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silicate gels
Colloids and Surfaces A: Physicochem. Eng. Aspects 246 (2004) 9–19
Diffusion of H+, H2 O and D2 O in polymer/silicate gels I. Lakatos∗ , J. Lakatos-Szab´o Research Institute of Applied Chemistry, University of Miskolc, P.O. Box 2, Miskolc-Egyetemv´aros, H-3515, Hungary Received 10 September 2003; accepted 16 June 2004 Available online 12 September 2004
Abstract General features of H+ , H2 O and D2 O diffusion in polymer/silicate gels applied in different industrial areas were analyzed. It was found that the cumulative mass transport curves consist of two sections: an unsteady-state and a steady-state one. The length of the transient period is strongly dependent on the silicate content of the gel and the concentration gradient of H+ ions. Using the length and the intersection point of these periods the effective diffusion coefficient, break-through time and the ion retention capacity of gel could be calculated. The obtained effective diffusion coefficients were very close to the values characteristic in aqueous solutions. The diffusion mass transport could be described by the modified Fick’s I law, however, it was stated that on account of the high ion retention in all gels the one-dimensional random-walk equation may not be used for prediction of the mean diffusion distance (break-through time). Using the formation factor of gel (effective “porosity” and tortuosity) the effective diffusion coefficient can be predicted with good accuracy and vice versa, the gel structure can be determined by the absolute and the effective diffusion coefficients characteristic in bulk aqueous phase and gel, respectively. The laboratory experiments provided valuable new information and data to design and formulation of the industrial “gel” technologies in both the enhanced oil recovery and the environmental protection. © 2004 Elsevier B.V. All rights reserved. Keywords: Diffusion; Gel; Polymer; Silicate; Hydrogen ion; Water; Heavy water
1. Introduction The use of electrolyte containing gels in batteries and electric cells goes back to more than a century. Similarly, the application of hydrated gels as flow profile modifiers and barrier forming agents opened up new vistas in enhanced oil recovery and environmental containment technology. During the past decades focusing on such topics a great variety of gelforming chemicals have been tested under laboratory and industrial conditions, and some of them proved very useful, indispensable and effective in practice. Analyzing the negative gel properties, however, it became evident that the transport properties being as main factors of the gelation, conductivity, block-forming capacity, long term usability, etc. are sometime poor and the mass transport mechanism in gels is not ∗
Corresponding author. Tel.: +36 46 565 255; fax: +36 46 363 349. E-mail address: [email protected] (I. Lakatos).
0927-7757/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2004.06.035
completely understood. Consequently, the diffusion of different chemical species in self-conforming phases formed prior to application or in situ in formation rocks, soils, etc. might be fundamental, and the transport process may influence significantly the efficiency of different industrial technologies to be applied in relevant areas. Despite these facts, the possible role of diffusion in such procedures—particularly in improved/enhanced oil recovery—sometimes are misunderstood or either not mentioned in literature at all [1], or it is considered as a negative factor resulting in a dilution or depletion of the reacting species in the mixing zones or the gel itself [2]. Fortunately, our knowledge on diffusion transport in gels proceeds well and these research areas (in material sciences, electrochemistry, etc.) are often revisited even today [3–6]. Initiating by the strong industrial demands, a complex gel system containing silicates and water-soluble polymers was developed with the aim at improving the efficiency of selective fluid shut-off treatments and containment technologies. The new methods were tested successfully at different Hun-
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Nomenclature an A Aeff D Deff F n t t0 t1 V x
specific species retention in gel (mol/m3 ) cross-section of gel (m2 ) effective cross-section of gel (m2 ) diffusion coefficient in the aqueous bulk phase (m2 /s) effective diffusion coefficient in gel (m2 /s) formation factor cumulative mass transport (mol) time (s) break-through time of a non-adsorbing species (s) break-through time of an adsorbing species (s) volume of gel (m3 ) thickness of gel (m)
Greek symbols c concentration difference between cells (mol/l) Φ porosity τ tortuosity garian oil fields for restriction of water production and gas coning in oil wells. Since multivalent cations were used as cross-linking agents the earlier investigations were concentrated on determination of the diffusion mass transport of different chromium ions in polymer/silicate gels [7,8]. Independent variables of the laboratory studies were the valency of chromium, and geometry and composition of gels. Simultaneously the effective diffusion coefficient, break-through time and retention of ions in gels were determined. Since the silicates were polymerized by pH regulation (addition of hydrochloric acid to gel-forming mixtures) the study of H+ diffusion in gels having different composition became also necessary. Realizing that not only the diffusion of H+ ions
in gels, but self-diffusion of solvent molecules together with D2 O molecules in counter-current system might be important for other scientific and engineering areas, the present paper gives a concise summary of the recent measurements, results and theoretical explanations with the aim at promoting the practical applications. 2. Experimental conditions The laboratory studies were focused on a complex gel containing a double network of cross-linked high molecular weight, partially hydrolyzed polyacrylamide and polysilicate. The measurements were carried out in a conventional dual cell apparatus with cell volume of 400 ml (Fig. 1). The gel phase having 30 mm diameter and 20 mm thickness was fixed into the connecting glass tube of cells by O-rings. A hydrostatic by-pass served to equilibrate the density difference between the test solutions minimizing thus the hydrodynamic mass transport between the cells. All measurements were carried out at constant temperature (298 K). The composition of the gel was the following. Polyacrylamide (g/l) Silicate (SiO2 ) (g/l) AlK(SO4 )2 (g/l) CaCl2 (g/l)
0–2.5 12.5–62.5 1.0 1.0
High molecular weight, partially hydrolyzed polyacrylamide (M = 9 × 106 g/mol and αH = 17%) was used as polymer, while commercial water glass with 17% SiO2 content was applied as stock silicate solution. The polymer/silicate and the cross-linking solutions were prepared separately and the mixed in 1:1 ratio. The cross-linking solutions contained always sufficient amount of HCl to set the gelation time within 2–10 min, and to maintain the pH in range of 7.5–8.5.
Fig. 1. Schematic of the dual cell diffusion apparatus.
I. Lakatos, J. Lakatos-Szab´o / Colloids and Surfaces A: Physicochem. Eng. Aspects 246 (2004) 9–19
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Fig. 2. General shape of pH change in Cell 2 as a function of time.
