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Polymer gels undergo volume phase transition in a thermodynamically poor solvent as a result of changes in molecular interactions. The osmotic pressure of gels, both synthetic and biological in nature, induces swelling and imparts the materials with the capacity to resist compressive loads. We have investigated the mechanical and swelling properties of poly(vinyl alcohol) (PVA) gels brought into the unstable state by changing the composition of the solvent. Chemically crosslinked PVA gels were prepared and initially swollen in water at 25 °C and then n-propyl alcohol (non-solvent) was gradually added to the equilibrium liquid. AFM imaging and force-indentation measurements were made in water/n-propyl alcohol mixtures of different composition. It has been found that the elastic modulus of the gels exhibits simple scaling behavior as a function of the polymer concentration in each solvent mixture over the entire concentration range investigated. The power law exponent n obtained for the concentration dependence of the shear modulus increases from 2.3 (in pure water) to 7.4 (in 35% (v/v) water + 65% (v/v) n-propyl-alcohol mixture). In the vicinity of the theta solvent composition (59% (v/v) water + 41% (v/v) n-propyl-alcohol) n ≈ 2.9. Shear and osmotic modulus maps of the phase separating gels have been constructed. It is demonstrated that the latter sensitively reflects the changes both in the topography and thermodynamic interactions occurring in the course of volume phase transition.
A polymer network is a macroscopic assembly of long polymer molecules joined at several connection sites (crosslinks). The crosslinks can be either chemical or physical bonds. Gels may involve different types of interactions, such as hydrogen bonds, electrostatic interactions (e.g., ion bridges), hydrophobic interactions, etc. In gels the crosslink distribution is always inhomogeneous. Upon swelling, the weakly crosslinked regions swell more, while the densely crosslinked regions swell less. The difference in swelling causes nonuniformity of the local elastic properties.
As an innate property of gels and soft tissues, the elastic modulus is often associated with the equilibrium mechanical response of the system. At the bulk material level, this quantity serves as a measure of small deformation behavior, and can be used, for example, to compare the compressive stiffness of tissues at different stages of development.1,2 Articular cartilage exemplifies the compositional and structural complexities found in soft tissues.3-6 The elastic modulus is also intrinsic to various mathematical models, both phenomenological and molecular in nature, that describe the load-deformation behavior of these materials. While the relationship between composition and material behavior is consequential at the bulk level, it becomes nontrivial at smaller length scales, where local variations in mechanical properties are affected by changes in the structure and interactions among the constituents. Even the most sophisticated constitutive models, however, consider composition only implicitly.
Despite significant differences in composition, synthetic and biological gels exhibit qualitatively similar responses (e.g., viscoelasticity and nonlinear stress-strain behavior at large deformation) under various loading conditions since they are governed by common physical principles.7 Biological tissues are always inhomogeneous, i.e., their local mechanical and osmotic properties strongly differ from those at the macroscopic level.
The atomic force microscope (AFM) is a powerful device to map the elastic and structural features of synthetic and biological gels at nanometer resolution.8-11 The force sensitivity of AFM cantilevers allows us to detect heterogeneities in gels that appear uniform at lower resolutions9,12-19 as well as to reveal the correspondence between mechanical and structural features in soft tissues (e.g., cells and their extracellular matrix can be delineated by their dissimilarity in stiffness20,21). However, previous AFM studies have shown that the shear modulus only weakly varies even in highly inhomogeneous biological tissues such as the extracellular matrix of mouse cartilage.21 This finding implies that the elastic modulus is not particularly sensitive to changes in either the interactions between the polymer and the solvent or the polymer molecules themselves.
In general, the swelling pressure of a gel is determined by the difference between the osmotic pressure of the crosslinked polymer and the elastic pressure of the swollen network.22 Changes in the thermodynamic quality of the solvent strongly affect the osmotic contribution of the swelling pressure while the elastic term is only slightly influenced.23,24 Thus, a sensitive indicator of changes in the thermodynamic properties should explicitly contain the contribution from the osmotic pressure.
