compatibility, convergence, stability Steady-state problems Page 9. Non-steady-state conditions Concentration-distance curves Sorption- and desorption-time curves Diffusion-controlled evaporation Effect of a surface skin Page A frame of reference when the total volume of the system remains constant Alternative frames of reference Intrinsic diffusion coefficients Methods of measurement Weighted-mean diffusion coefficients Radiotracer methods Glassy polymers Characteristic features Mathematical models Disperse phase in a continuum The mathematical problem of a two-phase system Discontinuous diffusion coefficients General problem of the moving boundary Radially symmetric phase growth Approximate analytic solutions Finite-difference methods Other methods Instantaneous reaction Irreversible reaction Reversible reaction A bimolecular reaction Reduced sorption curves Constant reaction rate Uptake of water by a textile fibre Two possible equilibrium conditions Propagation of two disturbances Equations for diffusion of heat and moisture Solution of the equations Surface temperature changes accompanying the sorption of vapours Page Tables Page References Page Author index Page Subject index No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means.
The book contains a collection of mathematical solutions of the differential equations of diffusion and methods of obtaining them. They are discussed against a background of some of the experimental and practical situations to which they are relevant. Little mention is made of molecular mechanisms, and I have made only fleeting excursions into the realms of irreversible thermodynamics.
These I hope are self-explanatory. A number of general accounts of the subject are already available, but very few mathematical solutions of the equations of non-equilibrium thermodynamics have been obtained for practical systems. During the last years the widespread occurrence of concentrationdependent diffusion has stimulated the development of new analytical and numerical solutions. The time-lag method of measuring diffusion coefficients has also been intensively investigated and extended.
Similarly, a lot of attention has been devoted to moving-boundary problems since the first edition was published. These and other matters have now been included by extensive revision of several chapters.
Also, the chapter dealing with the numerical solution of the diffusion equations has been completely rewritten and brought up to date. It seems unbelievable now that most of the calculations in the first edition were carried out on desk calculating machines. Two entirely new chapters have been added. In one are assembled some of the mathematical models of non-Fickian or anomalous diffusion occurring mainly in solvent-polymer systems in the glassy state.
The other attempts a systematic review of diffusion in heterogeneous media, both laminates and particulates. A succession of improved solutions are described to the problem of diffusion in a medium in which are embedded discrete particles with different diffusion properties. I have resisted the temptation to lengthen appreciably the earlier chapters. The enlarged edition of Carslaw and Jaeger's book Conduction of heat in solids contains a wealth of solutions of the heat-flow equations for constant heat parameters.
Many of them are directly applicable to diffusion problems, though it seems that some non-mathematicians have difficulty in makitfg the necessary conversions. For them I have included a brief 'translator's guide'. A few new solutions have been added, however, some of them in the context in which they arose, that is the measurement of diffusion coefficients.
I am deeply grateful to my academic colleagues who shared my administrative responsibilities and particularly to Professor Peter Macdonald who so willingly and effectively assumed the role of Acting Head of the School of Mathematical Studies. I am most grateful to Mrs. Joyce Smith for all the help she gave me, not least by typing the manuscript and checking the proofs.
Alan Moyse kept me well supplied with the seemingly innumerable books, journals, and photostat copies which I requested. I owe a great deal to friendly readers who have pointed out mistakes in the first edition and made helpful suggestions for the second. In particular I have benefited from discussions with my friend and former colleague, Dr. Geoffrey Park. I had an invaluable introduction to the literature on which Chapter 12 is based from Mr.
Woodcock, who came to me for help but, in fact, gave far more than he received. Finally, I have appreciated the understanding help and guidance afforded me by members of staff of the Clarendon Press.
Uxbridge October J. Faraday Soc; Fig. ScL; Fig. It deals with the description of diffusion processes in terms of solutions of the differential equation for diffusion. Little mention is made of the alternative, but less well developed, description in terms of what is commonly called 'the random walk', nor are theories of the mechanism of diffusion in particular systems included. The mathematical theory of diffusion is founded on that of heat conduction and correspondingly the early part of this book has developed from 'Conduction of heat in solids' by Carslaw and Jaeger.
These authors present many solutions of the equation of heat conduction and some of them can be applied to diffusion problems for which the diffusion coefficient is constant. I have selected some of the solutions which seem most likely to be of interest in diffusion and they have been evaluated numerically and presented in graphical form so as to be readily usable. Several problems in which diffusion is complicated by the effects of an immobilizing reaction of some sort are also included.
Convenient ways of deriving the mathematical solutions are described. When we come to systems in which the diffusion coefficient is not constant but variable, and for the most part this means concentration dependent, we find that strictly formal mathematical solutions no longer exist.
I have tried to indicate the various methods by which numerical and graphical solutions have been obtained, mostly within the last ten years, and to present, again in graphical form, some solutions for various concentration-dependent diffusion coefficients.
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