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CHEMICAL ENGINEERINGTRANSACTIONS
VOL. 43, 2015
A publication of
The Italian Association
of Chemical Engineering
Online at www.aidic.it/cet
Chief Editors:SauroPierucci, JiříJ. Klemeš
Copyright © 2015, AIDIC ServiziS.r.l.,
ISBN 978-88-95608-34-1; ISSN 2283-9216
Diffusion Phenomena in Gels
Boris G. Pокusaev, Sergey P. Karlov, Andrey V. Vyazmin*, Dmitry A. Nekrasov
Moscow State University of Mechanical Engineering, ul. Staraya Basmannaya 21/4, Moscow, 105066, Russia
av1958@list.ru
The formulation and solution of fundamental problems of the study of transport phenomena in micro-and
nano-porous gels are considered. To study the diffusion in the gels the optical methods are used. This method
allows a real-time visualization of the introduced impurities spreading in gels. It is found that the
decomposition of the spectrum of the reflected radiation in the optical range allows one to get more
information not only about the nature of the diffusion distribution of the considered impurities, but also on the
state of the gel itself. Investigated gels are of different lifetimes, whereby a strong influence of these
parameters on the rate of diffusion is identified. Relaxation processes must also be taken into account in
mass-transfer models describing diffusion in gels. The time-dependent model of diffusion laws is of particular
interest. Its application to the modeling of mass transfer processes will bring further understanding of cause
and effect relationship of relaxation and diffusion processes.
1. Introduction
Gel is a specific two-phase system. One phase represents a relatively sparse spatial network of polymer
molecules which are bonded at the intersections of intermolecular bonds. The second phase is a liquid (in
some cases - the water), which occupies the space between the polymer molecules. This fluid does not have
any chemical bond with the polymer molecules, i.e. does not form chemical compounds with them.
All these features create interesting applications of gels in industries and technologies. Gels are widely used in
production of polymers, catalysts, sorbents, membrane filters, service wellbore fluids, cosmetic and pharmacy
compositions (Kajiwara and Osada, 2000). Presently, the promising application of gels is the use in
regenerative medicine for growth of different biological issues, including stem cells in vitro (Westrin and
Axelsson, 1991). The gel capillary network can be used for transport of nutrients to separate cells with culture
broth and for removal of cell metabolism. One of variants of problem solution is creation of artificial hydrogel
matrix with complicated spatial system of microchannels: this task assumes formulation and solving of
fundamental problems for study of micro- and nano-porous systems in hydrogel matrix (Amsden, 1998).
The peculiarity of mass transfer of gels is diffusion movement in microchannels of complex shape and
structure, skirting macromolecules dispersed phase, the interaction with which strongly affects the rate of
transfer (Fang et al., 2008). Gel structure is spatially and temporary heterogeneous. For this reason, the
appearence of stagnitiion areas on the microscale is possible where the diffusion transfer occurs with a time
delay. Thus, diffusion at the micro level is always three-dimensional, and the velocity is locally unstady and it
is not isotropic in space (Pokusaev et al., 2013). This particularity of mass transfer is important for unsteady
processes when diffusive relaxation processes are not completed. In this case, the mass transfer at the
microscale is not described by a simple one-dimensional diffusion equation with constant coefficient.
The transfer processes in gels have such features as unsteady and anisotropy: they are determined the
nature, structure, and behavior of transfer medium (Kazenin et al., 2011). For this type of systems the critical
aspect is the effect of solute size in comparison with inner scales of microchannels within the transfer medium
(Ozturk et al., 2010). For characterization of the unsteady mass transfer in the gel on the macro scale, one
uses the effective diffusion coefficients, which differ from the real diffusion coefficients of the matter in fluid
(Lauffer, 1961).