The gel formed might be characterized by a double network: one of them consist of the flexible polymer chains crosslinked mainly by Al3+ ion, while the other formed by the more rigid poly-silicates. Precipitated calcium silicates were also present in the gel, which was indicated by light opaque character of the gel. In study of H+ diffusion the model tests were performed by hydrogen ion added as analytical grade HCl in concentration of 100 mmol/l. Thus, the initial concentration gradient between the cells was always constant (50 mol/cm4 ). The total diffusion mass transport was evaluated by the actual H+ ion concentration in cells determined continuously by precision pH-meter (Mettler-Toledo MP220 pH meter). It should also be mentioned, that actually both the H+ and its counter Cl− ions were migrating co-current in the gel, but the measurements were focused only the former one neglecting other simultaneous process. In case of
H2 O and D2 O diffusion study Cell 1 contained 300 ml heavy water with 95% purity, while Cell 2 contained a little more, roughly 330 ml distilled water to compensate the density difference. The counter-current migration of water and heavy water molecules was followed by precision density meter (Anton Paar DMA 60 density meter). The total mass balance in cells were calculated for samples taken from both cells by a special computer program developed by us. The diffusion coefficient and the break-through time were obtained by the slope and the intersection point of the linear section of the cumulative mass transport curves shown in Fig. 2, while the specific ion retention capacity was calculated by the following relationships: Deff =
1 n x A t c
Fig. 3. Time dependency of pH in Cell 2 as a function of SiO2 content of gel.
(1)
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Fig. 4. Time dependency of the cumulative H+ ion transport as a function of the SiO2 content of gel.
and an =
1 c Deff A V x
t1 t0
dt =
1 c Deff A (t1 − t0 ) V x
(2)
An independent diffusion test using the same gel and nonadsorbing CrO4 −6 ion served for determination of t0 breakthrough time. The reproducibility of this method was determined by repeated measurements and it was better than 5 rel.%. 3. Results and discussion 3.1. Effect of silicate content of gel on transport properties of H+ ions The basic studies were carried out with a gel containing 1 g/l polymer and 20–70 g/l SiO2 equivalent ortho-silicate.
The concentration gradient of H+ ion was kept constant during the measurements. Results of the primary pH measurements and the derived cumulative mass transport curves of H+ ions are shown in Figs. 3 and 4. The conclusions drawn by the experimental findings are as follows: 1. After break-through the cumulative transport of diffusing ions increases linearly in time if the absolute change of the original concentration gradient is negligible (Fig. 4). That fact proves indirectly that the transport phenomena might be described by the Fick’s I law. 2. The effective diffusion coefficient of H+ ions significantly depends on the silicate content of gel. As shown in Fig. 5, the calculated effective diffusion coefficient decreases linearly with the silicate content. 3. Although, the effective diffusion coefficients calculated by the slope of the cumulative curves, depending on the
Fig. 5. Effect of SiO2 content of gel on the effective diffusion coefficient of H+ ions.
I. Lakatos, J. Lakatos-Szab´o / Colloids and Surfaces A: Physicochem. Eng. Aspects 246 (2004) 9–19
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Fig. 6. Effect of SiO2 content of gel on the break-through time of H+ ions.
SiO2 content of gel, are different for H+ ions, they are only 10–50% less than the value (9.32 × 10−9 m2 /s) reported by Kollthoff and Lingane [9] for that ion in aqueous solutions. 4. The break-through times found for H+ ions show also remarkable silicate dependency (Fig. 6). This fact can be explained by the original silicate content, and hence pH of the gels. Since neutralization of the gel needs more acid as the silicate content increases, the break-through time should increase with the silicate concentration. 5. The ion retention capacity calculated by Eq. (2) is changing parallel with the break-through time. As predicted, the acid consumption (neutralization or setting the gel pH to 100 mmol/l in gel) also increases with the silicate content of gel (Fig. 7).
diffusion and in situ reaction/adsorption processes. This idea is also proper in the present case: since the main component of the gel (silicate in form of water glass) is highly alkaline in its original form and the free alkali present in the gel-forming mixtures is only partially neutralized by acid pre-addition, the governing effect is the full neutralization of the free alkali (Na2 O) remaining after gelation under the unsteady-state period of measurement. In addition, the precipitated and hydrated silica is well known as an adsorbent containing free –OH groups on their surface, which may interact with numerous diffusing species.
The phenomena are converging with the earlier findings reported by Janke and Radke [10] who stated that the migration of charged ions in gels can be explained jointly by
In the second series of measurements both the polymer and the silicate content of the gel were identical and the same original H+ concentration gradient was applied but the
3.2. Effect of the original HCl content of gel on transport properties of H+ ions
Fig. 7. Effect of SiO2 content on H+ ion retention in gel.
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Fig. 8. Time dependency of pH in Cell 2 as a function of the original HCl content of gel.
Fig. 9. Time dependency of cumulative H+ ion transport in Cell 2 as a function of the original HCl content of gel.
amount of hydrochloric acid added to initiate the gelation of silicates changed between 0.1550 and 0.1877 mol/l. The diffusion tests were carried out at eight different acid concentrations and the same evaluation technique and calculations were used as previously. The pH change in Cell 2 and the corresponding cumulative mass transport curves for selected cases, as typical examples, are illustrated in Figs. 8 and 9 and the main conclusions concerning the transport properties can be listed as follows: 1. As far as the trends are regarded the effect of original HCl content of gel has similar consequences as of the silicate content. Adding hydrochloric acid to the gel-forming mixture a pre-neutralization takes place, viz. higher the initial HCl content in the gel, less additional acid is needed to rich equilibrium condition or pH characterizing the gel upto the steady-state period. 2. As shown in Fig. 10, the effective diffusion coefficient linearly increases with the original HCl content. These
phenomena are quite obvious taking the interpretation of Iler [11] and Falcone [12] into account who reported a definite influence of pH environment on the structure of silica gels. 3. The original HCl content of gels has also a substantial impact on the break-through time of H+ ions. As illustrated in Fig. 11 the break-through time decreases with the HCl concentration. This observation can easily be explained by the gradually lowering pH of gel, which needs less and less amount of HCl in the transient period to reach the equilibrium condition. 4. Parallel with the statements in the previous paragraphs, the H+ retention is also decreases with the original HCl content of gel (Fig. 12). The reason of that fact is the same as mention in point 3). The effect of HCl content on transport properties forecasts a radical structural modification in gels. Obviously, the permeability of gel having identical polymer and silicate content
I. Lakatos, J. Lakatos-Szab´o / Colloids and Surfaces A: Physicochem. Eng. Aspects 246 (2004) 9–19
Fig. 10. Effect of the original HCl content of gel on the effective diffusion coefficient of H+ ions.
Fig. 11. Effect of the original HCl content of gel on break-through time of H+ ions.
Fig. 12. Effect of the original HCl content of gel on H+ ion retention.
