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Swelling of Random Copolymer Networks in Pure and Mixed Solvents: Multi-Component Flory-Rehner Theory Rutvik V Godbole, Fardin Khabaz, Rajesh Khare, and Ronald Carl Hedden J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b02194 • Publication Date (Web): 25 Jul 2017 Downloaded from http://pubs.acs.org on July 26, 2017

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The Journal of Physical Chemistry

Swelling of Random Copolymer Networks in Pure and Mixed Solvents: Multi-Component Flory-Rehner Theory Rutvik V. Godbole, Fardin Khabaz,1 Rajesh Khare, and Ronald C. Hedden* Department of Chemical Engineering, Texas Tech University, Lubbock, TX, 79409, USA

Abstract A generalized extension of Flory-Rehner (FR) theory is derived to describe equilibrium swelling of polymer networks, including copolymers with two or more chemically distinct repeat units, in either pure or mixed solvents. The model is derived by equating the chemical potential of each solvent in the liquid and gel phases at equilibrium, while assuming the deformation of the network chains is affine.

Simplifications of the model are derived for specific cases

involving homopolymer networks, copolymer networks, pure solvents, and binary solvent mixtures. With reasonable assumptions, the number of polymer-solvent interaction parameters

that must be determined by experiments can be reduced to two effective parameters ( and ), which describe the net interactions between water/copolymer ( ) and ethanol/copolymer ( ), respectively. Experimental measurements of the swelling of random copolymer networks of n-butyl acrylate and 2-hydroxyethylacrylate in water, ethanol, and a 100 g/L ethanol/water mixture are utilized to validate the model. For a random copolymer network, and can be

obtained by fitting the three-component FR model to equilibrium swelling data obtained in the pure solvents.

Predicted solvent volume fractions for swelling in water-ethanol mixtures

obtained by inserting fitted values of and into the four-component FR model are in reasonable agreement with experimental measurements.

1

Present Address: Department of Chemical Engineering, University of Texas at Austin, Austin, TX, USA

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I. INTRODUCTION Swelling of polymer networks in solvents plays an important role in numerous soft materials applications

including

chromatography,

membrane

separations,

polymer

thin

films,

superabsorbent hydrogels, and gel-based drug delivery systems.1 Swelling of homopolymer networks in pure solvents has long been modeled with the classical Flory-Rehner (FR) theory,2 a lattice-based model that considers the equilibrium swelling state to be characterized by a balance between osmotic swelling forces and the elastic retractive forces exerted by the deformed network chains.3 The Frenkel-Flory-Rehner hypothesis supposes that the elastic and mixing contributions to the free energy change of swelling can be treated independently,4-5 an idea which has been questioned,6-7 yet is supported by experimental evidence for some systems.8 For binary polymer-solvent systems, the FR model captures the material-specific polymer-solvent

interactions through a single polymer-solvent interaction parameter introduced in the

polymer solution theories of Huggins9 and Flory.10 The binary FR model has been successful in

modeling swelling of many homopolymer networks in pure solvents, and discussions of its applicability and limitations are found in the literature.8, 11-13 Rubbery copolymer networks are of interest in liquid-phase membrane separations, where inclusion of a co-monomer can advantageously affect transport properties and selectivity of the membrane for a desired permeant. Quantitative modeling of the swelling behavior of copolymeric membranes can provide an analytical framework for modeling the composition at the membrane-liquid interface, which is assumed to achieve a state of local thermodynamic equilibrium. Modeling the equilibrium swelling of copolymer networks when multiple species are present is a complex problem that requires a robust thermodynamic model. Calculating the free energy change of mixing of polymer segments and solvent(s) requires material-specific 2 ACS Paragon Plus Environment

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interaction parameters describing interactions between the species present. Such interaction

parameters should be readily accessible through experiments; the relative simplicity of the FR model is advantageous in this respect.

For a system having P components (solvents and chemically distinct polymeric repeat units

combined), the number of independent is equal to P(P-1)/2, meaning the number of

adjustable interaction parameters increases dramatically as the number of components increases.

