4. Diffusion. 4.1 Definition Spreading out, mixing. The diffusion of gases and liquids refers to...

4. Diffusion

4.1 Definition

Spreading out, mixing. The diffusion of gases and liquids refers to their mixing without an external force.

Diffusion is defined as a process of mass transfer of individual molecules of a substance, brought about by random molecular motion and associated with a concentration gradient.

4.2 .Mechanisms

Diffusion could occur: Throughout a single bulk phase (solution’s

homogeneity).

Through a barrier, usually a polymeric membrane (drug release from film coated dosage forms, permeation and distribution of drug molecules in living tissues, passage of water vapor and gases through plastic container walls and caps, ultrafiltration).

4.2 .Mechanisms

Diffusion through a barrier may occur through either:

Simple molecular permeation (molecular diffusion) through a nonporous membrane. This process depends on the dissolution of the permeating molecules in the barrier membrane.

Movement through pores and channels which involves the passage of the permeating molecules through solvent filled pores in the membrane rather than through the polymeric matrix itself.

4.2 .Mechanisms

A B

Diffusion through a homogenous nonporous membrane (A) and a porous membrane with solvent (usually water) filled pores (B)

4.2 .Mechanisms

Both mechanisms usually exist in any system and contribute to the overall mass transfer or diffusion.

Pore size, molecular size and solubility of the permeating molecules in the membrane polymeric matrix determine the relative contribution of each of the two mechanisms.

4.2 .Mechanisms

Passage of steroidal molecules substituted with hydrophilic groups in topical preparation through human skin involves transport through the skin appendages (hair follicles, sebum ducts and sweat pores in the epidermis) as well as molecular diffusion through the stratum corneum.

4.2 .Mechanisms

By combining the two mechanisms for diffusion through a membrane we can achieve a better representation of a membrane on the molecular scale.

A membrane can be visualized as a matted arrangement of polymer strands with branching and intersecting channels.

4.2 .Mechanisms

An illustration of the microstructure of a cellulose membrane used in the filtration processes showing the intertwining nature of fibers and aqueous channels

4.2 .Mechanisms

Depending on the size and shape of the diffusing molecules, they may pass through the tortuous pores formed by the overlapping strands of the polymer.

The other alternative is to dissolve in the polymer matrix and pass through the film by simple diffusion.

4.3 .Related Phenomena and Processes

Some processes and phenomena related to diffusion: Dialysis: A separation process based on unequal

rates of passage of solutes and solvent through microporous membranes (Hemodialysis).

Osmosis: Spontaneous diffusion of solvent from a solution of low solute concentration (or a pure solvent) into a more concentrated one through a membrane that is permeable only to solvent molecules (semipermeable).

4.3 .Related Phenomena and Processes

Ultrafiltration: A separation process for colloidal particles and macromolecules where a hydraulic pressure is employed to force the solvent through a membrane that prevents the passage of large solute molecules (albumin and enzymes).

Microfiltration: A process similar to ultrafiltration employing slightly larger pore size membranes (100 nm to several micrometers). Used to remove bacteria from intravenous injections, food and drinking water

4.4 .Fick’s First Law of Diffusion

Flux J is the amount, M, of material flowing through a unit cross section, S, of a barrier in unit time, t.

J = dM / S.dt

The units of the Flux J are g.cm-2.sec-1.


According to the diffusion definition, the flow of material is proportional to the concentration gradient.

Concentration gradient represents a change of concentration with a change in location.

Concentration gradient is referred to as dc/dx where c is the concentration in g/cm3 and x is the distance in cm of movement perpendicular to the surface of the barrier ( i.e. across the barrier).


J dc/dxJ = -D*(dc/dx) Fick’s First Law

In which D is the diffusion coefficient of the permeating molecule (diffusant, penetrant) in cm2/sec.

The negative sign in the equation signifies that the diffusion occurs in the direction of decreased concentration.

Flux J is always a positive quantity (dc/dx is always negative)


D is more correctly referred to as Diffusion Coefficient rather than constant since it does not ordinarily remain constant and may change with concentration.