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Fig. 13. Change of density of heavy water (Cell 1) and water (Cell 2) during the test period.
significantly increases with the original acid content. In other words, polymerization of water-soluble silicates in low pH environment yields a fast precipitation in the organic polymer network, and such a composite gel structure is more penetrable by diffusing species than that of those gels, which are prepared under neutral or slightly alkaline media. 3.3. Counter-current diffusion of H2 O and D2 O in polymer/silicate gel The diffusion phenomena of H2 O and D2 O in gel could be studied only in counter-current arrangement. Since the availability of heavy water was highly limited only one measurement could be performed. The gel used for that experiment had the following composition: cSiO2 = 50 g/l, cp = 2 g/l and cHCl = 0.155 mol/l. The heavy water was practically pure, since its density (1.10317 g/cm3 ) measured by precision density meter was nearly the same as the value (1.1045)
reported by handbooks for pure heavy water under identical conditions. According to the experimental arrangement Cell 1 and Cell 2 contained the heavy water and the double distilled water, respectively. Thus, the initial concentration gradient was 9.16 mmol/cm4 for water and 8.25 mmol/cm4 for heavy water. Since there is roughly 10% difference in densities, as a result of counter-current diffusion of species, the density in cells continuously changed and that phenomena served for calculation of the effective diffusion coefficient in the steady-state period. The main conclusions drawn by the experimental results can be summarized as follows: 1. As a result of the counter-current diffusion of H2 O and D2 O in gels the density in Cell 1 decreased, while the opposite tendency was observed in Cell 2 (Fig. 13). The absolute change in cells, however was different: the density of heavy water decreased by 2.94%, meanwhile the change in Cell 2 the density of water increased by 2.55%
Fig. 14. Cumulative mass transport of heavy water (Cell 2) and water (Cell 1) during the test period.
I. Lakatos, J. Lakatos-Szab´o / Colloids and Surfaces A: Physicochem. Eng. Aspects 246 (2004) 9–19 Table 1 Characteristic diffusion properties of H2 O and D2 O in polymer/silicate gel (cSiO2 = 50 g/l; cp = 2 g/l; cHCl = 0.155 mol/l) Species
Deff (×10−9 m2 /s)
Break-through time (h)
Retention capacity (mol/m3 )
H2 O D2 O
6.4 2.6
0 6
0 0
2.
3.
4.
5.
after 160 h test period. That fact forecast a measurable difference in the effective diffusion coefficients. Using the changes of densities in cells the cumulative mass transport of H2 O and D2 O could be calculated. As shown in Fig. 14 both curves have also an unsteady-state and a steady-state section. It deserves attention, however, that water molecules starts to appear in heavy water immediately after contacting the gel with the liquids in the cells. In contrast to this observation, D2 O molecules in Cell 2 were no detectable in the first measuring point (6 h). That experimental finding is explained by the fact that water molecules present in gel migrate to the heavy water without time-leg, but it takes a certain time for D2 O molecules to penetrate and pass a distance determined by the thickness of gel until they may appear in Cell 2 filled by water. Intersection points of the trend lines with the y-axes and the slope of the cumulative straight line indicate that more water molecules diffused into Cell 1 than D2 O ones into Cell 2. The faster diffusion of H2 O molecules is proved by the effective diffusion coefficient listed in Table 1. Comparing the values characteristic for H2 O and D2 O we may say that more than two times more H2 O molecules are crossing the gel than D2 O molecules under unit time. The relatively slow diffusion of D2 O molecules are partly explained by the existing difference in the molecular sizes, and presumably the different thermodynamic driving forces (activation energies of diffusion) and wall/particulate interactions being favorable to H2 O molecules play also an important role in shaping of the transport process. Comparing the effective diffusion coefficients obtained for H2 O molecules and H+ ions in the same gels it is surprising that the value characteristic for the former one is the higher. That fact is attributed to form of the diffusing species. Namely, a charged ion is always diffusing as a hydrated species, which may have much higher hydrodynamic diameter than a solvent molecule. The data in Table 1 provide also additional information for speciality of H2 O and D2 O diffusion. Since there is no detrimental chemical interaction between gel and the migrating species, the break-through time is zero (H2 O) or minimal (D2 O) and retention could not be calculated for either molecules.
Unfortunately, the limited availability of heavy water makes the statistically reliable results not possible. It should also be emphasized that the “concentration gradient” could not be considered as constant during the whole test time.
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Therefore, some points fall off the trend line in Fig. 14, and in addition, the calculation had to be made manually using the actual data in each points. Despite all efforts the calculated effective coefficients, particularly for D2 O, can be accepted only as approximate values. The performed measurements, however, clearly indicate that a significant difference might be among diffusing species, which are otherwise similar in physical and chemical properties. 3.4. Characterization of gel structure by diffusion of H+ ion Since the absolute diffusion coefficient of H2 O and D2 O molecules in eigensolvent has no practical meaning, thus its value is not known, the analysis of gel structure could be conveyed only by the data of the H+ ion transport in gel. Our starting point is that the effective diffusion coefficients of H+ ions found for the composite polymer/silicate gels having less than 10% solid content confirm the earlier findings by Jost [13]. Namely, the effective diffusion coefficient of charged ions might be maximum one order of magnitude less in gels than in aqueous solutions. Further, the general features of the diffusion transport in gels, particularly the existence of the simultaneous and competitive diffusion and retention processes, allow us also to predict that the one-dimensional random-walk equations can not be applied for calculation of the mean diffusion distance of chromium ions in gels. The experimental results, however, raise a question: is it possible to draw a correlation between the effective diffusion coefficient and the gel structure or not? Using the “formation factor” concept developed by Pirson [14], Schopper [15], Dullien [16] and others for description of the diffusion mass transport in porous media, a reliable relationship between the effective diffusion coefficients and the porosity and tortuosity was derived in earlier studies [7,8]. Following the same procedure first the formation factor was calculated. F=
τ2 D = Φ Deff
(3)