In polymer physics, the osmotic modulus K is used to quantify the volume resistance of gels to an applied compressive load that encompasses not only the elastic properties of the material, but also its composition and interactions with its environment.24 Construction of maps of the osmotic and elastic moduli that reflects the effect of spatial variations of concentration fluctuations on these quantities has great potential in understanding the balance between the osmotic and elastic forces in complex biological samples, such as tissues and cells that are intrinsically inhomogeneous. For example, the load-bearing capacity of cartilage is governed by the osmotic modulus of the highly charged aggrecan/hyaluronic complexes enmeshed in the collagen matrix.3-6,25 The collagen fibrils (collagen molecules packed into overlapping organized bundles) are arranged in different combinations and concentrations in various tissues to provide the required biomechanical properties. Knowledge of the local structure and physical properties of the collagen matrix as well as the mechanism of fiber formation is particularly important to better understanding how tissues are constructed in growth and repair, and change in development and disease.
In this paper we introduce a method to create osmotic modulus maps by combining swelling pressure measurements and high-resolution AFM force-indentation measurements. To illustrate the feasibility of this procedure we investigate the macroscopic and microscopic elastic and osmotic moduli of a model polymer gel in which phase separation is induced by gradually varying the thermodynamic quality of the solvent. The gel system studied in this work is crosslinked poly(vinyl alcohol) (PVA) swollen in water/n-propyl alcohol mixtures of various n-propyl alcohol contents [from zero to 65% (v/v)]. Water is a relatively good solvent for the PVA chains while n-propyl alcohol is a non-solvent. In solvent/non-solvent mixtures containing more than approximately 45% (v/v) n-propyl alcohol PVA gels become turbid and shrink. It will be shown that the osmotic modulus map reveals fine scale variations in the osmotic response due to changes in the molecular interactions among the components.
The paper is structured as follows: after a brief description of the theory of gel swelling we present experimental results obtained by macroscopic measurements with particular emphasis on the effect of solvent quality on the concentration dependence of the elastic (shear) modulus. A scaling formalism is employed to analyze the data. Imaging results and elasticity measurements made by the AFM are then discussed. The procedure proposed to determine the osmotic modulus maps from AFM force-indentation measurements in conjunction with macroscopic swelling pressure observations is described. Illustrative examples of osmotic and shear modulus maps are presented for lightly crosslinked PVA gels swollen in water/n-propyl alcohol mixtures of different composition.
Since the pioneering works of Dusek26,27 and Tanaka24,28 it is known that polymer gels may undergo a volume transition as experimental conditions, such as temperature, pressure, solvent composition, ionic strength and pH vary. Flory,22 James and Guth,29,30 Hermans31 and others proposed various statistical mechanical models to estimate the free energy change accompanied by the deformation and swelling of polymer networks. Experimental studies made on weakly crosslinked neutral gels [e.g., poly-N-isopropylacrylamide gels (PNIPAM)] as well as polyelectrolyte gels indicated a rather sharp volume change as a function of the temperature or ionic strength.32 These observations have been interpreted as a discontinuous volume transition. Continuous volume changes have also been reported for gels. For example, Li and Tanaka33 found that the discontinuous volume transition of PNIPAM hydrogels becomes continuous as the crosslink density increases.
On the microscopic level the polymer gel is composed of closely packed polymer coils kept together by the crosslinks. Gel swelling (or shrinking) is related to the expansion (or contraction) of the network chains. In general, the radius of gyration of the individual chains consisting of N monomeric units is given as
where ν is a universal exponent the value of which is governed by the effective interaction between the polymer segments and the solvent molecules.34 In good solvent condition (ν ≈ 3/5) the monomer-monomer interaction is repulsive and tends to swell the chain (strong excluded volume interaction). In a theta solvent where the attractive and repulsive forces compensate, the polymer exhibits ideal behavior (no excluded volume interaction), i.e., ν = 1/2. In a poor solvent the attractive interactions between polymer segments dominate, the chains collapse and ν = 1/3. In polymer solutions, the decrease in solvent quality leads to phase separation, and beyond the critical point a solvent rich phase will coexist with a polymer rich phase.