Diffusion will stop when the concentration gradient no longer exists (i.e., when dc/dx = 0).

4.5 .Fick’s Second Law of Diffusion

Fick’s first law examined the mass diffusion across a unit area of a barrier in a unit time.

Fick’s second law examines the rate of change of diffusant concentration with time at a point in the system.

The diffusant concentration c in a particular volume element changes only as a result of a net flow of diffusing molecules into or out of the specific volume unit.


C OutputInput

Volume element

Bulk medium


A difference in concentration results from a difference in input and output.

The rate of change in the concentration of the

diffusant with time in the volume element (c/t) equals the rate of change of the flux (amount diffusing) with distance (J/x) in the x direction.

dc/dt = dJ/dx


dc/dt = dJ/dx

J = -D*(dc/dx) Differentiating with respect to x

dJ/dx = D*(d2c/dx2) substituting dc/dt from the top equation

dc/dt = D*(d2c/dx2)


dc/dt = D*(d2c/dx2)

This equation represents diffusion in the x axis only. In order to describe diffusion in three dimensional space, Fick’s second law can be written as

dc/dt = D*(d2c/dx2 + d2c/dy2 + d2c/dz2 )


C OutputInput

Volume element

Bulk medium

4.6 .Steady state Diffusion

An important condition in diffusion is that of steady state

Fick’s first law gives the flux (or rate of diffusion throughout unit area) in the steady state of flow.

Fick’s second law refers in general to a change in concentration of diffusant with time, at any distance, x (i.e., a nonsteady state of flow).

Steady state may be described, however, in terms of the second law.


Diffusion Cells: In a diffusion cell, two compartments are

separated by a polymeric membrane. The diffusant is dissolved in a proper solvent

and placed in one compartment while the solvent alone is placed in the other.

The solution compartment is described as Donor Compartment because it is the source of the diffusant in the system while the solvent compartment is described as the Receptor Compartment.


In diffusion experiments, the solution in the receptor compartment is constantly removed and replaced with a fresh solvent to keep the concentration of the diffusant passing from the donor compartment at a low level. This is referred to as the Sink Condition.


Receptor Compartment

DonorCompartment

Flux in Flux out

Mem

brane

Flow of solvent to maintain sink condition

Diffusant Solution

)c(

Pure

solvent


As the diffusant passes through the membrane from the donor compartment (d) to the receptor compartment (r), the concentration in the donor compartment (Cd) will fall while the concentration in the receptor (Cr) will rise.

However, the concentration in the receptor compartment is always maintained at very low levels because of the sink condition.

This means that Cr << Cd


When the system has been in existence for a sufficient period of time (determined by the nature of barrier and the rate of removal of the diffusant by the sink system), the rate of change in concentration in the two compartments with time will become constant.

However, the concentration in the two compartments is not the same.


dc/dt = D*(d2c/dx2) = 0

Since D is not equal to (0), then d2c/dx2 should be 0.

Since d2c/dx2 is a second derivative, and is equal to (0) the first derivative dc/dx should be a constant.

This means that the concentration gradient dc/dx across the membrane is constant (linear relationship between concentration c and distance or membrane thickness h)


h0

Thickness of barrier

Donor Compartment

Receptor Compartment

Cr

Cd

C2

C1

High concentration

of diffusant molecules


In such systems (diffusion cells), Fick’s first law may be written as:

J = dM / S.dt = D *(C1-C2)/h

C1 and C2 are the concentrations within the membrane and are not easily measured.

However they can be calculated using the partition coefficient (K)and the concentrations on the donor (Cd) and receptor (Cr) sides which can be easily measured


K = C1/Cd = C2/Cr Replacing C1 and C2 with KCd and KCr

dM / S.dt = D (C1-C2)/h = D(K Cd -K Cr)/h

dM / S.dt = DK(Cd - Cr)/h

dM / dt = DSK(Cd - Cr)/h


If the sink condition holds in the receptor compartment Cd>>Cr 0 and Cr drops out of the equation which becomes

dM / dt = DSKCd /h

The term DK/h is referred to as the Permeability Coefficient or Permeability (P) and has the units of linear velocity (cm/sec).