The use of binary interaction parameters for polymer-solvent systems containing three or more components leads to multiple interaction parameters, increasing the chances that the model will fortuitously fit a small data set over a limited range of compositions. In part because of these challenges, multi-component equilibrium thermodynamic models for copolymer-solvent systems are still a subject of debate. Several previous studies of ternary systems (mainly consisting of a homopolymer network swollen in a mixture of two solvents) illustrate the complexities of applying the FR model when three or more components are present. Scott attempted to extend the classical binary FR theory to mixed-solvent systems for the first time in 1949.14 Since then, researchers have applied ternary modifications to estimate the equilibrium swelling of a polymer network in a mixture of solvents using various approaches.15-25 While ternary versions of FR theory provide satisfactory results for swelling of homopolymer networks in mixtures of non-polar solvents, the results have been found to be unsatisfactory for solvent mixtures of polar/polar or polar/nonpolar types. Bristow15 cited the semi-empirical nature of polymer-solvent interaction parameters as a likely reason for the observed discrepancies. Further developments in the multi-component FR theory have either added a ternary interaction parameter21,

26-29

as an empirical correction

term or utilized a concentration-dependent polymer-solvent interaction parameter, which can 3 ACS Paragon Plus Environment


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have a variety of functional forms.22-24, 30 However, both options limit the use of the theory to modeling particular homopolymer/solvent-1/solvent-2 systems. Comparatively little attention has been paid to modeling swelling of rubbery copolymer networks in a single solvent. In addition, polymer-solvent systems with four or more components have received little attention, despite the potential utility of copolymer networks in various separation processes where at least two solvents are present. Examples of such separation processes include the ethanol/water and acetone/n-butanol/ethanol/water mixtures obtained from aqueous fermentation processes, which usually contain additional components besides the solvents listed. To this end, we recently employed both experiments and molecular simulations to illustrate the potential value of random copolymer networks for pervaporation-based separation of dilute alcohol/water mixtures.31-33 These studies have shed light on the interrelationship between the solvent concentration, solvent structure in the network, polymer density, polymer glass transition temperature, polymer dynamics, and solvent dynamics in swollen copolymer networks. Combining these insights with a theoretical framework for modeling swelling in a multicomponent polymer-solvent system will facilitate modeling the separation of alcohol/water mixtures with copolymer membranes. This work begins with a derivation of a general multi-component extension of FR theory valid for any number of polymers and solvents, which proceeds with many of the same assumptions of the original FR theory. Furthermore, binary interaction parameters are assumed to be sufficient to model material-specific interactions between species when three or more components are present. We note that a similar approach of pairwise additivity of interactions is routinely followed in molecular thermodynamics theories, statistical mechanics and molecular simulations.34-36 The generalized model is then simplified to describe specific scenarios with 4 ACS Paragon Plus Environment

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three or four components. Emphasis is placed on describing swelling of random copolymer networks with two distinct types of repeat units in either pure solvents or binary solvent

mixtures. Although six are obtained for a quaternary system with two distinct polymeric

species and two solvents, it is shown that the number of polymer-solvent interaction parameters

that must be determined experimentally can be reduced to two ( and ), which describe the net interactions between copolymer/solvent-1 ( ) and copolymer/solvent-2 ( ), respectively.

A third input parameter describing solvent-solvent interactions ( ) is obtained independently from thermodynamic data for the excess Gibbs free energy of solvent mixtures (ethanol/water mixtures in this case).37 To examine the ability of the multi-component FR model to describe mixed solvent swelling

behavior of random copolymer networks, experimental data are obtained for a model system. An 11 x 11 matrix of polyacrylate homopolymer and random copolymer networks is studied, which has systematic variations in composition and crosslinker concentration. The test matrix consists of random copolymers of a hydrophobic monomer (BA ≡ n-butyl acrylate) and a hydrophilic monomer (HEA ≡ 2-hydroxyethyl acrylate). The large data set, which contains 121 networks of distinctly different compositions spanning a range of hydrophobic vs. hydrophilic character, permits evaluation of the ability of multi-component FR theory to model equilibrium swelling behavior without restricting the system to networks of a specific composition.