DOI: 10.3303/CET1543281
Please cite this article as: Pokusaev B., Karlov S., Vyazmin A., Nekrasov D., 2015, Diffusion phenomena in gels, Chemical Engineering
Transactions, 43, 1681-1686 DOI: 10.3303/CET1543281
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There are many publications relating to the study of the diffusion in gels, either by experimental
measurements of diffusion coefficients for gels of different chemical compositions (see, for example, research
of Muhr and Blanshard, 1982), or by different degrees of validity for the relation the calculation of the diffusion
coefficients (critical analysis of some of them is given, for example, in Amsden, 1998). In recent years, a
growing understanding of the mass transfer in microstructure environments cannot be described by Fick's law,
even when using the effective diffusion coefficients (Neuman and Tartakovsky, 2009).
The goal of this paper was experimental study of diffusion in gels with optical methods. That is why we studied
optical properties of initial solutions and the produced gels with different density and lifetime. One of objectives
was experimental prove of relation between the solute transfer rate and the local-time state of the gel, since
the gel is a medium with time-varying properties. At last, optical methods can be used for measuring the gel
structure anisotropy through visualization of distribution of solute diffusion transfer, and this can offer physical
explanation of this phenomenon.
2. Methods and objects
The study of transfer process in microsystems is often based on optical methods. For example, the optical
methods are instrumental for study of physical-chemical mechanisms of interfacial convection during laser
heating (Karlov et al., 2013), and for study of microflows in the brim meniscus of wetting (Karlov et al., 2002).
Optical tools help in visualization of transfer process in real time mode and useful for understanding the
physical and chemical features of the process. They are useful for obtaining of direct data on macrokinetic
transfer coefficients; in some cases these coefficients can be related to parameters describing the intrinsic
structure of the transport medium. The importance of the use of imaging techniques in modern science
confirmed the award of the Nobel Prize in Chemistry in 2014 for his work on the development of such
methods.
Optical techniques are commonly used to determine impurity transfer coefficients in a steady homogeneous
medium (see, for example, research of Leel et al., 2004) and in gels (Helseth, 2011). Determining of diffusion
coefficients was based on measuring of distribution of solute concentration over the distance, that is, we
tracked down the movement of iso-concentration planes. This method is known as a method of moving
boundaries, and the calculation of diffusion coefficient is based on the velocity of transport of a constant-
concentration solute inside a gel matrix (Malkin and Chalykh, 1979). Wherein the magnitude of the transfer
coefficient can be influenced by convection, surface effects, etc., the presence of which cannot be identified by
this method.
The optical method was used for estimating of the diffusion coefficient for a solute in a gel. This choice of
optical methods was supported by high accuracy of methods, contactless nature, and informative value of
tools. The experimental diagram is shown in Figure 1. The setup is a microscope with an illumination system
equipped with a device for placement and movement of a cell comprising a gel sample. The setup has an
option of placing an additional (external to the cell) box with immersion liquids for ameliorating the effect of cell
wall curvature. The microscope’s digital camera is focused on the boundary of transport substance (solute)
traveling in the gel volume. The setup was designed for recoding a flat front of the moving boundary of the
transport substance or a spherical front from a point source. The latter version was needed for estimating the
properties of diffusion anisotropy in gels.
To estimate the order of the diffusion coefficient, the following formula was applied:
)4(2 txD = , (1)
where D is the diffusion coefficient, x is the iso-concentration plane displacement velocity,
t is time. Eq(1) follows from the expression for the self-similar variable of the diffusion equation.
For study of diffusion process in a plain diffusion layer, we used gels on the basis of silicates. Gels of different
density were produced. The stock solution of sodium silicate was diluted with distilled water to the different
levels of density. The gel-generating mixture was prepared by adding of hydrochloric acid to the salt solution
and the mixture pH was adjusted to the range 3.5 − 4.5. The density of silicate aquatic solution was controlled
by refractometry method. Since the gel structure might be time-variable, we used both fresh and “aged” gels
(with lifetime more than 24 h) for study of aging on transfer processes.
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Figure 1: Experimental setup diagram: 1 ⎯ cell with gel sample, 2 ⎯ moving boundary of the marker
substance, 3 ⎯ system for marker feeding,4 ⎯ the additional cell with immersion fluid, 5 ⎯ light beams,
6 ⎯ illumination system, 7 ⎯ microscope, 8 ⎯ digital camera
The study of gel microstructure influence on diffusion in a plain layer was used a water solution of potassium
permanganate KMnO4, which is a “true” solution, being a mixture at molecular level. This substance is gel-inert
additives, that is, diffusion of this solute does not change gel properties.