In gels, however, permanent crosslinks prohibit the network chains from changing relative positions. As the solvent quality decreases the concentration fluctuations increase and the gel starts to shrink. The elastic stress due to chain deformation is coupled to the local concentration fluctuations. Polymer rich domains are formed as a result of the segregation process. The size and shape of these domains depend on the particular polymer-solvent system, the crosslink density and the actual geometry of the gel sample.
Simple scaling theory derives the behavior of neutral polymer gels from the properties of the corresponding solutions at the overlap concentration ϕ*. At ϕ* the average polymer concentration of the system is equal to that of an individual swollen coil.
The concentration of the fully swollen gel ϕe is proportional to ϕ*. The latter can be expressed as
In neutral gels interactions between the polymer chains are not expected to contribute significantly to the elastic properties, i.e., the elastic modulus is primarily governed by the concentration of the network chains. The shear modulus of a fully swollen gel Ge is given by
where A is a constant, kB is the Boltzmann factor and T is the absolute temperature. Since ϕe and N are related by equation 2, for gels differing in the crosslink density the concentration dependence of Ge should obey the scaling relation
Equation 4 predicts that the power law exponent n is ≈ 9/4 in good, and 3 in theta solvent condition. In the poor solvent regime n increases and diverges in the ν → 1/3 limit. However, in poor solvent condition, the validity of simple scaling relationships becomes questionable because of the high polymer concentration of the gel.
The elastic modulus defines the elastic pressure that contracts the gel while the osmotic pressure expands it. At equilibrium in excess amount of solvent the elastic pressure balances the osmotic pressure of the network
where the constant B depends on the thermodynamic quality of the solvent.
The concentration dependence of the elastic pressure can be given as
where G is the shear modulus of the gel.
Poly(vinyl alcohol) (PVA, Mn = 70,000-100,000, degree of hydrolysis > 99 mol%) gel cylinders (1 cm diameter, 1 cm height) were cast with initial polymer concentrations of 3, 6, 9 and 12% (w/w) for osmotic swelling and macroscopic compression tests. Gel slabs (> 2 mm thick) with initial polymer concentration of 6% (w/w) were cast for AFM nanoindentation. The gels were crosslinked with glutaraldehyde (one unit per 100 monomer units) at pH ≈ 1.5.37 After gelation the gel slabs were washed in pure water to remove sol fraction, and then equilibrated with water/n-propyl alcohol mixtures containing gradually increasing amount of n-propyl alcohol. Swelling and mechanical tests were performed on the gels after 2-3 days.
Shear modulus data were obtained from uniaxial compression measurements using a TA.XT2I HR Texture Analyser (Stable Micro Systems, UK). This apparatus measures the deformation (± 0.001 mm) as a function of an applied force (± 0.01 N). Stress-strain isotherms were determined for all gels in the fully swollen state and at different swelling ratios in pure water and in water/n-propyl alcohol mixtures. The PVA/water gels were equilibrated with poly(vinyl pyrrolidone) solutions of known osmotic pressure according to a method described previously.36,38 Gel samples were uniaxially compressed (at constant volume) between two parallel flat plates. The data were analyzed using the Mooney-Rivlin relation27,39
where σ is the nominal stress (related to the undeformed cross-section of the gel), Λ is the deformation ratio (Λ = L/L0, L and L0 are the lengths of the deformed and undeformed specimen, respectively), and C1 and C2 are constants. Equation 9 is generally valid for extension measurements made on rubber-like materials. The stress-strain data were determined in the range of deformation ratio 0.7 < Λ < 1.0. The value of C2 was negligibly small for the gel systems studied. In this situation, the constant C1 can be identified with the shear modulus (G) of the swollen network.
All swelling and mechanical measurements were made at 25 °C.
