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Thermophysics and Aeromechanics, 2013, Vol. 20, No. 6

Peculiarities of diffusion in gels*

B.G. Pokusaev, S.P. Karlov, A.V. Vyazmin, and D.A. Nekrasov

Moscow State University of Mechanical Engineering, Moscow, Russia

E-mail: [email protected]

(Received April 30, 2013)

An optical method was applied to study the peculiarities of diffusion in gel: this method provides real-time visualization of spreading of solutes brought into the gel. It was shown that spectral characteristics of reflected light give additional information about nature of diffusive spreading of solutes and about state of the gel. Gels with different densities and lifetime were studied. These parameters have strong influence on the velocity of diffusion. The study demonstrated critical differences for diffusion process in gels with true solutions and with solutions with nanoparticles. Experiments discovered the anisotropy in 3D diffusion of solutes in gels; physical explanation of this phenomenon was proposed.

Keywords: gels, optical methods, diffusion, nanoparticles, aging, anisotropy.

Introduction

The usual definition for gels is a dispersive system with liquid dispersing medium, and the dispersion phase makes up a spatial structured mesh due to intermolecular interaction in the contact sites. Gels have such inherent qualities like a tendency to retain its structure, plas-ticity, resilience, and also thixotropy, i.e., the quality of restoring the gel original structure after mechanical destruction under isothermal conditions. All these properties, including the loss of fluidity, are attained at low contents of dispersed phase in the gel: from fractions of per-cent to several percents only. Gels are subjected to aging, i.e., their physical and chemical properties do change due to recondensation and recrystallization, including the process of releasing of liquid phase because of settling of the structured mesh [1, 2].

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 [3, 4]. Presently, the promising appli-cation of gels is the use in regenerative medicine for growth of different biological issues, including stem cells in vitro [5, 6]. 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 [7].

* Research was financially supported by the Education and Science Ministry of RF (Project code 7.7787.2013) and by RFBR (Projects No. 11-08-00368_а and No. 12-08-31243 mol_a).

© B.G. Pokusaev, S.P. Karlov, A.V. Vyazmin, and D.A. Nekrasov, 2013


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The processes of diffusion and thermal conductivity in macro-objects have been studied in detail [8−10], but specifics of these processes in microporous and microchannel-filled bodies requires more focused consideration [11−16]. The problems similar in physical nature have been studied for the case of heat transfer in structured porous bodies (see, e.g., [17]). The trans-fer processes in gels have such features as nonstationarity and anisotropy: they are determined by the nature, structure, and behavior of transfer medium [18]. 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 [19].

The study of transfer process in microsystems is often based on optical methods [20]. The optical methods are instrumental in collecting of data about fluid dynamics for oil dis-placement with water [21], for study of physical-chemical mechanisms of interfacial instability during chemosorption, and for study of microflows in the brim meniscus of wetting [22, 23]. Optical tools help in visualization of transfer process in real time mode and are 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 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 proof 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. Gel is microporous medium, therefore, it is natural to expect a difference for diffu-sion velocity for true solutions and solutions with nanoparticles. However, this difference in diffusion velocities of solutes in gels has not been studied yet. Finally, optical methods can be used for measuring the gel structure anisotropy through visualization of maldistribution of solute diffusional transfer, and this can offer physical explanation of this phenomenon.

Investigation objects and methods

For study of diffusion process in a plain diffusional layer, we used gels on the basis of sili-cates. 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/cm

3. The gel-generating mixture was prepared by adding the 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 through measuring the refractive index with the Abbe refrac-tometer. As it is shown below, this minor difference in the initial density of sodium silicate solu-tion produces differences in properties of final gels (optical and diffusion-related properties). Since the gel structure might be time-variable, we used both fresh and “aged” gels (with life-time more than 24 hours) for study of aging on transfer processes.