Although, both the polymer and the silicate or silica exist in gel as a hydrated double network, the approximate porosity can be calculated by the solid content of the gel. Using this approach the data in Tables 2 and 3 were obtained. Calculation of the data in these tables was as follows: Deff is the primarily measured effective diffusion coefficient in gels and taking the absolute diffusion coefficient often referred in literature for H+ ions in bulk water phase (D = 9.337 × 10−9 m2 /s) the formation factor can easily be calculated by Eq. (3). Then, using the same relationship, the tortuosity can be obtained supposing that the “porosity” of gel is equal with the volume fraction of the given gel. As shown and expected, the data in this tables indicate a disproportionality between the solid content (porosity) of gel and the tortuosity. For instance, as the silicate content between 20 and 70 g/l results in a maximum 7% change in porosity, while the effective length of the migration path of H+ ions increases by nearly 40%. Even more stunning
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Table 2 Effect of silicate content on effective diffusion coefficient of H+ ion and properties of different SiO2 containing polymer/silicate gels (cp = 2 g/l; cHCl = 0.155 mol/l) SiO2 concentration (g/l)
Deff (×10−9 m2 /s)
Φ
F
τ
20 25 30 40 50 60 70
6.4 6.3 5.4 5.2 4.7 3.9 3.5
0.9810 0.9762 0.9715 0.9620 0.9525 0.9430 0.9335
1.459 1.482 1.729 1.796 1.987 2.239 2.395
1.193 1.205 1.299 1.308 1.382 1.495 1.586
is that in gels having the same silicate and polymer content the tortuosity drops by 20% when the original acid concentration increased from 0.155 to 0.183 mol/l. These data indicate a robust structural modification: extensive swelling and coiling (inter-molecular interactions) in the former case and shrinkage and ceasing intra- and inter-molecular interactions in the later ones. It must be pointed out, however, that the characteristic structural properties of gel determined by diffusion tests significantly depends on the diffusing species. That statement is well proved if we compare the formation factors and tortuosities obtained for the same gel by H+ and Cr3+ ions. In the latter case the relevant data were F = 1.36 and τ = 1.12 [8]. Obviously, the mobility (hydrodynamic size) of species diffusing in gel directly determines the accessible pore space in gel, viz. the ratio of active and dead volumes of gel is not only gel but also species-dependent. In other words, the diffusion and the hydrodynamic mass transport in gel show some similarity with the transport phenomena in porous media and that is a further argument for application of the same approach and model in description of the diffusion phenomena. The new results obtained by the detailed study of H+ ions in polymer/silicate gels prove again that the developed theoretical approach is correct and thus, the Fick’s I law can be rewritten in the following way: dn dc AΦ dc Aeff dc = −Deff A = −D 2 = −D 2 dt dx τ dx τ dx
(4)
Special advantage of Eq. (4) is two-fold: it provides an easy calculation technique for determination of the approximate Table 3 Effect of HCl content on effective diffusion coefficient of H+ ion and properties of a polymer/silicate gel having the same composition (cSiO2 = 25 g/l; cp = 2 g/l) HCl concentration (mol/l)
Deff (×10−9 m2 /s)
Φ
F
τ
0.155 0.159 0.164 0.173 0.178 0.183
6.3 7.4 7.7 8.5 9.0 9.2
0.9762 0.9762 0.9762 0.9762 0.9762 0.9762
1.482 1.262 1.213 1.098 1.037 0.995a
1.203 1.109 1.088 1.075 1.006 0.997a
a The value is practically 1.000. A value being less than the unity might be attributed to experimental error.
effective diffusion coefficient in gel if its “porosity” and tortuosity are known, or having the effective diffusion coefficient determined experimentally, the gel structure or its modification under special circumstances can be predicted. That was recently proved by determination of formation factor of a natural porous media (sandstone with 525 mD permeability) using the conventional and standardized conductometric technique and comparing the results with those obtained by the described method. The standard deviation of five consecutive measurements was within 15 rel.%, which is quite acceptable if two different methods are used. Similarly, comparing the formation factor and tortuosity (Table 2) at 25 g/l SiO2 and 2 g/l polymer content calculated by the effective and absolute diffusion coefficients obtained by H+ and Cr3+ ions (F = 1.356 and τ = 1.139, [8]) the results are also very close to each other. Consequently, the presented method might be important in both engineering and theoretical point of view in divers industrial and scientific areas when structure of solid or gels are investigated.
4. Conclusions 1. Transport properties of H+ ions were determined in polymer/silicate gels as a function of its silicate and HCl contents. 2. The effective diffusion coefficient of H+ ions decreased with the silicate content, while the break-through time and ion retention increased as the alkalinity of the gels strengthened. 3. The effective diffusion coefficient increased with the original HCl concentration of gels, meanwhile the breakthrough time and the ion retention changed according to an opposite trend. 4. The approximate effective diffusion coefficient obtained in counter-current diffusion for H2 O is more than two times higher than the value calculated for the D2 O molecules. 5. The relatively slow diffusion rate of D2 O molecules is partly explained by the larger size, different thermodynamic driving forces and wall/particulate. 6. The effective diffusion coefficients obtained for H2 O molecules and H+ ions in the same gels are different: the value characteristic for the former one is the higher. That fact indicates that the charged ions are always diffusing as a hydrated species, which may have much higher hydrodynamic diameter than a solvent molecule. 7. The close correlation was found between the transport properties and the gel properties characterized by the formation factor, effective porosity and tortuosity of gels. 8. The diffusion transport of H+ ions in polymer/silicate gels could be described by the modified Fick’s I law and this formula provides an appropriate method for prediction of the effective diffusion coefficient in gels if the properties of gels and the absolute diffusion coefficient in bulk aqueous phase are known.
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9. The structural properties (formation factor, tortuosity) is not only gel but also species-dependent, viz. not the whole pore space is taking place in the diffusion process. 10. The laboratory experiments provided valuable new information and data to formulation of different industrial “gel” technologies. References [1] B.J. Todd, G.P. Willhite, D.W. Green, SPE Reserv. Eng. 1 (1993) 51. [2] R.S. Seright, SPE Reserv. Eng. 3 (1991) 343. [3] R. Takahashi, S. Sato, T. Sodesowa, Y. Kamomae, J. Ceram. Soc. Jpn. 109 (2001) 840. [4] A. Walcarius, M. Etienne, J. Bessiere, Chem. Mater. 14 (2002) 2757. [5] R. Takahashi, S. Sato, T. Sodesowa, H. Nishida, Phys. Chem. Chem. Phys. 4 (2002) 3800.
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[6] R. Othman, A.H. Yahaya, A.K. Arof, J. New Mater. Electrochem. Syst. 5 (2002) 177. [7] I. Lakatos, K. Bauer, J. Lakatos-Szab´o, H.J. Kretzschmar, Land Contam. Reclam. 5 (1997) 165. [8] I. Lakatos, J. Lakatos-Szab´o, Colloids Surf. A., Physicochem. Eng. Aspects 141 (1998) 425. [9] I.M. Kollthoff, J.J. Lingane, Polarography, Interscience, New York, 1952, p. 52. [10] F.M. Janke, C.J. Radke, Ind. Eng. Chem. Res. 28 (1989) 347. [11] R.K. Iler, The Chemistry of Silica, Wiley, New York, 1979. [12] J.S. Falcone, Soluble Silicates, ACS Symposium Series 194, Washington, 1982. [13] W. Jost, Diffusion-Methoden der Messung und Auswertung, Steinkopff, Darstadt, 1957. [14] S.J. Pirson, Oil Reservoir Engineering, Mc Graw-Hill, New York, 1958. [15] J.R. Schopper, Geophys. Prospect. 14 (1966) 301. [16] F.A.L. Dullien, Porous Media: Fluid Transport and Pore Structure, Academic Press, New York, 1992.