Gel slabs were kept immersed in solution throughout the duration of the AFM measurements. General-purpose, oxide-sharpened, silicon nitride tips of pyramidal shape were used for the AFM measurements, performed using a commercial AFM (Bioscope SZ with Nanoscope V controller, Veeco). The spring constants of the cantilevers were measured by the thermal tune method. A raster scanning approach (“force-volume”) was applied to automatically perform indentations over areas ranging from 10 μm × 10 μm to 30 μm × 30 μm, at a resolution of 32×32 (1024 total indentations). Code written in Matlab was used to automatically process each dataset and extract values of shear modulus using an optimization-based approach.40 The shear modulus was extracted from each force curve using the relationship between force (F) and indentation depth (δ) for a pyramidal indenter derived by Bilodeau:41
where 2θ is the tip angle and μ is Poisson's ratio of the indented material.
Accuracy of elasticity measurements using the AFM is dependent upon methodology and modeling of the acquired force-deflection data.40 Tapered tips are generally better suited for indenting materials with a large linear elastic limit.40,42 They were selected over spherical probes in this study because of the greater resolution afforded. A tip velocity of approximately 800 nm/s was applied in all AFM indentation tests. We have shown previously21,40 that this rate produces elastic behavior comparable to macroscopic compression of PVA gels. Standard techniques of mechanical property measurement such as macroscopic compression are appropriate means of calibration or validation of AFM measurements. In Figure 1, representative datasets from the AFM indentation of two PVA gels (one swollen in water and the other in 65% (v/v) n-propyl alcohol + 35% (v/v) water) are compared to corresponding macroscopic compression data to demonstrate the validity of our approach. The results show that the shear moduli obtained by the two independent techniques are in reasonable agreement.
Previous indentation experiments on PVA gels using spherical indenters had indicated the linear elastic limit to exceed average strains of 5%.21 Although tapered indenters induce greater strains,42 it was found that at the maximum cantilever deflection of 100 nm applied in this study, the force-indentation behavior still conformed to linear elastic equation (10) as shown in Figure 1. Additionally, an indenter with a tip diameter of several nanometers as is typical of probes used in this study can be considered sharp without introducing unacceptably large errors. Finally, the AFM data showed tip-sample adhesion to be absent in the loading phase of the indentation, as is generally observed in soft samples.43-48 This obviated the need to account for adhesive effects.
As discussed in the Theoretical Section a characteristic feature of crosslinked polymers is their swelling ability, i.e., they can absorb (or expel) liquid due to changes in external conditions. In Figure 2 are shown typical swelling curves for PVA gels. These curves represent the volume fraction of the polymer in equilibrium with excess solvent (water/n-propyl alcohol mixture). On increasing n-propyl alcohol content the polymer volume fraction increases. On the macroscopic scale probed in this experiment the volume transition appears continuous.
When a gel is in equilibrium with the surrounding solvent its osmotic swelling pressure vanishes (ω = 0). The region above the swelling curve corresponds to states with ω > 0, while in region below the curve ω < 0. On changing the solvent composition the gel swells or shrinks until the new equilibrium state is attained. The inset illustrates schematically the phase diagram of a polymer gel. It appears similar to the miscibility curves usually observed for polymer solutions. In the hatched region the system is unstable. The solid line, often called the coexistence curve, represents the equilibrium condition. Systems inside this curve separate into two phases of compositions ϕ1 and ϕ2. The shape of the experimental swelling curves (dashed lines in Figure 2) is reminiscent of the phase diagram of gels.
As illustrated in Figure 2, the swelling degree (1/ϕe) decreases with increasing n-propyl alcohol concentration. Figure 3 shows the variation of the equilibrium shear modulus measured for PVA gels in different water/n-propyl alcohol mixtures. In each solvent mixture, the shear modulus Ge can be related to the polymer concentration ϕe by the scaling equation 4. The solid lines through the data points are least squares fits to equation 4. The prefactor AkBT varies from 3400 kPa (in pure water) to 4100 kPa (in 65% (v/v) n-propyl alcohol + 35% (v/v) water mixture). The inset shows the variation of n and ν with the solvent composition. In pure water n (≈ 2.3) is close to the theoretical value in good solvent condition and gradually increases with decreasing solvent quality. The excluded volume exponent ν decreases with increasing n-propyl alcohol content and approaches the value corresponding to collapsed polymer chains (ν → 1/3).