II. METHODS Experimental swelling data were used to obtain values of the interaction parameters. Additionally, molecular simulations were used to examine the validity of the volume additivity



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assumption used in development and application of the theory. A description of the methods used for the experimental and simulation parts of the work is provided in this section. Synthesis.

Random copolymers of HEA and BA were prepared by bulk free radical

polymerization with initiator azobisisobutyronitrile (AIBN) in the presence of the crosslinker pentaerythritol tetraacrylate (PETA). The monomers and crosslinker are miscible over the entire composition range, so addition of a mutual solvent was unnecessary. Details of the synthesis are described in a previous publication.31 In order to obtain networks having systematic variations in both hydrophilicity and crosslinker concentration, a combinatorial synthesis protocol was employed.

An 11 x 11 matrix of networks was prepared, into which orthogonal gradients in

composition (mole ratio of BA:HEA) and crosslinker concentration were programmed. Compositions of the 121 resulting network samples are summarized in Table S1. Sample names reflect the row and column of the sample; e.g., sample "BD" corresponds to row "B" and column "D" in the matrix. Rows contained a gradient in the mole ratio of BA:HEA, which controls hydrophobic vs. hydrophilic character, and columns contained a gradient in the mole fraction of PETA, which controls crosslinker concentration. For example, samples with names ending in "D", which appear in the same column, have the same mole ratio of BA:HEA and different amounts of crosslinker. Swelling Experiments.

All networks were swollen and extracted repeatedly in pure ethanol to

remove the soluble fraction, which was about 2 % by mass or less in all cases.31 After drying, equilibrium swelling measurements for the extracted networks in water, ethanol, and a single water-ethanol mixture (100 g/L ethanol) were obtained at 22 °C as described in Ref. [31]. For swelling in the ethanol/water mixture, the amounts of ethanol and water absorbed by each network were determined by a combination of gravimetry and HPLC analysis of the external 6 ACS Paragon Plus Environment

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liquid composition as described in Ref. [31]. Equilibrium swelling data for all networks are presented in Tables S2 and S3. Simulations. The multi-component FR model for copolymer networks developed in this work relies upon the assumption of additivity of the component volumes, much like the original binary FR model. Volume additivity was examined by performing molecular simulations. For this purpose, a random copolymer of 70 mol % BA and 30 mol % HEA was simulated both in bulk and in solutions with water and ethanol in order to examine the change in the partial molar volumes of polymer segments and solvents upon mixing. Simulations were performed for three different swelling scenarios, viz. swelling in pure water, swelling in pure ethanol, and swelling in a 100 g/L ethanol/water mixture. The general AMBER force field (GAFF)38-39 was used to quantify the interactions between the atoms. The binary interactions in the system were treated using the Lorentz-Berthelot combining rules, i.e. = + and = , where and are Lennard-Jones diameter and well

depth for the van der Waals interaction between atomic pair . These van der Waals interactions

between atoms were considered up to distance of 12 Å, and the tail correction approximation was used for calculating the interactions beyond this cutoff distance. The particle-particle particlemesh (PPPM) algorithm40 was applied to describe the long range Coulombic interactions. All simulations were carried out using a 1 fs time step at a temperature of 300 K and pressure of 1.0 atm. The temperature and pressure of the system were held constant by applying Nosé-Hover thermostat and barostat.41 Furthermore, the water molecules were represented by using the modified TIP3P model42 along with the SHAKE algorithm.



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The model structures were prepared using the simulated annealing polymerization technique that we have previously used for preparing structures of other amorphous polymeric systems including polyacrylates.32, 43-45 The required BA and HEA monomers, water molecules, and ethanol molecules were first simulated. The system was then “polymerized” in simulations by forming bonds between the spatially closest pairs of reacting atoms using the simulated annealing multivariable optimization technique.43-44,

46

The structure so prepared was relaxed

using a combination of constant NVT (constant number of molecules, constant volume, and constant temperature) and constant NPT (constant number of molecules, constant pressure, and constant temperature) MD simulations. The well-relaxed model structures so obtained were used for the subsequent production runs. All MD simulations were performed by using the LAMMPS47 package and the simulation results were averaged over five replicas. All of the model systems had at least 70,000 atoms. The GAFF force field and the simulation methodology used was validated in our previous work,32 where we showed that the thermal and volumetric properties (glass transition temperature and density) of linear homopolymers and random copolymers of BA and HEA as determined from simulations were in quantitative agreement with the experimental data.