Diffusion of H+, H2 O and D2 O in polymer/silicate gels I. Lakatos∗ , J. Lakatos-Szab´o Research Institute of Applied Chemistry, University of Miskolc, P.O. Box 2, Miskolc-Egyetemv´aros, H-3515, Hungary Received 10 September 2003; accepted 16 June 2004 Available online 12 September 2004
Abstract General features of H+ , H2 O and D2 O diffusion in polymer/silicate gels applied in different industrial areas were analyzed. It was found that the cumulative mass transport curves consist of two sections: an unsteady-state and a steady-state one. The length of the transient period is strongly dependent on the silicate content of the gel and the concentration gradient of H+ ions. Using the length and the intersection point of these periods the effective diffusion coefficient, break-through time and the ion retention capacity of gel could be calculated. The obtained effective diffusion coefficients were very close to the values characteristic in aqueous solutions. The diffusion mass transport could be described by the modified Fick’s I law, however, it was stated that on account of the high ion retention in all gels the one-dimensional random-walk equation may not be used for prediction of the mean diffusion distance (break-through time). Using the formation factor of gel (effective “porosity” and tortuosity) the effective diffusion coefficient can be predicted with good accuracy and vice versa, the gel structure can be determined by the absolute and the effective diffusion coefficients characteristic in bulk aqueous phase and gel, respectively. The laboratory experiments provided valuable new information and data to design and formulation of the industrial “gel” technologies in both the enhanced oil recovery and the environmental protection. © 2004 Elsevier B.V. All rights reserved. Keywords: Diffusion; Gel; Polymer; Silicate; Hydrogen ion; Water; Heavy water
1. Introduction The use of electrolyte containing gels in batteries and electric cells goes back to more than a century. Similarly, the application of hydrated gels as flow profile modifiers and barrier forming agents opened up new vistas in enhanced oil recovery and environmental containment technology. During the past decades focusing on such topics a great variety of gelforming chemicals have been tested under laboratory and industrial conditions, and some of them proved very useful, indispensable and effective in practice. Analyzing the negative gel properties, however, it became evident that the transport properties being as main factors of the gelation, conductivity, block-forming capacity, long term usability, etc. are sometime poor and the mass transport mechanism in gels is not ∗
Corresponding author. Tel.: +36 46 565 255; fax: +36 46 363 349. E-mail address: [email protected] (I. Lakatos).
0927-7757/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2004.06.035
completely understood. Consequently, the diffusion of different chemical species in self-conforming phases formed prior to application or in situ in formation rocks, soils, etc. might be fundamental, and the transport process may influence significantly the efficiency of different industrial technologies to be applied in relevant areas. Despite these facts, the possible role of diffusion in such procedures—particularly in improved/enhanced oil recovery—sometimes are misunderstood or either not mentioned in literature at all [1], or it is considered as a negative factor resulting in a dilution or depletion of the reacting species in the mixing zones or the gel itself [2]. Fortunately, our knowledge on diffusion transport in gels proceeds well and these research areas (in material sciences, electrochemistry, etc.) are often revisited even today [3–6]. Initiating by the strong industrial demands, a complex gel system containing silicates and water-soluble polymers was developed with the aim at improving the efficiency of selective fluid shut-off treatments and containment technologies. The new methods were tested successfully at different Hun-
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I. Lakatos, J. Lakatos-Szab´o / Colloids and Surfaces A: Physicochem. Eng. Aspects 246 (2004) 9–19
Nomenclature an A Aeff D Deff F n t t0 t1 V x
specific species retention in gel (mol/m3 ) cross-section of gel (m2 ) effective cross-section of gel (m2 ) diffusion coefficient in the aqueous bulk phase (m2 /s) effective diffusion coefficient in gel (m2 /s) formation factor cumulative mass transport (mol) time (s) break-through time of a non-adsorbing species (s) break-through time of an adsorbing species (s) volume of gel (m3 ) thickness of gel (m)
Greek symbols c concentration difference between cells (mol/l) Φ porosity τ tortuosity garian oil fields for restriction of water production and gas coning in oil wells. Since multivalent cations were used as cross-linking agents the earlier investigations were concentrated on determination of the diffusion mass transport of different chromium ions in polymer/silicate gels [7,8]. Independent variables of the laboratory studies were the valency of chromium, and geometry and composition of gels. Simultaneously the effective diffusion coefficient, break-through time and retention of ions in gels were determined. Since the silicates were polymerized by pH regulation (addition of hydrochloric acid to gel-forming mixtures) the study of H+ diffusion in gels having different composition became also necessary. Realizing that not only the diffusion of H+ ions
in gels, but self-diffusion of solvent molecules together with D2 O molecules in counter-current system might be important for other scientific and engineering areas, the present paper gives a concise summary of the recent measurements, results and theoretical explanations with the aim at promoting the practical applications. 2. Experimental conditions The laboratory studies were focused on a complex gel containing a double network of cross-linked high molecular weight, partially hydrolyzed polyacrylamide and polysilicate. The measurements were carried out in a conventional dual cell apparatus with cell volume of 400 ml (Fig. 1). The gel phase having 30 mm diameter and 20 mm thickness was fixed into the connecting glass tube of cells by O-rings. A hydrostatic by-pass served to equilibrate the density difference between the test solutions minimizing thus the hydrodynamic mass transport between the cells. All measurements were carried out at constant temperature (298 K). The composition of the gel was the following. Polyacrylamide (g/l) Silicate (SiO2 ) (g/l) AlK(SO4 )2 (g/l) CaCl2 (g/l)
0–2.5 12.5–62.5 1.0 1.0
High molecular weight, partially hydrolyzed polyacrylamide (M = 9 × 106 g/mol and αH = 17%) was used as polymer, while commercial water glass with 17% SiO2 content was applied as stock silicate solution. The polymer/silicate and the cross-linking solutions were prepared separately and the mixed in 1:1 ratio. The cross-linking solutions contained always sufficient amount of HCl to set the gelation time within 2–10 min, and to maintain the pH in range of 7.5–8.5.
Fig. 1. Schematic of the dual cell diffusion apparatus.
I. Lakatos, J. Lakatos-Szab´o / Colloids and Surfaces A: Physicochem. Eng. Aspects 246 (2004) 9–19
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Fig. 2. General shape of pH change in Cell 2 as a function of time.
The gel formed might be characterized by a double network: one of them consist of the flexible polymer chains crosslinked mainly by Al3+ ion, while the other formed by the more rigid poly-silicates. Precipitated calcium silicates were also present in the gel, which was indicated by light opaque character of the gel. In study of H+ diffusion the model tests were performed by hydrogen ion added as analytical grade HCl in concentration of 100 mmol/l. Thus, the initial concentration gradient between the cells was always constant (50 mol/cm4 ). The total diffusion mass transport was evaluated by the actual H+ ion concentration in cells determined continuously by precision pH-meter (Mettler-Toledo MP220 pH meter). It should also be mentioned, that actually both the H+ and its counter Cl− ions were migrating co-current in the gel, but the measurements were focused only the former one neglecting other simultaneous process. In case of
H2 O and D2 O diffusion study Cell 1 contained 300 ml heavy water with 95% purity, while Cell 2 contained a little more, roughly 330 ml distilled water to compensate the density difference. The counter-current migration of water and heavy water molecules was followed by precision density meter (Anton Paar DMA 60 density meter). The total mass balance in cells were calculated for samples taken from both cells by a special computer program developed by us. The diffusion coefficient and the break-through time were obtained by the slope and the intersection point of the linear section of the cumulative mass transport curves shown in Fig. 2, while the specific ion retention capacity was calculated by the following relationships: Deff =
1 n x A t c
Fig. 3. Time dependency of pH in Cell 2 as a function of SiO2 content of gel.