3. Results and discussion
The dependency of refractive index on gel density brings a conclusion about a difference of reflection and
transmission spectra for gel samples at different densities, and this phenomenon can be a foundation for
developing a spectral method for analysis of gel structure.
The regular visual observation for the condition of diffusive front is good for qualitative study only. A more
accurate approach is an experiment on measuring the relative transmittance (or absorbance) of white light
passing through the gel and the diffusing substance. Meanwhile the substances of the studied system have
different spectral absorption characteristics, so the data on light absorbance at different wavelength of light are
very informative. The data on KMnO4 in the sodium silicate gel are plotted in Figure 2, where we have shown
the dynamics of light transmittance for a diffusive component in different points of the gel sample. These
spectral studies demonstrated that measurement of diffusion in gel with a moving boundary method for
different wavelengths of probing light gave different values for diffusion coefficient for the same component;
this can be indication of selective interaction between the component and structures of the silica gel.
Therefore, the data for transmittance spectra for silica gel samples might be used for study and control of
sorption features of the gel matrix (when the multiphase and multicomponent filtration flows take place).
As it was mentioned above, the gel is not a steady structure. Processes of recondensation with variation and
thickening of the structured mesh occur in the gel; that is, the medium of diffusion changes with time (they call
this process as “gel aging”). Therefore it is meaningless to tell about transfer coefficients in gels beyond the
time limits. This fact was confirmed in experiments: we measured the diffusion coefficient for KMnO4 in silicate
gels with different densities and different lifetime (a fresh gel and one-day aged gel). Gels of different density
were produced. The stock solution of sodium silicate was diluted with distilled water to the levels of density
1.04, 1.068, and 1.09 g/cm3. The density of silicate aquatic solution was controlled by refractometry method
through measuring the refractive index with the Abbe refractometer. The results are plotted in Figure 3. As it is
shown below, this minor difference in the initial density of sodium silicate solution produces differences in
properties of final gels (optical and diffusion-related properties). This graph shows that the diffusion coefficient
decreases with the gel density (this is predictable due to condensing of gel mesh) and with gel lifetime. When
relaxation processes take place, the structured mesh also changes and becomes denser.
It is experimentally found that there are two characteristic stages in time for unsteady mass transfer in the gel
that is rendered in Figure 4. For the experiments an "old" gel designed with an open surface more than a day
is used. It is assumed that by this time it all relaxation processes are fully completed. Part of the dispersion
liquid evaporated, whereby the microscopically in homogeneous multiphase structure is formed along the
inside boundary. A part of the gel away from its borders remains in liquid-phase. In the border areas of the gel
liquid evaporated, resulting in a formation of air microchannels.
There is a fractal stage at the initial stage of unsteady mass transfer in a diffusing substance across the border
of the dried gel (object 3 in Figure 4). This stage lasts during the first minute from the starting of the process.
Concentration front has a pronounced fractal structure. Then the diffusion step begins (object 4 in Figure 4),
when the fractal boundary starts to blur. Far from the boundary of the gel, we observed the interaction
between processes of fractal and diffusion mass transfer (object 5 in Figure 4).
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Figure 2: Spectral-revealed curves of solute front movement in the gel. The abscissa axis plots the distance
from the initial solute front while diffusion into the gel (gel is on the right, and solvent is on the left), mm; the
ordinate axis is the relative transmittance intensity of light at given wavelength А (the complete transparency
corresponds to number 256, the complete absorbance stands for 0). Position of diffusion front: 1, 2 - initial, 3,
4 - intermediate, 5, 6 - final. Wavelengths: 1, 3, 5 - red light (650±2 nm), 2, 4, 6 - green light (530±2 nm).
