Key words

Water absorbing polymer, hydrogel swelling, sugar solution concentration, characteristic time 
INTRODUCTION

Hydrogel polymers commonly available in the market under various names such as magic pearl jelly, watergel or crystal soil etc. are often used as decorative as well as source of water for indoor flower pots, vases etc. These superabsorbent gels are of various chemical compositions but having almost the same swelling behavior. They have the capability to absorb water at a very fast rate and grow to a maximum of 150300 times their weight in the process. The swelling properties of watergels 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 deswell very slowly. Their water retentive property finds a number of applications in the industry. These 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. Hydrogels 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 deswelling behavior of hydrogels is used to detect certain analytes such as volatile organic compounds [4, 5], various gases [6, 7], biological molecules [8, 9, 10] etc. Hydrogels may be used in sensing applications in which the swelling or deswelling of the material is converted via a transducer into a measurable signal [5]. The sensitivity of hydrogels 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 hydrogel objects are strong candidates for sensors and actuators in microelectromechanical systems (MEMS) technology [12]. Electroresponsive ionic polymer gels have been employed for designing dust wipers, miniature robotic arms etc [13]. 
Scientists still do not fully understand why and how hydrogels manage to absorb so much 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 hydrogels to perform new functions. 
MATERIAL AND METHOD

In the present study instead of using the standard theories of diffusion, we have described absorption of water in sugar solution in terms of the phenomenon of attachment of ionized water molecules to charged sites on the polymer chains and thereby causing a mass growth of hydrogel. 
Spherical white hydrogel beads of most abundantly found and widely used in Kolkata were obtained from a standard source. The main ingredients of these gels are polyacrylate or polyacrylamide. They are a type of superabsorbent polymers commonly made from the polymerization of acrylic acid blended with sodium hydroxide in the presence of an initiator to form a polyacrylic 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 hydrogel beads (white in color and almost identical in size) were kept immersed in sugar solution of a certain concentration and the mass 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 and the mass was measured by a digital weighing machine having an accuracy of 0.001 gm. Stop watches, having an accuracy of 0.2 second, were used for time measurement. The concentration of the sugar solution was now increased and the experiment was repeated. In these ways we have varied the sugar concentration from 0 % to 25 % with a step of 5 %. 
Theoretical Modeling

A mathematical model has been developed regarding the swelling behavior of the beads [16], which has been found to strictly obey the experimental data. 
Hydrogels are highly porous and the pores are distributed all over its surface area. Let be the number of pores and r be their mean radius at any instant of time during the swelling process. 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. We also assume that 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 hydrogel bead. If f be the fraction of surface area covered by pores, we may write 
(3.1) 
Total surface area (A) in contact with water is the sum of areas on the spherical surface and insides of these conical pores. Thus we have [16] 
(3.2) 
with with 2 = 4 f ( + 4 f ) . Let be the thickness of active layer on this area (A) exposed to sugar solution. This layer, of volume A , contains active sites capable of getting attached to water molecules. At any time t, let n be the number of water molecules already attached to the hydrogel and N be the total number of such sites in that piece of hydrogel in dry state. Thus, the dynamics of absorption, by vacant site occupation, may be expressed mathematically by [16] (as according to our earlier assumption the rate of variation of n being equal to the number of vacant sites in the region exposed to water at time t 
(3.3) 
Since ? f , for a sufficiently large value of we may write = 2 f , which following “(3.1)” leads to 
(3.4) 
Hence (with p=3) 
(3.5) 
Instead of considering conical shaped pores, one can also consider the existence of cylindrical pores, each having an average depth of R and cross section r , which leads to p=6 in “(3.5)”. From micrographs of highly porous hydrogels, provided by various groups of researchers [17, 18], it is often found that almost the entire surface area of hydrogel is covered with pores, making f very close to unity. Taking f 1, we may neglect the first term on the right hand side of equations “(3.5)” and its new form would be 
(3.6) 
Here is the time constant or characteristic time for the process of swelling of hydrogel in pure water. 
Now to formulate the swelling behavior in sugar solution we consider that in sugar solution, a large number of water molecules are bound to sugar molecules and are thus unavailable for attachment at the active sites of the hydrogel. Hence the probability of attachment of water molecules to active sites may be expected to depend upon the fraction of water molecules not hindered in any way to get attached to active sites of watergel. For larger concentrations of sugar, this fraction will be smaller and consequently, the chance of attachment becomes smaller. The modified form of equation “(3.6)” may be expressed as 
(3.7) 
The parameter K denotes the fraction of water molecules capable of interaction with active sites of gel which may be defined in the following way. 
(3.8) 
where, w N is the number of water molecules present per unit volume of water. This is actually the number of water molecules per unit volume, which are all equally capable of interaction with active sites when the hydrogel bead is immersed in pure water. N2 is the number of water molecules which are prevented from having interaction with active sites, per unit volume, in a sugar solution of a certain concentration (say P% strength, i.e. P gm of sugar per 100 cc of water). 
Experimental Results

