Modeling Swelling Behavior of Cellulose Ether Hydrogels - ACS

Chapter 7

Downloaded by UNIV OF CALIFORNIA SAN DIEGO on December 16, 2015 | http://pubs.acs.org Publication Date: March 5, 1993 | doi: 10.1021/bk-1993-0520.ch007

Modeling Swelling Behavior of Cellulose Ether Hydrogels D. C. Harsh and S. H . Gehrke Department of Chemical Engineering, University of Cincinnati, Cincinnati, O H 45221

Successful application of responsive gels will require knowledge of factors that control the swelling degree, transition temperature, and sharpness of the volume transition. As a result of the failure of the Flory network theory in predicting lower critical solution temperature (LCST) phenomena, more sophisticated theories accounting for the system free volume and specific interactions are required. In this work, the compressible lattice theory of Marchetti et al. is modified to account for ionic content and the elastic behavior of highly swollen networks. The resulting model is used to describe the observed swelling behavior of cellulose ether gels. The predicted and observed effects of controllable synthesis parameters are compared. The compressible lattice theory successfully describes LCST behavior, but is extremely sensitive to the parameter values, limiting its potential as a predictive tool. Accounting for non-Gaussian chain statistics results in the introduction of an additional parameter which does not greatly affect predicted swelling behavior. Environmentally responsive hydrogels have been proposed for a variety of applications including artificial muscles and drug delivery devices (1,2). Successful applications will require both the magnitude and location ofthevolume change be well characterized and controllable. There has been much interest in relating the physical properties of the linear polymers to the swelling behavior of the crosslinked networks (3,4), but the predictive ability of such observations is limited beyond qualitative guidelines. Hence, it is desirable to make use of thermodynamic theories for gel swelling to model the swelling behavior. A successful theory would be able to predict gel swelling behavior based on the properties of the pure components (solvent and polymer). Development of thermodynamic theories is the subject of much current work (5-14), with particular interest in describing temperature-sensitive swelling behavior. In this chapter, the predictions of the classical Flory network theory will be presented, with discussion of how the controllable synthesis parameters affect these predictions. Then, the compressible lattice theory of Marchetti and Cussler (5-7) will be presented with modifications to account for ionic content and non-Gaussian network elasticity. 0097-6156/93/0520-0105$08.50/0 © 1993 American Chemical Society

In Polymeric Delivery Systems; El-Nokaly, M., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1993.

106

POLYMERIC DELIVERY SYSTEMS

Theoretical predictions of gel swelling with the modified Marchetti theory w i l l then be presented i n order to compare predictions of the two theories. Finally, the swelling behavior o f cellulose ether gels w i l l be modeled using the modified theory and the predictive utility of the modified theory w i l l be discussed.


Equilibrium

Swelling Thermodynamics

A gel w i l l swell until the solvent chemical potential is equal i n the gel and the free solution. For conceptual convenience, it is convenient to convert chemical potential to osmotic swelling pressure, which is zero at equilibrium. A c c o r d i n g to F l o r y , total osmotic swelling pressure is represented as the sum of the individual contributions of polymer-solvent mixing, network elasticity, ionic osmotic pressure, and electrostatic effects (75): n ix+nelas+n +nelec = 0 m

(1)

ion

Tlmix represents the contribution from polymer-solvent interaction, which, i n a good solvent, tends to increase gel swelling. I I represents the contribution due to the network elasticity w h i c h arises from the restraints on swelling imposed by the crosslinks. In gel that contains charged or ionizable groups, it is necessary to include e l a s

n

i o n

, which represents the osmotic pressure of the counter-ions. In addition, it may be

necessary to account for electrostatic effects, represented by I I

e l e c

. The theories used

to describe these contributions w i l l now be reviewed. Polymer Solution Theory. The Flory-Huggins theory uses a mean-field approach and introduces the interaction parameter, χ , to describe the solvent quality; χ increases w i t h decreasing solventy quality. However, the concentration and χ

λ

temperature dependence of % is highly non-linear so direct measurement is required. x

In addition, as originally defined, χ decreases as temperature increases, failing to predict the inverse solubility behavior of temperature-sensitive gels. Current theoretical work does not use classical polymer-solvent interaction parameters. Instead, by a corresponding states approach, equations of state have been developed based on pure component parameters w h i c h may be obtained from experimental data. These models attempt to account for the features that have been neglected i n the Flory-Huggins theory, allowing description o f volume changes o f m i x i n g and non-random m i x i n g processes. Several different approaches have been taken, which w i l l be now be described. Flory et al., in an early attempt to correct the deficiencies of the Flory-Huggins theory, used a corresponding states approach to develop an equation o f state accounting for non-zero volumes of mixing (76,77). The resulting theory has been useful i n modelling of hydrocarbon solutions, although there has been limited success in application to aqueous systems due to strong orientation-dependent interactions such as hydrogen bonds. Sanchez and Lacombe introduced the concept of a compressible lattice. The ternary version of the Flory-Huggins theory is adapted, adding holes to the lattice as a third component (18-21). The interaction parameter is defined as a function of the solvent and polymer cohesive energy densities. One adjustable parameter is added which characterizes the mixture: the deviation from the geometric mean mixing rule. The holes are treated as non-solvents for the polymer, with zero cohesive energy density. Solvent parameters are determined from P - V - T data; values for the polymer may be obtained by D S C measurements. This model w i l l satisfactorily predict the χ



