Keywords
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| Water absorbing polymer, hydro-gel swelling, porous hydro-gel |
INTRODUCTION
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| Hydro-gel or water-gel polymers commonly available in the market under various names such as Magic Pearl Jelly or Crystal Soil etc. are often used as decorative as well as source of water for indoor flower pots, vases etc. These hydro-gels are of many different chemical compositions but they all share almost the same swelling behavior. They have the capability to absorb water at a very fast rate and grow to a maximum of 150-300 times their weight in the process. The swelling properties of hydro-gels are mainly related to the elasticity of the polymer network, the presence of hydrophilic functional groups in the polymer chains, the extent of crosslinking, and porosity of the polymer [1]. Under dry conditions they de-swell very slowly. This water retentive property finds a number of applications in the industry. These polymer gels are used as moisture retaining soil additives in gardening; they also absorb liquid fertilizer just as they do water and serve as food reservoirs in the soil for plants. They have several major medical uses including tissue engineering, wound dressing and controlled drug release in their swollen state [2, 3]. Disposable nappies are made making use of the ability of hydrogels to take up and retain water. They are also used to make soft contact lenses. The swelling and de-swelling behavior of hydro-gels is used to detect certain analytes such as volatile organic compounds [4, 5], various gases [6, 7], biological molecules [8, 9, 10] etc. Hydro-gels may be used in sensing applications in which the swelling or de-swelling of the material is converted via a transducer into a measurable signal [5]. The sensitivity of hydro-gels to a large number of chemical and physical factors like temperature, light, electrical voltage, pH, ionic strength, biological, and chemical agents make them suitable for a broad range of applications [11]. Miniaturized hydro-gel objects are strong candidates for sensors and actuators in micro-electromechanical systems (MEMS) technology [12]. Electro-responsive ionic polymer gels have been employed for designing dust wipers, miniature robotic arms etc [13]. |
| Scientists still do not fully understand how hydrogels manage to absorb so much of water and there is still plenty of ongoing research into their properties and uses. Understanding their structure helps to explain their properties which in turn is used to design new hydro-gels to perform new functions. |
METHOD
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| In the present study instead of using the standard theories of diffusion, we have described absorption of water in terms of the phenomenon of attachment of ionized water molecules to charged sites on the polymer chains and thereby causing a volume expansion and mass growth of hydrogel. |
| Spherical hydro-gel beads of various colors most abundantly found and widely used in Kolkata were obtained from a standard source. The main ingredients of these gels are poly-acrylate or polyacrylamide. They are a type of Superabsorbent polymers (SAP) commonly made from the polymerization of acrylic acid blended with sodium hydroxide in the presence of an initiator to form a poly-acrylic acid sodium salt (referred to as sodium polyacrylate). This polymer is the most common type of SAP made in the world today [14, 15]. |
| These spherical beads of hydro-gel were kept immersed in distilled water and the mass and diameter of each bead were measured at intervals of 15 minutes. Each time, after taking out from water, a bead was put in contact with blotting paper for removing excess water from its surface. The diameter of a bead was measured by a screw gauge having an accuracy of 0.001 cm and the mass was measured by a digital weighing machine having an accuracy of 0.001 gm. Stop watches, having an accuracy of one second, were used for time measurement. The swelling behaviors of beads of white, yellow, orange, red, pink and green colored hydro-gels were studied. These are denoted in table 1 by their abbreviations. |
MODELING:
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| A mathematical model has been developed regarding the swelling behavior of the beads [16], which has been found to strictly obey the experimental data. |
| Let η be the number of pores and r be their average radii at any instant during the swelling. Let us also assume that a pore has the shape of a hollow cone having its base on the outer surface of the spherical hydrogel bead. Let R be the average depth of a pore (i.e. the height of such a hollow cone) where 0 < 1and R is the radius of hydro-gel bead. If f be the fraction of surface area covered by pores, we may write |
 (1) |
| Total surface area (A) in contact with water is the sum of the areas on the spherical surface and also those inside the conical pores. Thus we have [16] |
 (2) |
with  . Let is the thickness of active layer on this area (A) exposed to water. This layer, of volume A , contains active sites capable of getting attached to water molecules. At any instant t, let n be the number of water molecules already attached to the hydro-gel and N be the total number of such sites in that piece of hydro-gel in dry state. Thus, the dynamics of absorption, by vacant site occupation, may be expressed mathematically using [16] (as according to our earlier assumption the rate of variation of n being proportional to the number of vacant sites in the region exposed to water at time t |
 (3) |
| Since >> f , for a sufficiently large value of we may write = 2 f , which following (1) leads to |
 (3) |
| Hence (with p=3) |
 (4) |
| Hence (with p=3) |
 (5) |
| Instead of considering conical shaped pores, one may also consider the existence of cylindrical pores, each having an average depth of R and cross section 2 r , which leads to p=6 in (5). |
| We may now use the empirical relation [17] (which we have proved using a functional analysis in the appendix section) |
 (6) |
| and pluck it into (5) to obtain |
 (7) |
| Integrating it in the interval n = 0 to n ( t = 0 to t) and simplifying (noting that the final mass of a bead is given by 0 M(t) = M + mn , where 0 M and m are the initial mass of the bead and the mass of a single water molecule) we obtain |
| M=M |
 (8) |
| Clearly the above equation exhibits the behaviors at t = 0 and t . At t = 0 , 0 M(0) = M and at t , ( ) 0 M = M + mN . |
NUMERICAL ANALYSIS
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| We have fitted the experimental data of radius of hydro-gel bead vs. time to (11) as shown in fig.1. From this plot we have obtained max R and c. Using these parameters the mass vs. time data has been fitted to (13). This is depicted in fig.2. Thereby we have obtained the values of the parameters , , , r f and N as listed in table 1. |
| It is seen that the characteristic time c for radius is very much smaller than the characteristic time for mass [16], indicating that the equilibrium volume is reached earlier than the equilibrium mass. This is happening because the initial swelling of the hydrogel beads occurs at the surface only. Since the surface is stretched to its elastic limit the volume does not increase, yet the active sites embedded deep within the hydrogel keep absorbing water and mass increases. |
| It is worth noting that the pore radius r has the expected order of magnitude 10 μm[18]. Also, the values of N found in this case have good agreement with those listed in [16]. |
CONCLUSION:
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| The present theoretical model has been developed for conical or cylindrical shaped pores. This is an ideal case which is different from the real situation where pores of various sizes and shapes are present in the hydro-gel. Also, the predictions of the model are best verified with precise measurements of both mass and radius. In the present study mass measurement process has greater accuracy than that of volume measurement one, owing mainly to the deviations from spherical shape. Also swollen hydro-gels are soft objects that get depressed under pressure. This has affected the measurements. The unique feature of this study is that, on the basis of a completely new concept of occupation of vacant polymer sites by water molecules, the mechanisms of mass rise and volume expansion have been explained and it has also been shown that larger porosity induces faster absorption of water by hydrogels. |
APPENDIX:
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As we know that at t = 0 R(t) = R0 and at  , hence we may express the time dependent radius as the following |
 (9) |
| where f(t) is some time dependent function satisfying f (0) = 0, f () =1. Let us now define a new function (t) =1− f (t) . Hence (0) =1,() = 0. The simplest function that may satisfy the above relation will be t t k − ( ) = , where k is some positive constant. Hence |
 (9) |
| where1/c=ln k , is a new constant which finally leads to (6). |
Tables at a glance
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| Table 1 |
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Figures at a glance
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| Figure 1 |
Figure 2 |
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