III. THEORY Theory: Derivation of Generalized P-Component FR Model.

The classical binary FR

model3 computes the overall change in free energy (∆) of the system after swelling of a network to equilibrium by assuming that the free energy change of mixing (∆ ) and the elastic

free energy cost of deforming the network chains (∆ ) are additive (eq. 1).

∆ = ∆ + ∆ 1 8 ACS Paragon Plus Environment

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In the present derivation, the same idea is invoked to describe the equilibrium swelling in a

generalized -component system consisting of solvent species and − polymeric species.

The elastic chains are assumed to consist of one or more polymeric repeat units, and sequence distributions are assumed to be random (or nearly so) for copolymers. The network is assumed

to be free of glassy or crystalline domains and to swell uniformly. The free energy change of mixing (∆ ) upon swelling of a network is given by eq. 2. $

∆ = %

In eq. 2,

$'

$

ln # + # ( 2 % %&

is the number molecules of component 'i', # is its volume fraction, is the binary

interaction parameter between components 'i' and 'j', k is the Boltzmann constant, T is the absolute temperature, and the subscript '*' refers to the network (or membrane) phase. Like the original FR theory, eq. 2 assumes that molecules are placed randomly on a lattice, thus neglecting specific associations between molecules of different species. For instance, when multiple solvents are present, the model does not make provisions for selective solvation of the polymer chains by one of the solvents. According to Flory's modified affine model of swelling,48 the elastic free energy change (∆ ) upon swelling of the network is expressed by eq. 3. ∆ =

+, -. + .0 + .1 − 3 − ln./ .0 .1 3 3 2 /

In eq. 3, . is the principal extension ratio in the -direction relative to the reference state in

which chains assume random configurations corresponding to those of unperturbed free chains, and +, is the number of elastically effective network chains in the system. Alternative models for 9 ACS Paragon Plus Environment


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the elastic contribution to the free energy change have been proposed, which have been shown to offer improvements in some respects. These models include the phantom network model49 and the constrained junction model,48, 50-51 for instance. In this work, where experiments on weakly swelling networks are considered, we limit the derivations to the affine deformation model for

the sake of simplicity. Assuming isotropic swelling of the polymer network, i.e. ./ = .0 =

.1 = ., the principal extension ratio can be related to the polymer volume fraction in the swollen state according to eq. 4.

'

$

567899,: = = # ? ./ .0 .1 = . 4 = 5;&

4

By combining eqs. (1-4), the overall change in free energy (∆) for swelling of a network to equilibrium is expressed by eq. 5. ∆ = $

%

$'

$

ln # + # ( % %&

$ C +, + 3 = # ? 2B %>& A

'

4

F − 3 + ln = # ? 5 E %>& D $

In deriving eq. 5, the mixing terms were computed assuming the network has a truly random sequence distribution. For copolymers with blocky sequence distributions, the situation is more complicated due to the possibility that one block may not swell appreciably in the solvent, among other considerations. The case of block copolymer networks is therefore not considered here. The change in chemical potential for any solvent "k" in the network phase (with respect to

its chemical potential in the pure state, H I is represented by eq. 6. 10 ACS Paragon Plus Environment

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∆HJ K ∆ = L MN K J $,P,:

QRS

$

$'

J'

$

5J 5J 5J = 1 + ln #J − T # U − T # # U + T # J U 5 5 5 +

$

%J&;JW$

%

$'

$

% %&

# J + T # % %&

⁄4

$

+ X, 5J Y= # ? %>&

$

K U K J

%;JW

1 − = # ?[ 6 2 %>&

In eq. 6, 5 is the molar volume of component 'i' and X, is the molar concentration of elastically effective chains, which is equal to +, divided by the volume of the dry network. A similar

equation can be written for the change in chemical potential for the same solvent "k" in the external solution (eq. 7). ∆HJ 6 K ∆6 = L MN K J $,P,:

QRS

>

>'

>

J'