Probable mechanism of the fractal surface formation at the initial stage of mass transfer in the gel is the
movement of colored liquid on the dried capillaries under the influence of the Laplace pressure. The
movement of colored liquid in each capillary is stops, when the transferred substance merges with the gelling
liquid (water). Compare to the structural channels formed by dispersion phase, this is macroinhomogeneity a
part of which filled by the water. Thus, the diffusion process goes not only toward the gelled fluid from the
fractal surface, but also within the already colored volume filling macroinhomogeneity by diffusing substance.
Let us consider the process of diffusion in gel with the evolving structure. We assume that recondensation in
gel produces temporary local stagnation zones without any mass transfer to the ambient zones; the solute
concentration there becomes frozen for a time. This is a typical time of gel relaxation τ , which describes the
duration of gel restructuring processes.
For this kind of diffusion model, a 1D nonstationary diffusion equation for gel can be written in the following form:
)(
2
2
τ−−−∂
∂=
∂
∂
t
CCK
x
CD
t
C , (2)
Figure 3: Dependency of diffusion coefficient (D×10−10 m2/s) in sodium silicate gel on its density (ρ, g/cm3)
while gel aging: 1 ⎯ fresh gel, 2 ⎯ old gel.
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Figure 4: Various stages of water diffusion in the gel KMnO4 solution: 1 - the thin layer of gel; 2 – the border of
the gel; 3 – the initial (fractal) stage of droplets diffusion in the gel; 4 - diffusion stage; 5 – interection of fractal
and diffusion stages for drops inside the gel.
where С is the transfer substance concentration, t is the current time, x is the longitudinal coordinate, and D
is the diffusion coefficient of the transfer substance in the liquid phase of gel. Averaged over volume and time
τ , the transfer coefficient between the typical and dense structured mesh zones K depends on the gel density
(and this produces dependency on the current time of diffusion process t), on the volumetric fraction of gel
with dense structure, diffusion coefficient D, and the scale of developed dense structures. The value of K, the
same way as τ , is a characteristic of the intrinsic structure of gel. Note here that the concentration in the last
term of Eq(2) is taken at moment of t − τ , i.e., since the formation of a stagnant zone which has been
decomposed till this moment and released out the accumulated solute.
If we assume the relaxation time to be a small parameter (this is not a compulsory condition for gels), we can
decompose τ−tС into a series in t and keep only two main terms of the series, and then we obtain:
+
∂
∂−=− t
CtCC
t
ττ )( , (3)
Substitution of Eq(3) in Eq(2) gives us an equation
,
2
2
ef
x
CD
t
C
∂
∂=
∂
∂
(4)
where
.
1ef τK
DD
+
= (5)
This relation for diffusion coefficient in gels was developed by Lauffer (1961) from phenomenological ideas. The
Eq(5) for diffusion coefficient in gels is included into a single complex of values τK , which describes the
intrinsic structure of gel and can be calculated from direct experimental data. Thus, the optical-spectral study
of diffusion in gels gives information about dynamics of gel intrinsic structure.
We should note that Eq(5) was obtained for small relaxation times τ , but when this condition is not fulfilled,
the reverse problem of diffusion on the basis of Eq(2) should be considered. For this situation the parameters
K and τ are the functions of current time t, so time interpolation of these parameters can be done with
exponential laws. As the gel aging process continues, the transfer coefficient decreases. We should
emphasize that this type of time dynamics is inherent for diffusion processes in true solutions, but for diffusion of
nanoparticles in gels the time evolution of coefficients might be more complicated.
The physical meaning of the time delay in the present model of mass transfer is associated with the inertial
properties of the system that responds to the impact is not instantaneously, but rather with a time lag.
Experimental measurements show that the rate of mass transfer in the gel is less than for diffusion in a
homogeneous fluid (water). The probable reason for this is the heterogeneity of the microstructure of the
transfer medium. The rapid transfer of unsteady diffusion material in the direction opposite to the gradient of
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the initial concentration occurs along the longitudinal channels. Slow unsteady three-dimensional diffusion of
internal filling of the dead space with clean water occurs under the influence of small local concentration
gradients. The model relates the effective diffusion coefficient with the real diffusion coefficient for fluid.