(1)
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I. Lakatos, J. Lakatos-Szab´o / Colloids and Surfaces A: Physicochem. Eng. Aspects 246 (2004) 9–19
Fig. 4. Time dependency of the cumulative H+ ion transport as a function of the SiO2 content of gel.
and an =
1 c Deff A V x
t1 t0
dt =
1 c Deff A (t1 − t0 ) V x
(2)
An independent diffusion test using the same gel and nonadsorbing CrO4 −6 ion served for determination of t0 breakthrough time. The reproducibility of this method was determined by repeated measurements and it was better than 5 rel.%. 3. Results and discussion 3.1. Effect of silicate content of gel on transport properties of H+ ions The basic studies were carried out with a gel containing 1 g/l polymer and 20–70 g/l SiO2 equivalent ortho-silicate.
The concentration gradient of H+ ion was kept constant during the measurements. Results of the primary pH measurements and the derived cumulative mass transport curves of H+ ions are shown in Figs. 3 and 4. The conclusions drawn by the experimental findings are as follows: 1. After break-through the cumulative transport of diffusing ions increases linearly in time if the absolute change of the original concentration gradient is negligible (Fig. 4). That fact proves indirectly that the transport phenomena might be described by the Fick’s I law. 2. The effective diffusion coefficient of H+ ions significantly depends on the silicate content of gel. As shown in Fig. 5, the calculated effective diffusion coefficient decreases linearly with the silicate content. 3. Although, the effective diffusion coefficients calculated by the slope of the cumulative curves, depending on the
Fig. 5. Effect of SiO2 content of gel on the effective diffusion coefficient of H+ ions.
I. Lakatos, J. Lakatos-Szab´o / Colloids and Surfaces A: Physicochem. Eng. Aspects 246 (2004) 9–19
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Fig. 6. Effect of SiO2 content of gel on the break-through time of H+ ions.
SiO2 content of gel, are different for H+ ions, they are only 10–50% less than the value (9.32 × 10−9 m2 /s) reported by Kollthoff and Lingane [9] for that ion in aqueous solutions. 4. The break-through times found for H+ ions show also remarkable silicate dependency (Fig. 6). This fact can be explained by the original silicate content, and hence pH of the gels. Since neutralization of the gel needs more acid as the silicate content increases, the break-through time should increase with the silicate concentration. 5. The ion retention capacity calculated by Eq. (2) is changing parallel with the break-through time. As predicted, the acid consumption (neutralization or setting the gel pH to 100 mmol/l in gel) also increases with the silicate content of gel (Fig. 7).
diffusion and in situ reaction/adsorption processes. This idea is also proper in the present case: since the main component of the gel (silicate in form of water glass) is highly alkaline in its original form and the free alkali present in the gel-forming mixtures is only partially neutralized by acid pre-addition, the governing effect is the full neutralization of the free alkali (Na2 O) remaining after gelation under the unsteady-state period of measurement. In addition, the precipitated and hydrated silica is well known as an adsorbent containing free –OH groups on their surface, which may interact with numerous diffusing species.
The phenomena are converging with the earlier findings reported by Janke and Radke [10] who stated that the migration of charged ions in gels can be explained jointly by
In the second series of measurements both the polymer and the silicate content of the gel were identical and the same original H+ concentration gradient was applied but the
3.2. Effect of the original HCl content of gel on transport properties of H+ ions
Fig. 7. Effect of SiO2 content on H+ ion retention in gel.
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I. Lakatos, J. Lakatos-Szab´o / Colloids and Surfaces A: Physicochem. Eng. Aspects 246 (2004) 9–19
Fig. 8. Time dependency of pH in Cell 2 as a function of the original HCl content of gel.
Fig. 9. Time dependency of cumulative H+ ion transport in Cell 2 as a function of the original HCl content of gel.
amount of hydrochloric acid added to initiate the gelation of silicates changed between 0.1550 and 0.1877 mol/l. The diffusion tests were carried out at eight different acid concentrations and the same evaluation technique and calculations were used as previously. The pH change in Cell 2 and the corresponding cumulative mass transport curves for selected cases, as typical examples, are illustrated in Figs. 8 and 9 and the main conclusions concerning the transport properties can be listed as follows: 1. As far as the trends are regarded the effect of original HCl content of gel has similar consequences as of the silicate content. Adding hydrochloric acid to the gel-forming mixture a pre-neutralization takes place, viz. higher the initial HCl content in the gel, less additional acid is needed to rich equilibrium condition or pH characterizing the gel upto the steady-state period. 2. As shown in Fig. 10, the effective diffusion coefficient linearly increases with the original HCl content. These
phenomena are quite obvious taking the interpretation of Iler [11] and Falcone [12] into account who reported a definite influence of pH environment on the structure of silica gels. 3. The original HCl content of gels has also a substantial impact on the break-through time of H+ ions. As illustrated in Fig. 11 the break-through time decreases with the HCl concentration. This observation can easily be explained by the gradually lowering pH of gel, which needs less and less amount of HCl in the transient period to reach the equilibrium condition. 4. Parallel with the statements in the previous paragraphs, the H+ retention is also decreases with the original HCl content of gel (Fig. 12). The reason of that fact is the same as mention in point 3). The effect of HCl content on transport properties forecasts a radical structural modification in gels. Obviously, the permeability of gel having identical polymer and silicate content
I. Lakatos, J. Lakatos-Szab´o / Colloids and Surfaces A: Physicochem. Eng. Aspects 246 (2004) 9–19
Fig. 10. Effect of the original HCl content of gel on the effective diffusion coefficient of H+ ions.
Fig. 11. Effect of the original HCl content of gel on break-through time of H+ ions.
Fig. 12. Effect of the original HCl content of gel on H+ ion retention.