5J 5J 5J = 1 + ln #J6 − T #6 U − T #6 # 6 U + T #6 J U 5 5 5 +

>

%J&;JW>

%

>'

>

% %&

# 6 J + T # 6 % %&

%;JW

K U 7 K J

In eq. 7, subscript "s" refers to the external solution phase, which is assumed not to contain any polymeric species, and (∆G) s is the change in Gibbs free energy in the external solution phase upon swelling of the network. At equilibrium, the chemical potential of any solvent "k" in the

network with respect to its chemical potential in the reference state, HJ − HJ I , is equal to the chemical potential of the solvent "k" in the external solution with respect to its chemical potential 11 ACS Paragon Plus Environment


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in the same reference state, HJ 6 − HJ I . Therefore, the generalized form of the -component FR

model for any solvent "k" is obtained by equating these chemical potentials, as in eq. 8. $

$'

J'

$

5J 5J 5J 1 + ln #J − T # U − T # # U + T # J U 5 5 5 %

+

$

%J&;JW$

% %&

$'

$

# J + T # % %&

⁄4

$

+ X, 5J Y= # ? %>&

$

%;JW

K U K J

1 − = # ?[ 2 %>&

>

>'

>

J'

5J 5J 5J = 1 + ln #J6 − T #6 U − T #6 # 6 U + T #6 J U 5 5 5 +

>

%J&;JW>

%

>'

>

% %&

# 6 J + T # 6 % %&

%;JW

K U 8 K J

The equilibrium conditions are met when eq. 8 is satisfied for each of the k solvent species present; thus, there are k equilibrium conditions in the form of eq. 8. The derivation for the change in chemical potential makes no assumptions except isotropic swelling of the rubbery polymer network and random sequence distributions in the case of a copolymer. Eq. 8 can be truncated for a system consisting of a specific number of polymer and solvent species by retaining the relevant terms. This truncated form of eq. 8 can be applied to experimental data either to extract the polymer-solvent thermodynamic interaction parameters ( ) or to predict the

equilibrium volume fractions of the components from assumed values of the . These aspects

are further discussed in the ensuing sections. In particular, we begin with the general,


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multicomponent FR model developed above and consider its application to specific cases involving homopolymers, random copolymers, pure solvents, and binary solvent mixtures. Case 1: Swelling of a Homopolymer Network in a Mixture of Two Solvents. The threecomponent extension of the FR model has previously been applied with various modifications to describe swelling of homopolymer networks in a mixture of two solvents.15-25 Derivation of the three-component extension of the generalized model for homopolymer/solvent-1/solvent-2 systems is provided in the Supporting Information (eqs. S1 and S2). Thermodynamic interaction

parameters between the two solvents ( ) and the solvent-homopolymer pairs (4 and 4 )

describe the material-specific interactions. In order to improve the agreement between theory and experiment, some researchers have either introduced a ternary interaction parameter (P ) as

an empirical correction21 or assumed concentration dependence of the according to one of

several functional forms.18,

24, 52

For the model developed in this work, no such empirical

modifications are invoked. Case 2: Swelling of a Random Copolymer Network with Two Distinct Repeat Units in a Pure Solvent. Applying the generalized FR model to a random copolymer network with two distinct repeat units (3 & 4) swollen in either pure solvent (1) or pure solvent (2) leads to eq. 9 or eq. 10, respectively. ′ ′ ln # + 1 − #

−

+T

5 ′ 5 ′ 5 ′ ′ ′ ′ ′ #4 − #_ + #4 4 + #_ _ 1 − #

− T #4 #_ 4_ U 54 5_ 54

′ #4

K4 K_ ′ U + T #_ U+T K K

′ ′ + X, ∙ 5 a#4 + #_

⁄4

−

′ 4 #_

K4_ U K

′ ′ #4 + #_

b = 0 9 2



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" " ln # + 1 − #

−

5 " 5 " " " " #4 − #_ + #4 4 + #_ _ 1 − #

54 5_

5 " " − T #4 #_ 4_ U + T 54 +

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" X, ⋅5 a#4

+

" #4

⁄4 " #_

K4 U+T K

" #_

K_ U+T K

" 4 #_

K4_ U K

" " + #_

#4 − b = 0 10 2

Although there are only three components present, the original indices (1 to 4) are retained here to avoid a change in nomenclature when a comparison between the ternary system parameters and the quaternary system (random copolymer in a mixed solvent) parameters is ′ made. In eq. 9, the # are the volume fractions of each component in the gel phase when

