1.4.1 Diffusion coefficient
One of the most important parameters involved in eq.(4.93) (the others being connected to the
generative term Gi) is surely the diffusion coefficient Dim. However, up till now, nothing
about this parameter has been specified if not a generic definition. It is now necessary to be
more precise and to provide some more information on this topic. At this purpose, let’s focus
the attention on a fluid mixture composed by only two components named i and j,
respectively [Crank], being the following considerations easily extensible to a mixture
composed by r components. Once an appropriate frame of reference is chosen, this system
may be described in terms of the mutual diffusion coefficient Dij (diffusivity of i in j and
viceversa). Unfortunately, however, unless i and j molecules are identical in mass and size, i
mobility is different with respect to that of j. This leads to a hydrostatic pressure gradient
compensated by a bulk-flow (convective contribution to species transport) of the i - j mixture.
Consequently, the mutual diffusion coefficient is the combined result of the bulk flow and the
molecules random motion. Accordingly, an intrinsic diffusion coefficient (Di and Dj) can be
defined to account only for molecules random motion. Finally, using radioactively-labelled
molecules, it is possible to observe the rate of diffusion of i in a mixture, composed by
labelled and not labelled i molecules, where uniform chemical composition is attained. In so
doing, the self-diffusion coefficient ( Di* ) can be defined. It is possible to verify that,
theoretically, both Di and Di* are concentration and temperature dependent. Indeed, the force
f acting on a i molecule at point X is proportional the i chemical potential gradient [1 MD]:
f  μ i
(4.93)
Consequently, the total force fT acting on all molecules is proportional to:
f T  Ciμ i
(4.94)
where Ci represent i concentration. Assuming that the flux Fi is proportional to the total force,
we have:
1
Fi  
Ci
μ i
σi η
(4.95)
being i a resistance coefficient connected with diffusing molecules mobility [1,54 MD].
On the basis of eq.(4.38) we have:
Fi  
RT dln a i 
C i
σ i η dln C i 
(4.96)
Accordingly, Di expression is:
Di 
RT dln ai 
σ i η dln Ci 
(4.97)
This equation clearly states that Di is temperature and concentration dependent. Only in the
case of an ideal solution, the concentration dependence disappears as ai coincides with Ci. An
analogue dissertation developed on Di* leads to:
Di* 
RT
σ i η'
(4.98)
as, obviously, now ideal solution conditions hold. If i and i’ can be retained equal, we
have:
Di  Di*
dln ai 
dln Ci 
(4.99)
1.4.1.1 Diffusion in solid or solid like medium
In the case of diffusion in a solid phase or in a solid like phase such as gels (polymeric
networks entrapping a solvent), it is convenient to assume the solid, stationary phase as a
fixed reference and consider only diffusing solutes as mobile components [Flynn]. In this
light, the distinction between mutual and intrinsic diffusion coefficient is no longer needed
and we can simply speak about the diffusion coefficient of a solute in a known solid or
semisolid structure being eq.(4.97) still valid. To further particularize the scenario, it is
important to remember that, as mentioned before, a typical drug delivery system can be
2
thought as a three components system (or, at least, it can be idealized as a three components
system). The first one is the fixed structure that, usually, identifies with a three-dimensional
polymeric network stabilized by the presence of junction zones (called crosslinks and that can
be of physical or chemical nature) among different polymeric chains. Obviously, its diffusion
coefficient is zero. The second component is a solvent unable to disassemble network
structure because of the crosslinks presence. The last is the pharmacological active principle,
namely the drug. Of course, for the sake of clarity, at this stage, the presence and the
influence of possible excipients are neglected. As a consequence, we are interested in three
diffusion coefficients, namely that of the solvent phase in the dry polymeric network Ds, that
of drug in the pure solvent phase D0 (this is the mutual drug diffusion coefficient in the pure
solvent) and, finally, that of the drug in the polymeric network-solvent system D [PhD].
Indeed, although more complicated situations can take place as discussed in chapter 6,
usually, the interest is focused on dry polymeric network swelling by means of an external
solvent and on the diffusion of drug from an already swollen polymeric network.
Regardless the diffusion coefficient we are dealing with (Ds, D0 and D), its evaluation is
usually performed recurring to the hydrodynamic, kinetic and statistical mechanical theories.
While the first two approaches can be classified as molecular theories, as they are
mathematical expressions of the system (liquid, polymeric solution or crosslinked netywork)
physical view at the molecular scale, the last one can be defined as simulations theories as
they are based on atomistic simulations and they do not yield to a mathematical equation (or
set of equations) for diffusion coefficient evaluation. While, theoretically, this last category is
the most promising for what concerns diffusion coefficient prediction, due to huge
computational resources needed, at present, it is applicable only to very simple molecules
[Tonge 01]. In addition, up till now, no rigorous theoretical approaches exist for the
description of liquids and dense fluids, this being particularly true for polymeric solutions and
3
swellable crosslinked polymeric networks [Reis]. This is the reason why it is usual to study
liquids extending, in a semi-theoretical way, the above mentioned approaches [Liu 98].
According to the most famous hydrodynamic theory, the well-known Stokes-Einstein
equation, D0 can be evaluated according to:
D0 
KT
6 πηRH
(4.100)
where K is the Boltzam constant, T is absolute temperature,  is solvent (continuum medium)
shear viscosity and RH is the solute molecules hydrodynamic radius. This equation holds for
large, spherical shaped solute molecules immersed in a continuous solvent provided that the
resulting solution is a diluted solution. For non-spherical polyatomic molecules (freely
jointed Lennard-Jones chain fluids), Reis and co-workers [Reis 05] propose to evaluate D0
combining the kinetic theory of Chapman - Enskog model for dense fluid [Champan 70] with
van der Waals mixing rules [Grunde, 72].
The theoretical estimation of D must account for the fact that polymer chains slow down
solute movements by acting as physical obstructions thereby increasing the path length of the
solute, by increasing the hydrodynamic drag experienced by the solute and by reducing the
average free volume per molecule available to the solute. Accordingly, in addition to
hydrodynamic and kinetic theories, obstruction theory has been proposed [MD82]. Models
based on obstruction theory assume that the presence of impenetrable polymer chains causes
an increase in the path length for diffusive transport. Polymer chains, acting as a sieve, allow
the passage of only sufficiently small solutes. Carman [Mhur ref], schematizing the network
as an interconnected bundle of tortuous cylindrical capillaries having constant cross section,
demonstrates that D is given by:
D 1
 