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I. Lakatos, J. Lakatos-Szab´o / Colloids and Surfaces A: Physicochem. Eng. Aspects 246 (2004) 9–19
Fig. 13. Change of density of heavy water (Cell 1) and water (Cell 2) during the test period.
significantly increases with the original acid content. In other words, polymerization of water-soluble silicates in low pH environment yields a fast precipitation in the organic polymer network, and such a composite gel structure is more penetrable by diffusing species than that of those gels, which are prepared under neutral or slightly alkaline media. 3.3. Counter-current diffusion of H2 O and D2 O in polymer/silicate gel The diffusion phenomena of H2 O and D2 O in gel could be studied only in counter-current arrangement. Since the availability of heavy water was highly limited only one measurement could be performed. The gel used for that experiment had the following composition: cSiO2 = 50 g/l, cp = 2 g/l and cHCl = 0.155 mol/l. The heavy water was practically pure, since its density (1.10317 g/cm3 ) measured by precision density meter was nearly the same as the value (1.1045)
reported by handbooks for pure heavy water under identical conditions. According to the experimental arrangement Cell 1 and Cell 2 contained the heavy water and the double distilled water, respectively. Thus, the initial concentration gradient was 9.16 mmol/cm4 for water and 8.25 mmol/cm4 for heavy water. Since there is roughly 10% difference in densities, as a result of counter-current diffusion of species, the density in cells continuously changed and that phenomena served for calculation of the effective diffusion coefficient in the steady-state period. The main conclusions drawn by the experimental results can be summarized as follows: 1. As a result of the counter-current diffusion of H2 O and D2 O in gels the density in Cell 1 decreased, while the opposite tendency was observed in Cell 2 (Fig. 13). The absolute change in cells, however was different: the density of heavy water decreased by 2.94%, meanwhile the change in Cell 2 the density of water increased by 2.55%
Fig. 14. Cumulative mass transport of heavy water (Cell 2) and water (Cell 1) during the test period.
I. Lakatos, J. Lakatos-Szab´o / Colloids and Surfaces A: Physicochem. Eng. Aspects 246 (2004) 9–19 Table 1 Characteristic diffusion properties of H2 O and D2 O in polymer/silicate gel (cSiO2 = 50 g/l; cp = 2 g/l; cHCl = 0.155 mol/l) Species
Deff (×10−9 m2 /s)
Break-through time (h)
Retention capacity (mol/m3 )
H2 O D2 O
6.4 2.6
0 6
0 0
2.
3.
4.
5.
after 160 h test period. That fact forecast a measurable difference in the effective diffusion coefficients. Using the changes of densities in cells the cumulative mass transport of H2 O and D2 O could be calculated. As shown in Fig. 14 both curves have also an unsteady-state and a steady-state section. It deserves attention, however, that water molecules starts to appear in heavy water immediately after contacting the gel with the liquids in the cells. In contrast to this observation, D2 O molecules in Cell 2 were no detectable in the first measuring point (6 h). That experimental finding is explained by the fact that water molecules present in gel migrate to the heavy water without time-leg, but it takes a certain time for D2 O molecules to penetrate and pass a distance determined by the thickness of gel until they may appear in Cell 2 filled by water. Intersection points of the trend lines with the y-axes and the slope of the cumulative straight line indicate that more water molecules diffused into Cell 1 than D2 O ones into Cell 2. The faster diffusion of H2 O molecules is proved by the effective diffusion coefficient listed in Table 1. Comparing the values characteristic for H2 O and D2 O we may say that more than two times more H2 O molecules are crossing the gel than D2 O molecules under unit time. The relatively slow diffusion of D2 O molecules are partly explained by the existing difference in the molecular sizes, and presumably the different thermodynamic driving forces (activation energies of diffusion) and wall/particulate interactions being favorable to H2 O molecules play also an important role in shaping of the transport process. Comparing the effective diffusion coefficients obtained for H2 O molecules and H+ ions in the same gels it is surprising that the value characteristic for the former one is the higher. That fact is attributed to form of the diffusing species. Namely, a charged ion is always diffusing as a hydrated species, which may have much higher hydrodynamic diameter than a solvent molecule. The data in Table 1 provide also additional information for speciality of H2 O and D2 O diffusion. Since there is no detrimental chemical interaction between gel and the migrating species, the break-through time is zero (H2 O) or minimal (D2 O) and retention could not be calculated for either molecules.
Unfortunately, the limited availability of heavy water makes the statistically reliable results not possible. It should also be emphasized that the “concentration gradient” could not be considered as constant during the whole test time.
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Therefore, some points fall off the trend line in Fig. 14, and in addition, the calculation had to be made manually using the actual data in each points. Despite all efforts the calculated effective coefficients, particularly for D2 O, can be accepted only as approximate values. The performed measurements, however, clearly indicate that a significant difference might be among diffusing species, which are otherwise similar in physical and chemical properties. 3.4. Characterization of gel structure by diffusion of H+ ion Since the absolute diffusion coefficient of H2 O and D2 O molecules in eigensolvent has no practical meaning, thus its value is not known, the analysis of gel structure could be conveyed only by the data of the H+ ion transport in gel. Our starting point is that the effective diffusion coefficients of H+ ions found for the composite polymer/silicate gels having less than 10% solid content confirm the earlier findings by Jost [13]. Namely, the effective diffusion coefficient of charged ions might be maximum one order of magnitude less in gels than in aqueous solutions. Further, the general features of the diffusion transport in gels, particularly the existence of the simultaneous and competitive diffusion and retention processes, allow us also to predict that the one-dimensional random-walk equations can not be applied for calculation of the mean diffusion distance of chromium ions in gels. The experimental results, however, raise a question: is it possible to draw a correlation between the effective diffusion coefficient and the gel structure or not? Using the “formation factor” concept developed by Pirson [14], Schopper [15], Dullien [16] and others for description of the diffusion mass transport in porous media, a reliable relationship between the effective diffusion coefficients and the porosity and tortuosity was derived in earlier studies [7,8]. Following the same procedure first the formation factor was calculated. F=
τ2 D = Φ Deff
(3)