Addition of a non-solvent decreases the thermodynamic quality of the solvent and thus reduces the swelling capacity of the network. However, changes in the thermodynamic interactions may also affect the structure and the elastic properties of the gel. To determine the effect of the composition of the solvent/non-solvent mixture on the shear modulus of PVA gels, we plotted G/Ge as a function of ϕ/ϕe (Figure 4) where G and ϕ are the shear modulus and the polymer volume fraction in the mixed solvent, respectively, and Ge and ϕe are the corresponding quantities in water. In the figure are also displayed the values of G/Ge determined in water at different PVA concentrations (open sysmbols). It is clear from this double logarithmic representation that all data fit the power law relationship
where the exponent (m ≈ 0.42) slightly exceeds the value 1/3 predicted by the theory of rubber elasticity.39 The observed universal behavior implies that the elastic modulus is primarily determined by the overall polymer concentration, i.e., changes in the polymer-solvent interactions do not affect the number of elastic chains. This finding is consistent with previous results obtained for various polymer gels in different solvent conditions.35,49
As we have seen, changes in the solvent quality affect the osmotic swelling pressure and thus the equilibrium concentration of polymer gels. The osmotic modulus K [= ϕ(ω/ϕ)] defines the resistance of the gel to changes in the swelling degree. As shown above (see equation 8) ω contains two contributions: (i) the osmotic pressure of the network chains that is the driving force of the swelling process, and (ii) the shear modulus that counteracts the osmotic pressure and preserves the shape of the gel. It follows from equation (8) that the osmotic modulus is given by
Figure 5 shows the variation of K and G for PVA gels in different n-propyl alcohol/water mixtures. It can be seen that K sharply increases with decreasing solvent quality in the transition region while the variation of G is much less pronounced.
So far we have discussed the effect of solvent composition on the macroscopic swelling and elastic behavior of PVA gels. We have seen that both the polymer concentration and the shear modulus increase continuously through the transition. This result is consistent with the picture that in the phase separating gel the nuclei of the new phase are randomly formed and distributed in space and the two phases coexist locally over the entire sample. In what follows we investigate the spatial variation of the shear and osmotic moduli by combining AFM measurements with macroscopic swelling pressure observations. To gain insight into the effect of changes in the structure and interactions occurring at high-resolution, we construct the elastic and osmotic modulus maps of the PVA gels at different stages of phase separation.
In the region below the spinodal decomposition little is known about the structure and dynamics of polymer gels. The transport of solvent molecules between the gel and the surrounding liquid phase is a diffusion-controlled process, during which concentration inhomogeneities gradually develop. De Gennes discussed theoretically the phase behavior of polymer gels.34 Onuki50 constructed a model for gels undergoing spinodal decomposition and showed that in the late stage the growth rate slowed down due to the elastic forces imposed by the network. It has been found that on the mesoscopic level the gel is composed of highly concentrated collapsed domains separated by lower concentration swollen regions. These domains can slowly coalesce and segregate. Domain formation strongly affects the local physical properties (e.g., elastic modulus) of the system.
Shear and osmotic modulus maps for PVA gels in different n-propyl-alcohol/water mixtures are shown in Figures 6 to to8,8, where the 3D plots have been scaled identically to facilitate comparison between the different cases. The shear modulus maps were directly obtained from the AFM force-indentation measurements. The elasticity-concentration relationships displayed in Figures 3 and and44 are representative of the prerequisite data for generating maps of K: for each point on the shear modulus map the local polymer concentration in the gel was estimated from the measured value of the local shear modulus assuming that the macroscopic scaling relationship applies on the microscopic level.
Figure 6 shows that in water the surface of the PVA gel is smooth and both Ge and K exhibit only small local variations (4 < Ge < 5 kPa and 5 < K < 15 kPa) at the micrometer length scale.14 As the solvent quality decreases nonuniformities in polymer concentration arise throughout the gel; K, and to a much lesser extent, G, reflect these inhomogeneities. This trend is clearly visible in comparing the AFM topographaphical images and shear modulus maps in Figures 6 to to8.8. In the case of the theta solvent (Figure 7), surface topography and local shear moduli remain relatively uniform. However, the increased amplitude of K is an indication of the onset of phase separation.
