Swelling of polymer networks

When a polymer network is placed in a solvent, it will swell and reach an equilibrium swelling in the presence of excess solvent. The free energy of a swollen network can be assumed to consist of the sum of the free energy of mixing the polymer network 

An elastic polymer chain strand is defined as a chain attached to the network by cross-Unks at its two ends. The free energy of a swollen netwoik or gel is then given by the sum of Equations 2.14 and 2.15 as 

When the netwoik swells to equilibrium in excess solvent, the chemical potential (partial molar free energy) of the solvent inside the gel (p-i) is equal to the chemical potential of the pure solvent (p ) outside the gel. Thermodynamics gives the quantity (pi - p ) as the derivative of with respect to Af, at constant T, P, and N2. Setting the derivative of the expression in Equation 2.16 with respect to Af, equal to 0 gives the following relation between Vj, %. and N, where Af is the number of moles of elastic polymer chain strands per unit volume of dry network  

This equation can be thought of as the result of the balance between two types of forces thermodynamic forces due to the solvent swelling the network (left-hand side of the equation) and elastic forces resisting the network expansion (right-hand side). 

FIGURE 2.3 The interaction parameter X as a function of polymer volume fraction Vj for various PDMS networks of different structures and modulus swollen in benzene. The data points are calculated from the swelling results using Equation 2.17. The line represents data obtained for solutions of uncross-linked PDMS chains in benzene. (Data from Flory, P. J., and Shih, H., Macmmolecules, 5,761,1972 Patel, S. K., Malone, S., Gilmor, J. R., and Colby, R. H., Macwmolecules, 25, 5241, 1992.) 

Another way to deform an elastomer network is to put it in contact with a solvent. In this case molecules of solvent are absorbed in the network, giving rise to a phenomenon known as swelling. Swelling of a network by the 

Figyre 3.16 Mooney-Rivlin plots [Eq. (3.38)] showing the effect of the temperature on stress-strain isotherms for model PDMS networks (15,16). The filled circles represent the reversibility of the elastic measurements, and the vertical lines locate the fracture points. (From Ref. 15.) 

In other words, swelling equilibrium is reached when the drop in the chemical potential of the solvent in contact with the polymer is compensated for the rise in chemical potential undergone by the solvent due to the elastic pressure of the network. 

The classical theory of swelling developed by Flory and Rehner in 1953 (1) provides the following expression for swelling equilibrium  

In a more recent theory of swelling equilibrium developed by Flory in 1979 (19), the extent to which the swelling equihbrium deformation is non-affine is taken into account. The nonaffine behavior depends on the looseness with which the cross-links are embedded in the network and consequently is related to the network structure and the degree of swelling at equilibrium. The final expression is 

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A further test of the validity of ideal network theory can be obtained through studies of the equilibrium swelling of polymer networks (Eq. 11-22). The maximum amount of information can be extracted by inducing changes in the equilibrium swelling preferably combined with unidirectional stress-strain data (Eqs. III-21, 26 and 27). 

Hermans, J. J. Deformation and swelling of polymer networks containing comparatively long chains. Trans. Faraday Soc. 43, 591 (1947). 

Mijnlibff, P. F., and W. J. M. Jaspers Thermodynamics of swelling of polymer-network gels. Analysis of excluded volume effects in polymer solutions and polymer networks. J. Polymer Sci. A-2 (in press). 

Using Flory-Huggins theory it is possible to account for the equilibrium thermodynamic properties of polymer solutions, particularly the fact that polymer solutions show major deviations from ideal solution behavior, as for example, the vapor pressure of solvent above a polymer solution invariably is very much lower than predicted from Raoult s law. The theory also accounts for the phase separation and fractionation behavior of polymer solutions, melting point depressions in crystalline polymers, and swelling of polymer networks. However, the theory is only able to predict general trends and fails to achieve precise agreement with experimental data. 

Griffiths PR, de Haseth JA (2007) Fourier Transform Infrared Spectroscopy. Wiley, Hoboken Gunzler H, Gremlich HU (2002) IR Spectroscopy. Wiley-VCH, Weinheim Guo Y, Peng Y, Wu P (2008) A two-dimensional correlation ATR-FTIR study of poly(vinyl methyl ether) water solution. J Molec Stmcture 87 486-492 Hermans JJ (1947) Deformation and swelling of polymer networks containing comparatively long chains. Trans Faraday Soc 43 591-600... 

Gradient IPN. In this case, the overall composition or crosslink density of the material varies from location to location on the macroscopic level. One way of preparing these materials involves partial swelling of polymer network I by the monomer II mix, followed by rapid polymerization before diffusional equilibrium takes place. Films can be made with polymer network I predominantly on one surface, and polymer network II predominantly on the other surface with a gradient composition existing throughout the interior. 