The dependence of the mass transfer rate in gels on the geometry of the diffusion zone is experimentally
detected. On this basis, it can be assumed that in the gels the convective mechanism taking place due to
various physical and chemical mechanisms. However, due to the high resistance the velocity of the gelled fluid
is much lower than in a pure fluid.
4. Conclusions
Spectral studies show that the measurement of gel diffusion method, by the technique of the moving boundary
for different wavelengths of the probing radiation, gives different values of the diffusion coefficient. This
indicates the selective interaction of the diffusible component with microstructural elements of silica gel.
Visualization of unsteady filling in the horizontal thin layer of "old" (dried) gel by colored substances allows one
to find a fractal structure of the original concentration front. The reason for its occurrence is the movement of
colored liquid on thin dry microchannels due to capillary forces due to Laplace pressure. There is a possibility
of forming in gel at the initial moment of a mass transfer of inhomogeneous volume zones with different
concentrations of the substance.
Based on the analysis of the experimental data, a new model of non-stationary diffusion with a time delay in
micro-inhomogeneous medium is presented. The model takes into account the heterogeneity of the gel
structure related to the existence of local stagnant zones. In these zones, the diffusing material for some time
is not communicating with the environment and therefore its concentration is kept constant. Here the
characteristic time delay is taken into account. This time delay characterizes the duration of the internal
transitional processes of mass transfer. The effective value for the diffusion coefficient in gels is obtained by
comprising a single set of parameters characterizing the processes micro transfer in a structured environment.
Acknowledgements
This research was supported by the Russian Foundation for Basic Research (projects 14-08-00626a).
References
Amsden B., 1998, Solute diffusion within hydrogels: mechanisms and models, Macromolecules, 31, 8382–
8395
Fang H., Wan R., Gong X., Lu H., Li S., 2008, Dynamics of single-file water chains inside nanoscale channels:
physics, biological significance and applications, Journal of Physics D: Applied Physics, 41, 103002.
Helseth L.E., 2011, A simple experiment for visualizing diffusion, European Journal of Physics, 32, 1193–1197
Kajiwara K., Osada Yo., Eds., 2000, Gels Handbook. Four-Volume Set. Elsevier, Amsterdam, Netherlands.
Karlov S.P., Kazenin D.A., Vyazmin A.V., 2002, The time evolution of chemo-gravitational convection on a
brim meniscus of wetting, Physica A, 315, 236-242.
Karlov S.P., Kazenin D.A., Vyazmin A.V., Pirogova O.V., Okhotnikova K.A., 2013, Convection during the fluid
heating by laser emission, Chemical Engineering Transactions, 32, 565-570, DOI: 10.3303/CET1332095.
Kazenin D.A., Karlov S.P., Nekrasov D.A., Pokusaev B.G., 2011, Express-analysis of diffusion properties of
gels for arrangement of filtration flows in soils, Ekologiya & Promyshlennost Rossii, 1, 13−15 (in Russian).
Lauffer M., 1961, Theory of diffusion in gels, Biophysical Journal, 1, 205−213.
Leel S., Lee H., Lee I., Tseng C., 2004, Ink diffusion in water, European Journal of Physics, 25, 331–336.
Malkin A.Ya., Chalykh A.E., 1979, Diffusion and Viscosity of Polymers, Measurement Methods, Khimia,
Moscow, Soviet Union (in Russian).
Muhr A.H., Blanshard J.M.V., 1982, Diffusion in gels, Polymer, 23, 1012−1026.
Neuman S.P., Tartakovsky D.M., 2009, Perspective on theories of non-Fickian transport in heterogeneous
media, Advances in Water Resources, 32, 670–680.
Ozturk S., Hassan Ya.A., Ugaz V.M., 2010, Interfacial complexation explains anomalous diffusion in
nanofluids, Nano Letters, 10, 665−671.
Pokusaev B.G., Karlov S.P., Vyazmin A.V., Nekrasov D.A., 2013, Peculiarities of diffusion in gels,
Thermophysics and Aeromechanics, 20, 749-756.
Westrin B.A., Axelsson A., 1991, Diffusion in gels containing immobilized cells: a critical review, Biotechnology
and Bioengineering, 38, 439–446.
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