It was our experimental observation that the mass of 10 cc of 15% sugar solution is 10.626 gm. We know that 10cc of 15% solution contains 1.5 gm of sugar. Therefore, this solution contains (10.6261.5) gm = 9.126 gm of water whose volume is nearly 9.126 cc. Hence the volume occupied by sugar in this solution is (10 − 9.126)c.c. = 0.874 cc. Number of sugar molecules in 1.5 gm of sugar being 23 21 (1.5 / 342)×6.023×10 = 2.641×10 (as the molecular weight of sugar 12 22 11 C H O is S M = 342). Therefore the volume occupied by each sugar molecule is, say, 21 0.874 / (2.641 10 ) S v = × cc 22 3.309 10− = × cc. The volume of one water molecule is 23 / ( ) 2.989 10 w w w A v M N cc − = = × , A N being the Avogadro number. Here, the molecular weight of water is 18 w M = with density 1 w ? gm.cc −1 . 
Now for 1 cc of solution one may write 
Nsvs + Nsvs =1 (4.1) 
Where s , w N N are respectively the numbers of sugar molecules and water molecules per cc in a solution of a certain concentration (say P%). Thus from the above equation we have 
(4.2) 
For smaller values of P we obtain 
Now the number of sugar molecules, per unit volume, in a solution of P% concentration is given by 
(4.3) 
Let C1 be the average number of water molecules hindered by each sugar molecule, in different possible ways, in the solution. Hence we have 
(4.4) 
Hence following “(3.8)” we obtain 
(4.5) 
Now using “(4.5)” in “(3.7)” we get 
(4.6) 
with 
(4.7) 
Here, T is the new time constant, characteristic of the swelling process of hydrogel in a solution of strength P%. 
For pure water P = 0 and we have K = 1 and T = . Then “(4.6)” reduces to “(3.6)” which has already been solved to derive the mass versus time relation for the hydrogel swelling process in pure water [16]. Modifying this masstime relation accordingly (using “(4.6)”) we get 
(4.8) 
Where M0 is the initial mass or dry mass, n is the number of water molecules attached to the active sites of the hydrogel and m is the mass of a water molecule, 23 m 2.99146 10 gm − = × . Equation “(4.8)” may also be expressed as 
(4.9) 
Here, has been calculated for different samples of hydrogel. 
By a rearrangement of terms and using the relation “(4.7)”, “(4.8)” may be expressed as 
(4.10) 
Equation “(4.10)” suggests that if its left hand side is plotted as a function of t, it will be a straight line passing through the origin. The slope of this line is K. 
Fitting our experimental data numerically to “(4.8)”, we have determined the values of T for different concentrations of sugar solutions (figure 1). These values are listed in table 1. 
Now from “(4.5)” and using the value of obtained from the experimental study of the sample we get 
(4.12) 
Hence, the average number of water molecules ( 1 N ) not free to get attached to active sites in 1cc of solution of strength P% is 
(4.11) 
According to our theoretical formulation (as described by “(4.11)” above), a plot of 1/ T versus P will be a straight line with a negative slope ( − / ) and the corresponding intercept is 1/ . Figure 2 shows a plot of 1/ T as a function of concentration (P) for our sample regarding the swelling of hydrogel in sugar solution. This plot is in very good agreement with “(4.11)”. The slope and intercept of the best fitting straight line are 4 2.155 10− − × and 3 7.240 10− × respectively. The value of for the sample is 138.122 minutes, which should ideally be the value of T for P = 0 . The value we have from the table 1 is 4 / 2.155 10 − = × and hence we obtain 2 2.977 10 − = × . 
(4.13) 
And the fraction of free water molecules in a solution of strength P% is 
(4.14) 
Concluding Remarks