Swelling of Polymer Networks—The Frenkel-Flory-Rehner Hypothesis... 

Kramer, O. (ed.), Biological and Synthetic Polymer Networks , Elsevier, New York, 1988. Selected papers from the 8th Polymer Networks Group Meeting, Elsinore, Denmark, on the formation, characterization and properties of polymer networks, with special attention to biological networks and to the swelling of polymer networks. 

SD Sedimentation equihbritrm SW Swelling of polymer network VP Vapor pressirre incltrding manometry, isopiestic... 

The parameters of neutron scattering theory of polymer networks are A, the macroscopic stretching of the sample, or linear degree of swelling, f, the network functionality, K. which accounts for restricted junction fluctuations and a, a measure of the degree to which chain extension parallels the macroscopic sample deformation. The functionality is known from knowledge of the chemistry of network formation, and A is measured. Both K and a must be extracted from experiments. 

Note 2 For a polymer, the value of the solubility parameter is usually taken to be the value of the solubility parameter of the solvent producing the solution with maximum intrinsic viscosity or maximum swelling of a network of the polymer. 

Note 2 Semi-interpenetrating polymer networks may be further described by the process by which they are synthesized. When an SIPN is prepared by a process in which the second component polymer is formed or incorporated following the completion of formation of the first component polymer, the SIPN may be referred to as a sequential SIPN. When an SIPN is prepared by a process in which both component polymers are formed concurrently, the SIPN may be referred to as a simultaneous SIPN. (This note has been changed from that which appears in ref [4] to allow for the possibility that a linear or branched polymer may be incorporated into a network by means other than polymerization, e.g., by swelling of the network and subsequent diffusion of the linear or branched chain into the network.). 

The experiments were performed for the samples of neutral PAA gel, into which cations of poly(4-vinylpyridine) quaternized by ethyl bromide were incorporated. The degree of quaternization was 90%. The solvent was the mixture of water with ethanol. The degree of swelling of the networks was characterized by the parameter N/0 and <1>N is the volume fractions of the polymer in the gel at the preparation condition and after swelling in the mixture, respectively. 

Take a gel swollen in a solvent. We denote the number of chains contained in a gel network as Nc, the number of solvent molecules contained as Ns, and the concentration (volume fraction) of polymer as < >. To represent the degree of swelling of the network, it is convenient to introduce the linear swelling ratio a defined as... 

There is only an alternative way of determining crosslink density by a non empirical method, using the theory of equilibrium swelling of a network in a solvent (Flory and Rehner, 1943). This method, usual in the domain of rubbers, needs the knowledge of a polymer solvent interaction coefficient proportional to (8p — 8S)2, which is not very easy to determine accurately. Furthermore, damaging by swelling stresses and the need to work at elevated temperatures complicate the analysis seriously for the usual thermosets... 

Erman B, Flory PJ (1978) Theory of elasticity of polymer networks. II The effect of geometric constraints on junctions. J Chem Phys 68 5363—5369 Erukhimovich IYa, Irzhak VI, Rostiashvili VG (1976) On concentration dependence of swelling coefficient of weakly non-Gaussian macromolecules. Polym Sci USSR 18 1682-1689... 

The problem of determination of the partition function Z(k, N) for the iV-link chain having the fc-step primitive path was at first solved in Ref. [17] for the case a = c by application of rather complicated combinatorial methods. The generalization of the method proposed in Ref. [17] for the case c> a was performed in Refs. [19,23] by means of matrix methods which allow one to determine the value Z(k,N) numerically for the isotropic lattice of obstacles. The basic ideas of the paper [17] were used in Ref. [19] for investigation of the influence of topological effects in the problem of rubber elasticity of polymer networks. The dependence of the strain x on the relative deformation A for the uniaxial tension Ax = Xy = 1/Va, kz = A calculated in this paper is presented in Fig. 6 in Moon-ey-Rivlin coordinates (t/t0, A ), where r0 = vT/V0(k — 1/A2) represents the classical elasticity law [13]. (The direct Edwards approach to this problem was used in Ref. [26].) Within the framework of the theory proposed, the swelling properties of polymer networks were investigated in Refs. [19, 23] and the t(A)-dependence for the partially swollen gels was obtained [23]. In these papers, it was shown that the theory presented can be applied to a quantitative description of the experimental data. 

Important theoretical and experimental considerations of the use of macromolecular theories for the description of coal network structures have been recently analyzed (1). Relevant equations describing the equilibrium swelling behavior of networks using theories of modified Gaussian distribution of macromolecular chains have been developed by Kovac (2 ) and by Peppas and Lucht (3) and applied to various coal systems in an effort to model the relatively compact coal network structures (1 4). As reported before (1), Gaussian-chain macromolecular models usually employed in the description of polymer networks (such as the Flory... 

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