" swollen in pure solvent 1. Similarly, in eq. 10, the # are the volume fractions of each

component in the gel phase when swollen in pure solvent 2. The right hand sides of eq. 9 and 10 are equal to ∆HJ ⁄ , which is set to zero because the chemical potential for a pure solvent

outside the swollen network is the same as its chemical potential in the reference state. Eq. 9 can further be reduced to eq. 11 by applying two assumptions. In accordance with the original FR

theory, the effective molar volumes of polymers (54 and 5_) are assumed to be overwhelmingly large compared to the molar volume of pure solvent (5), so that both 5 ⁄54 and 5 ⁄5_ terms

can be neglected. We note that although random copolymers are broken up into short runs of similar repeat units, the parameters V3 and V4 are treated as large compared to the molar volume of a solvent because all of the polymer repeat units are attached to the macromolecular network

structure. Additionally, polymer-solvent interaction parameters (4 and _ ) and the polymer-

polymer interaction parameter (4_ ) are assumed to be independent of concentration so that the fghi f:h

terms can be neglected.


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′ ′ ′ ′ ′ ln # + 1 − #

+ #4 4 + #_ _ 1 − #

′ ′ + X, 5 a#4 + #_

⁄4

−

′ ′ + #_

#4 b = 0 11 2

In eq. 11, 5, the molar volume of solvent (1), is obtained from measurements or from

′ + simulations. Further mathematical modifications of eq. 11 are obtained by substituting #4 ′ ′ #_ = 1 − # and rearranging terms, yielding the remaining unknown parameters, as

described by eq. 12.

j j j j ' #4 4 + #_ _ #4 + #_

j 1 − # j ' j ⁄4 j j n 12 klX, 5 a − 1 − # bm − ln # − 1 − # = 1 − # 2

A similar equation is written for equilibrium swelling condition in pure solvent (2) as in eq. 13. " " " " #4 4 + #_ _ #4 + #_

'

= 1 −

' " #

=X, 5 l

" 1 − #

" − 1 − #

2

⁄4

" " m? − ln # − 1 − #

( 13

The left hand side of eq. 12 can be taken as volume fraction-weighted interaction parameter between solvent (1) and the copolymer (composed of monomers 3 and 4) as given by eq. 14.

≡ p#′3* + #′4* 14 r #′3* + #′4*

−1

13

(14)

With this definition, eq. 12 is now similar in form to eq. S3 with θ1 in place of χ1p. Similarly, a volume fraction-weighted interaction parameter between solvent (2) and the copolymer is defined by eq. 15.

≡ p#"3* + #"4* 24 r #"3* + #"4* −1

23

With this definition, eq. 13 is similar in form to eq. S4 with θ2 in place of χ2p.


(15)


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If θ1 and θ2 can be considered as constants (i.e., they are concentration-independent), then the ternary FR model for a random copolymer network swollen in a single pure solvent (Case 2, eqs. 12 and 13) has the same form as the ternary FR model for a homopolymer network swollen in a mixture of two solvents (Case 1, eqs. S3 and S4). Otherwise, eqs. 12 and 13 differ from eqs. S3 and S4 in the form of the left hand side terms, so that Case 1 and Case 2 are not equivalent. Case 3: Swelling of a Copolymer with Two Distinct Repeat Units in a Mixture of Two Solvents. Applying the generalized FR model to a quaternary system consisting of a random copolymer network with two distinct repeat units (3 & 4) swollen in a mixture of two solvents (1 & 2) leads to two equilibrium conditions, one for each solvent. At equilibrium, the chemical potential of each solvent molecule in the gel phase is equated with the chemical potential of the same solvent in the external solution. Mathematical representations of the two equilibrium conditions are provided in the Supporting Information (eqs. S5 and S6). Although five

appear in eqs. S5 and S6, χ4 and χ_ appear together in only one term. Thus, the definition of