The equation simplifies further to becomedM / dt = PSCd


Permeability can be calculated from experimental data obtained from diffusion cells.- If Cd remains relatively constant throughout

time, P can be obtained from the slope of a linear

plot of

M versus t. M=PS Cd t


- If Cd changes changes appreciably with time, then P can be obtained from the slope of log Cd versus t.

log Cd = log Cd(0) - (PSt/2.303Vd)

4.7 .Controlled release coated tablets

Saturated solution in equilibrium with solid core

Unit Activity

Solid Core

Polymeric membrane Dissolution

Diffusion


If the excess solid in the dosage form is depleted, the activity () decreases as the drug diffuses out of the system and the release rate falls exponentially (First order release).

A constant activity dosage form may not exhibit a zero order release in the initial stages (membrane hydration, dissolution of part of the solid core to create the saturated solution)

and a lag time may be observed.


Lag time tL

Nonsteady state

Steady state

Time

Am

ou

nt D

iffused


The straight line in the last figure can be presented by the following equation:

M = SDKCd(t-tL)/h and the lag time is given by:

tL = h2/6D

When diffusivity can be calculated, presuming a knowledge of the membrane thickness, h Also, knowing P, the thickness h can be

calculated from

tL = h/6P

4.8 .Diffusion and Molecular Properties

Diffusion depends on the resistance to passage of a diffusing molecule and is a function of the molecular structure of the diffusant as well as the barrier material.

Gas molecules diffuse rapidly through air and other gases. Diffusivities in liquids are smaller

and in solids still smaller.

4. .Diffusion and Molecular Properties

Effect of the Prtial Molar Volme on Diffusion Coefficient

0

5

10

15

20

0 50 100 150 200 250Partial Molar Volume (cm 3/Mole)

DX10

6 (c

m2/s

ec)

5.7 .Percutaneous Absorption of Drugs:


Percutanous absorption include: Transcellular diffusion Diffusion through channels between the

cells Diffusion through sebaceous ducts Transfollicular diffusion Diffusion through sweat ducts


It is an oversimplification to assume that one route prevails under all conditions. Yet after steady state conditions have been established, transcellular diffusion through the stratum corneum most likely predominates.

In the early stages of drug penetration, diffusion through skin appendages ( hair follicles, sebaceous and sweat ducts) may be significant.

These “shunt” pathways are even important in the steady state diffusion in the case of large polar molecules e.g. some polar corticosteroids.


The factors influencing the penetration of a drug into the skin include: The concentration of the dissolved drug Cs.

The partition coefficient Kvs between the skin and the vehicle, which is a measure of the relative affinity of the drug for skin and vehicle.

Diffusion coefficients, which represent resistance to drug molecule movement through vehicle (Dv) and skin (Ds).


The relative magnitude of the two diffusion coefficients, Dv and Ds, determines whether release from vehicle or passage through the skin is the rate-limiting step.


In cases where the skin is the rate-limiting barrier, the diffusional equation may be written as:

- dCv/dt = (SKvsDsCv/Vh) Where:

Cv is the concentration of the dissolved drug in the vehicle (gm/cm3)

S is the surface area of application (cm2) Kvs is the skin – vehicle partition coefficient Ds is the diffusion coefficient of the drug in the

skin (cm2/sec) V is the volume of the drug product applied

(cm3)


The diffusion coefficient and skin barrier thickness may be replaced by the resistance to diffusion in the skin Rs.

Rs = h/Ds

And

dCv/dt = SKvsCv / VRs


In the experimental evaluation of percutaneous absorption, the drug diffusing through a skin barrier from an applied dosage form is measured in the receptor compartment.

The rate of loss of the drug from the donor compartment (vehicle) is equal to the rate of gain of the drug in the receptor compartment.

- V (dCv/dt) = VR (dCR/dt)so

dMR/dt = SKvsCv / Rs

anddMR = (SKvsCv / Rs) dt


By integrating the previous equation for the amount of the drug in receptor solution as a function of time

MR = SKvsCv t / Rs

The flux is this system is

J = MR/S.t = KvsCv / Rs

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