Although, both the polymer and the silicate or silica exist in gel as a hydrated double network, the approximate porosity can be calculated by the solid content of the gel. Using this approach the data in Tables 2 and 3 were obtained. Calculation of the data in these tables was as follows: Deff is the primarily measured effective diffusion coefficient in gels and taking the absolute diffusion coefficient often referred in literature for H+ ions in bulk water phase (D = 9.337 × 10−9 m2 /s) the formation factor can easily be calculated by Eq. (3). Then, using the same relationship, the tortuosity can be obtained supposing that the “porosity” of gel is equal with the volume fraction of the given gel. As shown and expected, the data in this tables indicate a disproportionality between the solid content (porosity) of gel and the tortuosity. For instance, as the silicate content between 20 and 70 g/l results in a maximum 7% change in porosity, while the effective length of the migration path of H+ ions increases by nearly 40%. Even more stunning
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Table 2 Effect of silicate content on effective diffusion coefficient of H+ ion and properties of different SiO2 containing polymer/silicate gels (cp = 2 g/l; cHCl = 0.155 mol/l) SiO2 concentration (g/l)
Deff (×10−9 m2 /s)
Φ
F
τ
20 25 30 40 50 60 70
6.4 6.3 5.4 5.2 4.7 3.9 3.5
0.9810 0.9762 0.9715 0.9620 0.9525 0.9430 0.9335
1.459 1.482 1.729 1.796 1.987 2.239 2.395
1.193 1.205 1.299 1.308 1.382 1.495 1.586
is that in gels having the same silicate and polymer content the tortuosity drops by 20% when the original acid concentration increased from 0.155 to 0.183 mol/l. These data indicate a robust structural modification: extensive swelling and coiling (inter-molecular interactions) in the former case and shrinkage and ceasing intra- and inter-molecular interactions in the later ones. It must be pointed out, however, that the characteristic structural properties of gel determined by diffusion tests significantly depends on the diffusing species. That statement is well proved if we compare the formation factors and tortuosities obtained for the same gel by H+ and Cr3+ ions. In the latter case the relevant data were F = 1.36 and τ = 1.12 [8]. Obviously, the mobility (hydrodynamic size) of species diffusing in gel directly determines the accessible pore space in gel, viz. the ratio of active and dead volumes of gel is not only gel but also species-dependent. In other words, the diffusion and the hydrodynamic mass transport in gel show some similarity with the transport phenomena in porous media and that is a further argument for application of the same approach and model in description of the diffusion phenomena. The new results obtained by the detailed study of H+ ions in polymer/silicate gels prove again that the developed theoretical approach is correct and thus, the Fick’s I law can be rewritten in the following way: dn dc AΦ dc Aeff dc = −Deff A = −D 2 = −D 2 dt dx τ dx τ dx
(4)
Special advantage of Eq. (4) is two-fold: it provides an easy calculation technique for determination of the approximate Table 3 Effect of HCl content on effective diffusion coefficient of H+ ion and properties of a polymer/silicate gel having the same composition (cSiO2 = 25 g/l; cp = 2 g/l) HCl concentration (mol/l)
Deff (×10−9 m2 /s)
Φ
F
τ
0.155 0.159 0.164 0.173 0.178 0.183
6.3 7.4 7.7 8.5 9.0 9.2
0.9762 0.9762 0.9762 0.9762 0.9762 0.9762
1.482 1.262 1.213 1.098 1.037 0.995a
1.203 1.109 1.088 1.075 1.006 0.997a
a The value is practically 1.000. A value being less than the unity might be attributed to experimental error.
effective diffusion coefficient in gel if its “porosity” and tortuosity are known, or having the effective diffusion coefficient determined experimentally, the gel structure or its modification under special circumstances can be predicted. That was recently proved by determination of formation factor of a natural porous media (sandstone with 525 mD permeability) using the conventional and standardized conductometric technique and comparing the results with those obtained by the described method. The standard deviation of five consecutive measurements was within 15 rel.%, which is quite acceptable if two different methods are used. Similarly, comparing the formation factor and tortuosity (Table 2) at 25 g/l SiO2 and 2 g/l polymer content calculated by the effective and absolute diffusion coefficients obtained by H+ and Cr3+ ions (F = 1.356 and τ = 1.139, [8]) the results are also very close to each other. Consequently, the presented method might be important in both engineering and theoretical point of view in divers industrial and scientific areas when structure of solid or gels are investigated.
4. Conclusions 1. Transport properties of H+ ions were determined in polymer/silicate gels as a function of its silicate and HCl contents. 2. The effective diffusion coefficient of H+ ions decreased with the silicate content, while the break-through time and ion retention increased as the alkalinity of the gels strengthened. 3. The effective diffusion coefficient increased with the original HCl concentration of gels, meanwhile the breakthrough time and the ion retention changed according to an opposite trend. 4. The approximate effective diffusion coefficient obtained in counter-current diffusion for H2 O is more than two times higher than the value calculated for the D2 O molecules. 5. The relatively slow diffusion rate of D2 O molecules is partly explained by the larger size, different thermodynamic driving forces and wall/particulate. 6. The effective diffusion coefficients obtained for H2 O molecules and H+ ions in the same gels are different: the value characteristic for the former one is the higher. That fact indicates that the charged ions are always diffusing as a hydrated species, which may have much higher hydrodynamic diameter than a solvent molecule. 7. The close correlation was found between the transport properties and the gel properties characterized by the formation factor, effective porosity and tortuosity of gels. 8. The diffusion transport of H+ ions in polymer/silicate gels could be described by the modified Fick’s I law and this formula provides an appropriate method for prediction of the effective diffusion coefficient in gels if the properties of gels and the absolute diffusion coefficient in bulk aqueous phase are known.
I. Lakatos, J. Lakatos-Szab´o / Colloids and Surfaces A: Physicochem. Eng. Aspects 246 (2004) 9–19
9. The structural properties (formation factor, tortuosity) is not only gel but also species-dependent, viz. not the whole pore space is taking place in the diffusion process. 10. The laboratory experiments provided valuable new information and data to formulation of different industrial “gel” technologies. References [1] B.J. Todd, G.P. Willhite, D.W. Green, SPE Reserv. Eng. 1 (1993) 51. [2] R.S. Seright, SPE Reserv. Eng. 3 (1991) 343. [3] R. Takahashi, S. Sato, T. Sodesowa, Y. Kamomae, J. Ceram. Soc. Jpn. 109 (2001) 840. [4] A. Walcarius, M. Etienne, J. Bessiere, Chem. Mater. 14 (2002) 2757. [5] R. Takahashi, S. Sato, T. Sodesowa, H. Nishida, Phys. Chem. Chem. Phys. 4 (2002) 3800.
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[6] R. Othman, A.H. Yahaya, A.K. Arof, J. New Mater. Electrochem. Syst. 5 (2002) 177. [7] I. Lakatos, K. Bauer, J. Lakatos-Szab´o, H.J. Kretzschmar, Land Contam. Reclam. 5 (1997) 165. [8] I. Lakatos, J. Lakatos-Szab´o, Colloids Surf. A., Physicochem. Eng. Aspects 141 (1998) 425. [9] I.M. Kollthoff, J.J. Lingane, Polarography, Interscience, New York, 1952, p. 52. [10] F.M. Janke, C.J. Radke, Ind. Eng. Chem. Res. 28 (1989) 347. [11] R.K. Iler, The Chemistry of Silica, Wiley, New York, 1979. [12] J.S. Falcone, Soluble Silicates, ACS Symposium Series 194, Washington, 1982. [13] W. Jost, Diffusion-Methoden der Messung und Auswertung, Steinkopff, Darstadt, 1957. [14] S.J. Pirson, Oil Reservoir Engineering, Mc Graw-Hill, New York, 1958. [15] J.R. Schopper, Geophys. Prospect. 14 (1966) 301. [16] F.A.L. Dullien, Porous Media: Fluid Transport and Pore Structure, Academic Press, New York, 1992.