θ1 can be inserted into eq. S5, yielding eq. 16. ln # + 1 − # −

5 # + # 1 − # + 1 − # #4 + #_ 5

5 − T # U #4 + #_ + l 5 + X, 5 a#4 + #_ ⁄4 −

= ln #6 + #6 −

#

K m K

#4 + #_ b 2

5 # + #6 + l 5 6

#6


K m 16 K

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Similarly, χ4 and χ_ in eq. S6 are present in only one term. Following similar logic, we insert the definition of θ2 into eq. S6 to obtain eq. 17. ln # + 1 − # − −T

5 5 # + T # U 1 − # + 1 − # #4 + #_ 5 5

5 # U #4 + #_ + l 5

+ X, 5 a#4 + #_ ⁄4 − = ln #6 + #6 −

#

K m K

#4 + #_ b 2

5 5 #6 + T #6 U + l 5 5

#6

K m 17 K

Simplifying eqs. S5 and S6 to yield eqs. 16 and 17 reduces the number of adjustable fit

parameters to three: , , and . In eqs. 16 and 17, the volume fractions of solvent in the external solution (#6 and #6 , the molar volumes of solvents (5 and 5), and the concentration

of elastically effective chains (X, ) are known or calculated parameters. On the other hand, volume fractions at equilibrium swelling (# ), effective interaction parameters ( and ), and

the solvent-solvent interaction parameter ( ) can be determined from equilibrium swelling experiments or from computations. Estimation of the effective interaction parameters (vw and vx ).

Two approaches were considered for evaluation of the interaction parameters in order to

predict swelling behavior for the random copolymer networks in mixed solvents. First, the

determined by swelling of homopolymer networks in pure solvents (as in Fig. 2) could be inserted directly into eqs. S5 and S6 to predict the swelling of random copolymer networks in 17 ACS Paragon Plus Environment


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solvent mixtures. However, this approach immediately presents difficulties, as the in Fig. 2

clearly vary with solvent concentration. As the nature of the concentration dependence is unknown in mixed-solvent conditions, this approach is difficult to implement for the copolymer networks. Alternatively, the random copolymer networks could be swollen in pure solvents, and

and could be determined by applying eqs. 12 and 13. The values of and so determined could then be inserted into eqs. 16 and 17 to predict mixed solvent swelling behavior

for the same copolymer networks. Here, we pursue the latter approach in order to avoid making assumptions about the concentration dependence of the .

In order to employ eqs. 16 and 17, the ratio of volume fractions of polymer species

(#_ ⁄#4) is assumed to remain the same in all swelling scenarios, as in eq. 18.

j " #_ #_ #_ #_; = j = " = 18 #4 #4 #4 #4;

j j and #4 are the volume fractions of polymer species in solvent (1), In eq. 18, #_

" " #_ and #4 are the volume fractions of the polymeric species in solvent (2), and #_; and #4;

are the volume fractions of the polymeric species in the dry state. We note that if volumes can

reasonably assumed to be additive, #_; ⁄#4; in eq. 18 can be calculated from the composition of the dry polymer network. This assumption of additivity of volumes can be examined by molecular simulations, as is done in this work. Estimation of the Interaction Parameter Between Two Solvents (ywx ). The solvent (1)solvent (2) interaction parameter is a model input parameter that can be determined

independently of any experiments involving polymer networks. Bristow15 calculated for

mixtures of non-polar solvents by first measuring partial pressures of the two components in


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equilibrium with their liquid mixtures and then applying classical Flory-Huggins theory3, 53 to the data according to eq. 19.

∆H 6 { 5 = ln T I U = ln1 − #6 + #6 ∙ T1 − U + #6 ∙ 19 z { 5

In eq. 19, { is the partial pressure of solvent (1) in the gaseous mixture and { I is its vapor

pressure in the pure state. However, eq. 19 is not appropriate for liquid solutions involving strong hydrogen bonding.52, 54 For such systems, alternative methods based on excess Gibbs free energy (∆ ) data have been proposed55-57 as described by eqs. 20 and 21.