The present model has been developed with the assumption of conical or cylindrical shape of the pores. This is an ideal case which is different from the real situation where pores of various sizes and shapes are present in the hydrogel. The model may be improved by considering a suitable statistical distribution of sizes and shapes of the pores. The unique feature of this study is that, on the basis of a completely new concept, based on the phenomenon of occupying vacant polymer sites by water molecules, the mechanisms of mass rise has been rigorously studied. 
Tables at a glance


Table 1 


Figures at a glance



Figure 1 
Figure 2 


References

 Chavda H. V. and Patel C. N. Preparation and Characterization of Swellable Polymer Based Superporous Hydrogel Composite of Poly (AcrylamidecoAcrylic Acid), Published online 27 January (www.sbaoi.org), 2010.
 Kawaguchi, H., “Functional polymer microspheres”. Progress in Polymer Science 25, 11711210, 2000.
 Pal K, et. al., “Polymeric Hydrogels: Characterization and Biomedical Applications –A mini review”, Designed Monomers and Polymers, 12, 197220, 2009.
 Lonergan, M.C., et. al., “ArrayBased Vapor Sensing Using Chemically Sensitive Carbon BlackPolymer Resistors”. Chem. Mater. 8, 22982312, 1996.
 Porter, T.L. et. al., “Sensor Based on Piezoresistive Microcantilever Technology”. Sensors and Actuators A, 88, 4751, 2001.
 Kooser, A., et. al., “Gas Sensing Using Embedded Piezoresistive Microcantilever Sensors”. Sensors and Actuators B, 99, 430433, 2004.
 Porter, T.L., et. al., “A SolidState Sensor Platform for the Detection of Hydrogen Cyanide Gas”. Sensors and Actuators B, 123, 313317, 2007.
 Porter, T.L., et. al., “Viral Detection Using an Embedded Piezoresistive Microcantilever Sensor”. Sensors and Actuators A, 107, 219224, 2003
 Gunter, R.L., et. al., “Investigation of DNA Sensing Using Piezoresistive Microcantilever Probes”. IEEE Sensors Journal, 4, 430433, 2004.
 Kooser, A., et. al. [2003] “Investigation of the Antigen Antibody Reaction Between AntiBovine Serum Albumin and Bovine Serum Albumin Using Piezoresistive Microcantilever Sensors”. Biosensors and Bioelectronics, 19, 503508.
 Kost J, Pulsed and selfregulated drug delivery, (CRC, Boca Raton, Fla. Chapter 2), 1990.
 Chatterjee A. N., et. al., “Mathematical Modeling and Simulation of Dissolvable Hydrogels”, Journal of Aerospace Engineering, 5564, 2003.
 Cohen Y. B., “Electro active polymers as artificial musclesreality and challenges”, Proc. of the 42nd AIAA Structures, Structural Dynamics, and Materials Conference (SDM), Pasadena, Calif, 2011.
 Horie K. et. al., “Definitions of terms relating to reactions of polymers and to functional polymeric materials (IUPAC Recommendations 2003)”, Pure and Applied Chemistry, 76, 889906, 2004.
 Buchholz F. L. and Graham A. T., ed. Modern Superabsorbent Polymer Technology (1 ed.), (John Wiley & Sons, ISBN 0471194115, New York), 1997.
 Das, P. et. al., “An experimental study of the swelling properties of the spherical hydrogel immersed in water and the formulation of a theoretical model for its explanation”, International J. of Drug Delivery. 4, 2030, 2012.
 Dinu M., et. al., “Polyacrylamidedextran Polymeric Networks: Effect of Gel Preparation Temperature on Their Morphology and Swelling Properties”, Cellulose Chem. Technol., 45, 197203, 2011.
 Gemeinhart R. A., et. al., “Pore Structures of Superporous Hydrogels”, Polymers for Advanced Technologies, 11, 617625, 2000.
