Volume 8 Issue 7 - May 1, 2009
Contact measurement of internal fluid flow within poly(n-isopropylacrylamide) gels
Wei-Chun Lin1,*, Kenneth R. Shull1, Chung-Yuen Hui2, Yu-Yun Lin3

1Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA
2Department of Theoretical and Applied, Mechanics, Cornell University, Ithaca, NY 14853, USA
3Department of Civil Engineering, National Cheng Kung University, Tainan 701, Taiwan
*w-lin5@northwestern.edu
3cyylin@mail.ncku.edu.tw

Journal of Chemical Physics, 127(9), 2007, Art. No 094906

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Associate Professors, Yu-Yun Lin
Gels are porous materials composed of subunits that are able to bond with each other to form a macroscopic network. An important characteristic of gels is that their mechanical behavior depends on the time scale of the measurement used to probe the material. At sufficiently short times a gel behaves as an incompressible material because there is no time for fluid to flow out of the network. At extended times, however, the gel acts as a compressible material. Polymer gels, consisting of a network of cross-linked polymer chains, swollen with a small molecule solvent, are another important class of gels, with widespread applications in medicine and biomedical engineering. In the eye, for example, the lens and the vitreous body, the large space behind the lens, are gels. Changes in the viscoelastic properties of the lens can lead to the loss of accommodation, or its refractive power, while changes in its permeability can create the onset of a mature cataract. Liquefaction of the vitreous body—i.e., when the gel deteriorates to a liquid phase—is believed to play an important role in retinal detachment. Measuring the transport properties and mechanical properties of these gels could therefore have potential clinical applications. Additionally, an understanding of bulk properties in gels is applicable to the development of controlled drug delivery systems that are used in pharmaceutical, and cosmetic applications. Quantifying the bulk transport properties of gels is essential in each of these situations.

Measurement of the transport properties of polymer gels has traditionally been accomplished by light scattering or by direct mechanical measurement. However, the nature of these measurements is limited to transparent samples since light scattering is not possible for opaque gels, such as in the case of a cataract that affects the lens of the eye. These types of gel can be tested using mechanical method. For extremely flexible samples (moduli of several kilopascals), this study utilizes a flat, circular punch, in addition to flat, rectangular punch, to characterize the transport and mechanical properties of a polymer gel. These geometries are depicted in Figs. 1(a) and 1(b), respectively.
Fig. 2
Fig. 1(a), 1(b)


The model system chosen for this study is a poly(N-isopropylacrylamide) (pNIPAM) gel. pNIPAM is used because of its unique phase behavior and physical properties. These gels have generated much interest in the medical community, largely because of their phase behavior. pNIPAM gels undergo a reversible phase transition at a lower critical solution temperature (LCST) of ~33℃ in aqueous solutions, a temperature close to the human body. Below the LCST the polymer networks swell with solvent, resulting in clear, homogeneous gels, with water molecules forming cagelike structures around hydrophobic groups of the pNIPAM macromolecules. Above the LCST, the gel dehydrates and the phase separates, forming a heterogeneous, opaque structure. Phase separation occurs when the structure of the water molecules around the hydrophobic groups is disrupted, causing these hydrophobic groups to associate. The collapse of the network leads to polymer-rich and solvent-rich regions throughout the gel. The transition from a swollen, homogeneous gel at T=22℃ to a shrunken, heterogeneous gel at T=39℃ is depicted in the photographs in Fig. 2.

In the indentation experiments an indenter is brought into contact with a thin layer of a pNIPAM gel, using a probe tack apparatus that is shown schematically in Fig.3. We subject the gel to a two-step displacement history in the linear elastic regime. Figure 4 illustrates the time dependence of the load and displacement for a typical experiment.
Fig. 4
Fig. 3


First, the rigid indenter is brought into contact with the sample until a predefined maximum compressive load is reached. The displacement is then fixed for 1000  s as the gel relaxes due to solvent flow and viscoelastic relaxation of the gel network. Immediately following this period of stress relaxation, displacement oscillations with a 20µm amplitude are applied at frequencies ranging from 1.0  to  0.002  Hz. Elastic moduli obtained from oscillatory tests performed at 22 and 39℃are shown in Fig. 5. The shear moduli plotted in Fig. 5 were approximated by neglecting the effects of solvent flow, i.e., by taking veff=0.5. This is a very good approximation when the phase angles are small, as is indeed the case in our experiments. These phase angles are plotted in Fig.6.
Fig. 6
Fig. 5


In the case where the energy dissipation is entirely due to the solvent flow, a predictive model for the phase angles can be developed. We begin by defining the following response function Y(t), which describes the response of the material to an instantaneous displacement increment δ0:


where Y0 describes the instantaneous response and λ describes the fractional relaxation due to the solvent flow. The specific functional form of the unit force response Y(t) depends on the loading geometry and is not generally available. However, for an indenter with circular cross section against a porous half-space (a/h«1, a is the punch dimension and h is the gel thickness), Lin and Hu have obtained the following expression for Y(t):


where and  the normalized time T=Dct/a2.
If we consider the following steady state oscillatory displacement functionδss(t)=δωeiωt, the resulting steady state force is


where Y=Y0(1-λ). The phase angle is given by


Combination of the above equations leads to the following approximate expression for the phase angle for a flat circular punch,
(*)


Where Ω as the normalized angular frequency,


We define an effective pore size , Dpis the permeability of the gel. Equation (*) is a central theoretical result of this paper, which relates a measurable phase angle to relaxed Poisson’s ratio (through λ) and the pore size (through Ω).
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