The study of gel microstructure influence on diffusion in a plain layer was conducted with three substances with very different physical and chemical properties. One substance was a water solution of potassium permanganate KMnO4, which is a “true” solution, being a mixture at molecular level. Another sample was water-based ink, which is an aquatic suspension of nano-particles with wide dispersion composition. The third sample was inkjet ink ⎯ nanoparticle suspension with a narrow dispersion available in methylethylketone (organic solvent). All these substances are gel-inert additives, that is, diffusion of these solutes does not change gel properties.

For study of diffusion in volume, we tested the sample of hydrogels based on hyaluronic acid modified with a cross-linker; the cross-linker might be alfa-, omega-diamines. The cross-linker provides additional chemical bonds between different polymer chains making a hydrogel matrix, and this cross-linking creates a special 3D structure of gel. As an optical indicator (the substance for transfer) we used a gel-neutral aquatic dye solution.


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The optical method was used for estimating 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 setup diagram is shown in Fig. 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 mi-croscope’s digital camera is focused on the boundary of transport substance (solute) traveling in the gel volume. The setup was designed for recording 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.

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 diffu-sion coefficient is based on the velocity of transport of a constant-concentration solute inside a gel matrix [24]. To estimate the order of the diffusion coefficient, the following formula was applied:

2 ( ),D x tα= (1)

where D is the diffusion coefficient, x is the iso-concentration plane displacement velocity, t is time, and α is the coefficient that accounts for interaction of diffusive substance with the me-dium (in our calculations, we used α = 6).

Results and discussion

For the first stage of experiment, the refractive index of sodium silicate solutions was measured at different concentrations of corresponding gels. Measurements were conducted at the temperature of 21 °С. This data helped to control reproducibility of gel state during mea-surements. The results are shown in Fig. 2. One can see from this graph that the refractive in-dex is higher for original solution than for derived gel. This effect is linked to a change in the molecular structure of silicate sodium and development of structured mesh: this structure gives us a gel. Besides, we noticed a slower increment of refractive index with the density growth for a gel than for a solution.

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 phenome-non can be a foundation for developing a spectral method for analysis of gel structure.

Figure 3 depicts a picture of a fragment (corresponding to the same time moment) of a posi-tion of the concentration front in a plain cell for the case of vertical diffusive motion of solutes through a sodium silicate gel with a concentration 1.04 g/cm3. The pictures from left to right show the cases of diffusion of KMnO4 solution, inkjet ink, and water ink. The process of diffusion

Fig. 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.


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was initiated in three cells simultaneously. Figure 3 demonstrates that diffusion of KMnO4 solution in the gel matrix takes place with a high velocity; the diffusive smearing of inter-face is visible, and the visual pattern of diffusion fits the classic idea of diffusion dyna-mics. As for diffusion of two types of inks in gel, their diffusion velocities are close. The smearing of interface for spreading water-based ink is visible. On the opposite, the diffusion of inkjet ink in gels creates a distinct boundary; this might be explained by presence of organic solvent in the ink which forms the interface due to surface tension.

Here we use the collected experimental data on diffusion velocity and the method of iso-concentration planes for estimating the diffusion coefficient on the basis of relation (1) for all

tested substances: the diffusion coefficient for KMnO4 in gel was 2.9×10−9 m2/s, inkjet ink ⎯

1.1×10−9 m2/s, and for water-based ink ⎯ 0.96×10−9 m2/s. The difference in diffusion coefficients of true solution (KMnO4) and nanoparticle suspension (ink) is determined by a difference in sizes of transported particles, which are critical for particle-matrix interaction.

The Einstein formula [10] for the diffusion coefficient of a Brownian particle (and nanopar-ticles suspended in liquid belong to this class) tells that the diffusion coefficient is inversely pro-portional to the hydrodynamic drag of a particle in viscous fluid. We know that the nanoparticle size in inks is about the size of structural mesh of the gel (or possibly bigger), this drag force may be rather high and it prevents spreading of particles in gel (on the molecular level for KMnO4 diffusion, this makes impact on the value of transfer coefficient due to reduction of mole-cular mobility).