In gels inhomogeneous structures develop during volume phase transition. The size and shape of the polymer rich domains, as well as the volume ratio between the swollen and shrunken regions of the sample is a function of the solvent composition. The structural changes induced by the transfer of solvent molecules from the surrounding liquid phase into the network first take place at the gel surface and then gradually extend into the bulk. However, the gel surface attains equilibrium only after the bulk. The present results show that surface and bulk patterns strongly affect the local elastic and osmotic response of the PVA gels. This becomes evident in the poor solvent (Figure 8) where even the surface topography and the shear modulus exhibit significant spatial variations. The map of K contains peaks and valleys, demonstrating the strong dependence of the local value of K on polymer concentration distribution. The shear and the osmotic moduli vary in the range 5 < Ge < 20 kPa and 10 < K < 90 kPa, respectively.
The importance of gaining better physical understanding of the phase behavior of gels is obvious if we note the occurrence of gel-like materials in nature, particularly in living systems. The structure and properties of many composite materials are controlled by their phase behavior. In chemically crosslinked polymer gels various patterns of volume phase transition have been observed. The osmotic modulus map provides a detailed, high-resolution picture of swelling and elastic properties of gels along with a quantitative assessment of local inhomogeneity.
AFM nanoindentation is a widely applied technique for mapping the elasticity of inhomogeneous samples, such as biological cells and tissues, at high resolution.8,10,11,14,20,51-55 One of the chief advantages of this method is its capability to simultaneously measure the topography of the scanned region. To the best of our knowledge, the work presented herein represents the first time that the AFM has been used in tandem with osmotic swelling pressure measurements to map the osmotic modulus of gels, which is more reflective of local changes in composition and thermodynamic interactions than the elastic properties alone.
Swelling of polymer films is important in applications ranging from coatings and sensors to microelectronics and biomedical devices. Soft contact lenses represent an example where performance, comfort, and even eye health are influenced by local mechanical and hydration properties.56 In this case, a means of mapping the elastic and osmotic moduli at high resolution is essential in assessing local inhomogeneities and improving performance.
The osmotic modulus map allows us to predict both elastic response and recovery capability of complex viscoelastic materials and is particularly important to estimate the compressive behavior of biological tissues. For instance, it is known that a decrease in the shear modulus of articular cartilage results in reduced compressive resistance and hence, compromised load-bearing capacity.57 A relatively smooth map of K (Figure 6) indicates that resorption of water occurs consistently throughout the sample while large disparities in K (Figure 8) signify nonuniform volumetric recovery. It can be deduced from Figure 8 that regions of slightly different shear modulus can exhibit drastic differences in recovery behavior. For example, in articular cartilage of the knee and hip, which can experience large amplitude repetitive loading during physical activity (e.g., walking), rapid recovery of the bulk tissue is crucial to normal function. A decrease in the osmotic pressure at the tissue level is associated with degenerative conditions such as osteoarthritis.4,5,25,58 We surmise that changes in microscopic osmotic properties are complicit as a pathological feature of osteoarthritis.
The osmotic modulus is a measure of the contribution of osmotic pressure and elastic modulus to the compressive resistance of polymer gels. By relating K and G through concentration-dependent scaling laws, we have demonstrated that K is significantly more sensitive to polymer concentration gradients than G. The two quantities are complementary, serving to describe the swelling and shape preservation tendencies of soft gel-like materials. The osmotic mapping technique proposed herein allows local variations in K to be determined by combining nano- and microindentation instrumentation with macroscopic osmotic swelling pressure observations. The osmotic modulus map provides insight into the effect of structural changes (e.g., self-assembly) on the swelling and elastic properties of gel systems at high-resolution. This knowledge is essential in biomechanical studies, such as tissue characterization and the development of biomimetic tissue analogs, and may also help to assess the performance of gellike materials used in biomedical devices.
This work was supported by the Intramural Research Program of the NIH, NICHD.