D0  τ 
2
(4.113)
4
where  is tortousity, defined as the mean increase of the diffusion path due to the presence of
obstructions. For solute molecule of the same size as polymer segments, Mackie and Meares
[MD86], assuming a lattice model for the water-polymer hydrogel where polymer occupies a
fraction φ of the whole sites and, thus, solute transport occurs only within the free sites,
suggest:
D 1  


D0  1   
2
(4.114)
where φ represents also the polymer volume fraction. When solute molecules are much
bigger than polymer segments, the Ogston approach has to be considered [MD 87]. He
assumes that solute diffusion occurs by a succession of directionally random unit steps whose
execution takes place on condition that the solute does not meet a polymer chain. While
solute is assumed to be a hard sphere, the crosslinked polymer is thought as a random
network of straight long fibers of vanishing width. The unit step length coincides with the
root-mean-square average diameter of spherical spaces residing between the network fibers.
The resulting model is:
 rs  rf 1 2 

 

rf



D
 e
D0
(4.115)
where rs and rf are, respectively, solute and fiber radius. Deen [PHD] schematizes the
network as an ensemble of parallel fibers each one constituted by a series of non tangent
consecutive spheres of diameter rf. Applying the dispersional theory of Taylor [ref] to this
network, he estimates the ratio D/D0 in the case of solute diffusion perpendicular with respect
to fibers direction. The result of this analysis is approximately given by:
12
D
 e α 
D0
(4.116)
5
α  5.1768 - 4.0075 λ  5.4388 λ 2  0.6081λ 3
λ
(4.117)
rs
rf
Amsden [ref] assumes that solute movement through the polymeric network is a stochastic
process. Motion occurs through paths constituted by a succession of network openings large
enough to allow solute molecule transit (openings must me larger than solute hydrodynamic
radius). In addition, supposing that openings size distribution can be described by Ogston’s
equation relative to straight, randomly oriented polymer fibers [AM 31], he gets:
D
e
D0