7. HARSH AND G E H R K E

Swelling Behavior of Cellulose Ether Hydrogels

107

phase behavior of some linear polymer solutions. Application to crosslinked systems has been accomplished theoretically (5); modelling experimental data has been accomplished by fitting the cohesive energy densities to observed phase behavior (6). In addition, this model predicts volume transitions in response to pressure changes, which has been experimentally confirmed (7). Prausnitz and co-workers have taken a significantly different approach (8-12). Arguing that L C S T behavior in aqueous systems is due to order-disorder transitions involving orientation dependent interactions such as hydrogen bonds, free-volume effects are neglected since the systems studied are at temperatures far from the solvent critical point. A partition function is derived based on a lattice model with three categories of interaction sites: hydrogen-bond donors, hydrogen-bond acceptors, and dispersion force contact sites. The resulting model qualitatively predicts L C S T without the use of temperature dependent parameters in linear polymer solutions, although the parameter values must be fit to experimental data for accurate quantitative predictions. Although appealing from a conceptual standpoint, this theory has two major problems. First, free volume effects are totally neglected; volume changes in mixing and pressure induced volume changes are not described. Secondly, the model uses three "interaction energies" which are not easily accessible experimentally: the hydrogen bond strengths between like molecules, hydrogen bond strengths between unlike molecules, and the dispersion contacts. In addition, determining the solution o f the equilibrium swelling equations is quite complicated numerically. Although the theories discussed in die previous pages are the most commonly used, others have also been proposed. Otake et al. have introduced a separate contribution from hydrophobic interactions, which they use with a conventional virial expression to describe free energy o f m i x i n g (13). This approach, however, introduces three adjustable parameters to characterize the variation of hydrophobic interactions with temperature. A theory proposed by Painter et al. (14) accounts for the free energy change due to hydrogen bond formation by addition o f a separate term based on self-association models used i n alcohol-hydrocarbon systems. The extent of hydrogen bonding is quantified by an equilibrium constant, which must be determined experimentally. Network Elasticity. Elastic contributions arise due to the configurational (entropie) changes in the swollen gels. The network entropy is decreased, since there are fewer possible configurations for the swollen network. A t moderate deformations, the elasticity may be represented by an ideal network model. The ideal network assumes Hookean elasticity (retractive force proportional to deformation), with freely jointed polymer chains, tetrafunctional crosslinks of zero volume, crosslinked i n the bulk state, and a Gaussian distribution of chain end-to-end distances. The elasticity may be expressed as follows for a network that swells isotropically (75): Π ^ = -Ρχ*Τ[φ ^-φ /2] 2

(2)

2

p = Number o f elastically effective network chains per unit volume dry polymer. Equation 2 neglects non-Gaussian contributions which arise due to physical restraints imposed by the crosslinks as swelling increases. Equation 2 may be modified to account for crosslinking i n solution; elastic force is relative to the unstrained state of die network. In this case, the "relaxed" state x

of the network is i n solution, hence the introduction o f the factor f, the polymer volume fraction at network formation (22): 2

n ias = -Ρχ^φ [(φ /φ )ΐ/3-(φ /φ^)/2] e

2ί

2

2£

2


(3)

POLYMERIC DELIVERY SYSTEMS

108

Note that equation 3 reduces to equation 2 i n the case o f network formation i n the absence o f solvent, i.e. φ ί=1. A s the degree of swelling increases, the distribution o f end-to-end distances o f the network chains begins to deviate from a Gaussian distribution as the polymer chains begin to reach their maximum extensions. Hence, the validity o f the Gaussian distribution becomes i n v a l i d , requiring use o f non-Gaussian chain statistics. Alternatively, i f the polymer chain is not freely jointed, then it may be necessary to account for the flexibility of the polymer backbone. Several models for non-Gaussian networks have been proposed to approximate the exact distribution o f ideal chain 2

extensions (25-25). These models use power series i n φ and require knowledge o f the number o f effective links per network chain, or equivalently, the number o f structural units in an effective chain segment (N). G a l l i and Brumage have developed a function which closely matches the exact distribution o f freely jointed random chains (25) :


2

n ias = " R T p B e

(4)

x

where Β = φ /2 - φ ^ Ι - Ι / Ν + (2/5)/N +(8/25)/N ] 2

2

3

+φ -ΐ/3[-1/Ν +(13/5)/N - (43/25)/N ] 2

2

3

4ψ2-ΐ[-(11/5)/Ν + (221/25)/N ] + φ - /3[-(171/25)/Ν ]. 2

3

5

2

3

Use of non-Gaussian chain statistics results in a "levelling o f f o f predicted swelling at high swelling degrees (low φ) as opposed to the continued increase i n swelling as solvent quality increases. This is due to stress increasing faster than strain at high swelling degrees. The non-Gaussian expression presented i n equation 4 is written for networks prepared i n the bulk state. Thus, the expression must be modified to account for the dilution at the time of crosslinking. Although there are other non-Gaussian models for solution crosslinked networks (23,24), the G a l l i and Brumage model (25) comes closest to describing the exact distribution of chain extensions o f the models that have been proposed. A l l non-Gaussian models introduce an effective chain link, or an equivalent expression for the number o f links per network chain. The expression developed by G a l l i and Brumage is a more complicated mathematical expression, but uses no more parameters than the other non-Gaussian models. The added complexity is not a concern for numerical calculations. The modification, as developed by Harsh (26) , introduces the factor φ in a manner analogous to equations 2 and 3 to account for die relaxed network state in equation 5. 2 ί

Ilelas = ^ ρ φ Η 0 . 5 ( ^ ) - ( ^ ) ( ΐ 4 χ

3

2

4fef

+

Modeling Swelling Behavior of Cellulose Ether Hydrogels - ACS

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