When diffusion of water-based ink takes place in gel, only nanoparticles of finest fraction spread out. However, motion of the painted front of inkjet ink took place in out experiments. The possible explanation is the following. We observe diffusion of free water molecules from the gel into the organic phase. This transition of water from one phase to another phase creates a local osmotic rarefaction. This, on the one hand, creates the additional motive transfer force revealed as excessive pressure in organic phase which would promote interface (together with colorant nanoparticles) inside the gel; on the other hand, there is a possibility of changing the gel structure near the interface (the structured mesh becomes loose).

The regular visual observation for the condition of diffusive front is good for qualitative study only. A more accurate approach is an expe-riment on measuring the relative transmittance (or absorbance) of white light passing through the gel and the diffusing substance. Meanwhile the sub-stances of the studied system have different

Fig. 2. Refractive indices for sodium silicate solutions and corresponding silicate gels.

1 ⎯ liquid solutions, 2 ⎯ fresh gel with the same density.

Fig. 3. Position of a concentration front in a flat cell while vertical diffusive movement of the solute in a gel. Pictures from left to right: KMnO4, inkjet ink, water-based ink.


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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 Fig. 4,

where we have shown the dynamics of light transmittance for a diffusive component at differ-ent 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 inte-raction between the component and structures of the silica gel. Therefore, the data for transmit-tance 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 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 meaning-less 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). These results are plotted in Fig. 5. 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.

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−−−∂

∂=∂∂

tCCKx

CD

t

C, (2)

where С is the transfer substance concentration, t is the current time, x is the longitudinal coor-dinate, 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 struc-tured 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. Thus, 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 equation (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 accu-mulated solute.

Fig. 4. Spectral-revealed curves of solute front movement in the gel. The abscissa axis plots the distance H from the ini-tial solute front while diffusion into the gel (gel is on the right, and solvent is on the left); the ordinate axis is the relative intensity A of transmitted light at given wavelength (the complete transparency corresponds to num-ber 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; 2, 4, 6 ⎯ green light.


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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, then we obtain:

+∂∂−=− t

CtCC t ττ )( . (3)

Substitution of (3) in equation (2) gives us an equation

2

ef 2,

C CD

t x

∂ ∂=∂ ∂

(4)

where

ef 1

DD

Kτ=

+. (5)

This relation for diffusion coefficient in gels was developed in paper [15] from phenomeno-logical ideas. Relation (5) for diffusion coefficient in gels is included into a single complex of values Kτ, 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 relation (5) was obtained for small relaxation times τ, but when this condition is not fulfilled, the reverse problem of diffusion on the basis of equation (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 in diffusion processes in true solutions, but for diffusion of nanoparticles in gels the time evolution of coefficients might be more complicated.

Fig. 5. 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.

Fig. 6. Process of diffusion and time dependence of boundary position.

On the left ⎯ picture of colorant distribution in the gel volume while 3D diffusion; on the right ⎯ time- fixed (after two minute interval) positions of the colorant zone within the gel volume.


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One of interesting features of gels is anisotropy, which means a difference in diffusion velocity for different directions of matter transfer. To study this phenomenon, we use a sample of hyaluronic acid based gel thickened with a cross-linker. An inert colorant was injected into this sample via a syringe. For this situation, a three-dimensional diffusion of die took place. For every two minutes, the colorant boundary position was recorded with a video-camera; this gave a 2D image of the iso-concentration surface vs. time. A picture of diffusion process and image of time dependency of the boundary position is shown in Fig. 6. The experimental data are presented in detail in paper [18].

After processing of experimental data for tested hydrogel samples with formula (1), the following average values of diffusion coefficients have been obtained: in vertical direction

Dx = 0.41×10−9 m2/s, and for horizontal direction Dy = 0.98×10−9 m2/s. The latter value of diffusion coefficient exactly fits the result published in [14] for this medium. The aniso-tropy in diffusion is explained by anisotropy of gel properties developed due to features of gel-ling regime: polymerization conditions for longitudinal direction are different from those for transverse direction. The possible reason for this difference for impact of gravity: gravity force enhances settling of polymerized gel particles and gel densification in the longitudinal direction.

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