 π  rs  rf
 
 4  r  rf





2





(4.118)
where r is openings average radius that is related to the average end-to-end distance between
polymer chains ξ according to:
r  0.5 ξ  0.5 k s 0.5
(4.119)
where ks is a constant for a given polymer-solvent couple. In virtue of the straight polymer
hypothesis, this model is applicable to network characterized by strong crosslinks typical of
chemically crosslinked polymeric networks. Nevertheless, by using scaling laws for the
description of the average distance between polymer chains, it is possible to render the model
suitable also for weakly crosslinked network as it happens for physically crosslinked
polymeric networks. In this case, indeed, mesh openings are neither constant in size and
location as, on the contrary, happens for strongly crosslinked polymeric network (see chapter
6 for a more detailed discussion about chemically and physically crosslinked polymeric
networks).
Hydrodynamic theory assumes that solute mobility, and thus its diffusion coefficient,
depends on the frictional drag exerted by liquid phase molecules entrapped in the network
[MD67]. In particular, polymer chains are seen as centers of hydrodynamic resistance as they
6
reduce the mobility of the liquid phase and this, in turn, reflects in an increased drag effect
exerted by liquid phase molecules on solute. The starting point of this theory is the StokesEinstein equation (eq.(4.100)):
D0 
KT
KT

f
6πηRH
(4.100’)
where f is the friction drag coefficient. Accordingly, this category of models is concerned
with the calculation of f. For strongly crosslinked gels (rigid polymeric chains), Cukier
[MD85] suggests:
 
 12
3Lc N A
 rs  

2 rf  

 
  M ln  L
D
 e  f c
D0
(4.120)
where Lc and Mf are, respectively, the polymer chains length and molecular weight, NA is
Avogadro number and rf is the polymer fiber radius. For weakly crosslinked gels (flexible
polymeric chains), the same author proposes:
D
D0
0.75
 e kcrs  
(4.121)
where kc is a parameter depending on the polymer solvent system.
In order to diffuse, a solute molecule must acquire the energy necessary to win the attraction
forces exerted by surrounding solvent molecules. In this manner, it can jump into adjacent
voids formed in the liquid space due to liquid molecules thermal motion. According to kinetic
Eyring theory [TranPhe], the most important step is the first one, while, for the kinetic free
volume theory [MD83], voids formation is the rate determining step. According to Eyring
theory, the expression of the solute diffusion coefficient D0 in a pure liquid reads:
D0  λ 2 k
(4.122)
where  is the mean diffusive jump length and k is the jump frequency defined by:
7

ε 


KT
k
Vf1/3e  KT 
2 π mr KT
(4.123)
where K is Boltzman constant, mr is the solvent-solute couple reduced mass, Vf is the mean
free volume available per solute molecule while  is an energy per molecule representing the
difference, between the energy molecule in the activated state and that at 0°K. The extension
of eq.(4.123) to the case of solute diffusion in a swollen polymeric network reads [PHD 1]:
1
 ε -ε' 
D  λ'   Vf  3  KT 
    e
D0  λ   Vf' 
2
(4.124)
where superscript refers to solvent-polymer properties. Unfortunately, the difficulty of
parameters estimation makes this equation not so useful in practical applications.
According to the kinetic free volume theory, solute diffusion depends on jumping distance,
solute thermal velocity and on the probability that there is an adjacent void (free volume)
sufficiently large to host solute molecule. Physically speaking, the free volume of a solvent
coincides with the difference between solvent volume (evaluated at fixed pressure and
temperature) and the volume occupied by all its molecules perfectly packed to be tangent
each other. The probability ph that a sufficiently large void forms in the proximity of the
diffusing solute molecule is given by:
ph  e
 V* 
 γ 
 V 
f 

(4.125)
where Vf is the mean free volume available per solute molecule, V* is the critical local hole
free volume required for a solute molecule to jump into and  is a numerical factor used to
correct for overlap of free volume available to more than one molecule (0.5 ≤  ≤ 1).
Accordingly, we have:
D0  vT λ e
 V* 
 γ 
 V 
f 

(4.126)
8
where vT is solute thermal velocity and  is jump length. Assuming negligible mixing effects,
the free volume Vf of a mixture composed by solvent, polymer and drug is be given by:
Vf  Vfd ω d  Vfs ωs  Vfp ω p
(4.127)
where Vfd, Vfs and Vfp represent, respectively, drug, solvent and polymer free volume, while
d, s and p are, respectively, drug, solvent and polymer mass fraction. Starting from
eq.(4.126) and (4.127), Fujita [PHD1], for small polymer volume fraction , finds the
following relation:
D
e
D0


 1

q
 P








(4.128)
where P and q are  independent parameters. Always for small  values, Peppas and
Reinhart [MD84], resorting to the free volume theory, arrive to:
 M c  M *c   k2rs2  1  
D


e 
 k1 
 M  M *c 
D0
 n

(4.129)
where k1 and k2 are two constants, M c is the number average molecular weight between
polymer cross-links, M n is the number average molecular weight of the uncross-linked
*
polymer, M c is a critical molecular weight between cross-links,  is polymer volume
fraction and rs is solute radius. Lustig and Peppas [Amdsen 16], introducing the idea of the
scaling correlation length between crosslinks , suppose that solute molecules can move
inside the three-dimensional network only if rs < . Accordingly, on the basis of the free
volume theory and assuming (1-rs/) as sieve factor, they suggest the following model:

  
D  rs   Y  1  
 1  e
D0   
(4.130)
9
where Y  γπλrs2 Vfs . Consequently, Y represents the ratio between V* (critical local hole
free volume required for a solute molecule to jump into times ) and the average free volume
per molecule of solvent. The same authors suggest that, for correlation purposes, Y can be
considered equal to one. According to Amsden [MD82], free volume and hydrodynamic
theories should be used to deal with weakly crosslinked networks, while for strongly
crosslinked networks obstruction theory is more consistent with the experimental data.
Table 4.1, showing some examples of solute radius rs and D0 (in water) values, makes clear
that, usually, the larger the solute molecule, the lower the corresponding D0. However, it can
be seen that also chemical properties play an important role as, for example, PEG 3978 is
characterised by a D0 that is one order of magnitude higher than those corresponding to
smaller solutes (ribonuclease, myoglobin, lysozyme, and pepsin). In addition, table 4.2 and
table 4.3 show the best fitting results obtained by considering three models deriving,
respectively, from the hydrodynamic, free volume and obstruction theory. While
hydrodynamic (eq.(4.121)) and free volume (eq.(4.130)) models are tested on weakly
(physically) crossliked network, the obstruction one (eq.(4.118)) is tested on strongly
(chemically) crosslinked network. While in the first case (hydrodynamic and free volume
models) polymer volume fraction  ranges between 0 and 0.5, in the second one,  ranges in
a smaller range (0   < 0.06). As for eq.(4.121), eq.(4.130) and eq.(4.120) the correlation
coefficient ranges, respectively, between 0.87 – 0.99, 0.75 – 0.99 and 0.77 – 0.98, a
reasonably good data fitting is achieved for all models.
Duda and Vrentas [PHD 12] apply the free volume theory to the polymer – solvent mixtures
assuming temperature constant thermal expansion coefficients, no mixing effects (solvent and
polymer specific volumes concentration independent), that solvent chemical potential s is
given by the Flory theory [ref] (see also chapter 6, paragraph 1.3.5.1):

μ s  μ s0  RT ln 1       χ 2

(4.131)
10
where μ s0 is the solvent chemical potential in the reference state and  is the Flory interaction
parameter, and that the following relations hold:
Ds 
Dss ρ s
RT
Dss  D0s e
  μs 


 ρ s  T, P
(4.132)
 ωs Vs*  ωp Vp*ξ 
 γ



VFH


D0s  D0sse
 E 


 RT 
(4.133)
where s, s, s and Vs* are, respectively, solvent density, chemical potential, mass fraction
and specific critical free volume, p and Vp* are, respectively, polymer mass fraction and
specific critical free volume, D0ss is a pre-exponential factor,  is a numerical factor used to
correct for overlap of free volume available to more than one molecule (0.5 ≤  ≤ 1), VFH is
the specific polymer-solvent mixture average free volume while  is the ratio between the
solvent and polymer jump unit critical molar volume (for small solvents, the jump unit
coincides with solvent molecule, while polymer jump unit coincides with the smallest
polymer chain rigid segment that, sometimes, can be the monomeric unit). On the basis of
these assumptions, they get:
Ds  1 -  1 - 2χD0s e
2
 ωs Vs* ωp Vp*ξ 




VFH γ


(4.134)
where:
ωs 
ρ s
ρ s 1     ρ p
ω p  1  ωs
VFH K11
K

1 K 21  T  Tg1   12 2 K 22  T  Tg2 
γ
γ
γ
(4.135)
(4.136)
where eq.(4.136) parameters (K11/, K12/, (K21-Tg1) and (K22-Tg2)), for several polymer –
solvent systems, can be found in literature [Seong, Duda 82, Vrentas 98]. Grassi and coworkers [JCR 68], studying drug release from swellable polyvynilpirrolidone, applies a
11
simplified form of eq.(4.134) and he finds, on data fitting basis, that water diffusion
coefficient in the polymeric matrix ranges between 10-10 cm2/s (dry state) and 10-7 cm2/s
(swollen state).
12