1 | | } = ∙ a| ∙ ln T U + | ∙ ln T U + b 20 # # z | # } = 21 z| |

In eqs. 20 and 21, | and | are mole fractions of solvents (1) and (2), respectively, in the liquid

phase. If the molar volumes of the two components in the mixture are the same, then eqs. 20 and 21 are equivalent. However, when the molar volumes of the components differ significantly, it is

more appropriate to use eq. 20. ∆ data are generally obtained from vapor-liquid equilibrium experiments55 that are preferably carried out over the entire composition range.

Estimation of the Concentration of Elastically Effective Chains (~ ). The concentration of elastically effective network chains (X, ) is an additional model parameter that describes the

elastic component of the free energy change of mixing. For highly swelling networks, care must

be taken in evaluating X, in order to employ the FR model or any of its multi-component forms.

Interaction parameters extracted from swelling data via the FR model become increasingly sensitive to the assumed value of X, as the equilibrium swelling ratio increases. The contribution 19

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Page 20 of 44

of trapped entanglements to X, therefore should not be ignored in strongly swelling network-

solvent systems. One experimental approach to estimation of X, involves measurement of the

equilibrium shear modulus in the rubbery state. For model PDMS networks swollen in good

solvents (toluene and benzene), Patel et al. showed that X, could reasonably be approximated by , ⁄z, where , is the equilibrium rubbery shear modulus of the polymer network.58 The value

of X, estimated in this fashion includes the effects of trapped entanglements. The binary FR

model with the phantom network model of elasticity49 provided good agreement with experimental swelling measurements when X, was estimated from shear modulus measurements.

In the present study, the copolymer networks swell weakly in both water and ethanol, and

the equilibrium volume swelling ratio never exceeds 5. Because a large data set consisting of 121 networks was examined, measurement of the equilibrium shear moduli was considered impractical.

For the sake of simplicity, X, was approximated to the first order as the

concentration of elastic chains based upon idealized covalent bonding arrangements in the networks. Trapped entanglements and defects such as loops and dangling chains were neglected.

The approximate values of X, so obtained are listed in Table S1 for all networks. Figure S1 illustrates that uncertainty in X, has no significant effect on the calculated polymer-solvent

interaction parameters when the equilibrium swelling ratio is less than about 5.

Had the

networks swelled to a greater degree, it would have been necessary to obtain accurate measurements of X, and also to critically examine the assumption of affine deformation, which

is known to have shortcomings.59 Thus, the choice of a weakly swelling network system here

permits examination of the multi-component FR theory without introducing complications that become relevant in highly swelling systems.


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IV. RESULTS AND DISCUSSION The multi-component FR model was applied to pure solvent swelling data to extract values

of the effective interaction parameters and for each copolymer network in pure water and

pure ethanol, repectively. In the ensuing discussion, we demonstrate that the swelling of the copolymer networks in mixtures of water and ethanol can be predicted with nearly quantitative accuracy from experimentally determined values of and .

Evaluation of polymer-solvent interaction parameters ( χip ). The random copolymer networks of HEA and BA studied are described in Table S1. A large number of network compositions was studied such that the applicability of the model could be tested without restricting the study to either hydrophobic or hydrophilic networks.

Copolymers of BA and

HEA were chosen for evaluating the multi-component FR model for the following reasons. The monomers are miscible over the entire composition range, so it was straightforward to prepare networks without crosslinking in solution. All copolymer networks produced were rubbery at room temperature and did not crystallize, simplifying the experiments. Finally, the reactivity ratios of the monomers are nearly equal and are close to 1.0, suggesting that nearly random copolymers are formed. Mun et al.60 reported the following reactivity ratios for the monomers:

(HEA) = 0.95 and (BA) = 0.91. Water and ethanol are hereafter labeled as components (1) and (2), respectively, while BA and HEA repeat units in the copolymer network are labeled as components (3) and (4), respectively. The mass equilibrium swelling ratios of all the networks in pure ethanol and pure water

, and 7 , defined as [swollen mass]/[dry extracted mass]), were reported in our previous (

work.31 Volumetric swelling ratios of each network in pure ethanol and pure water (, and 7 ,

defined as [swollen volume]/[dry extracted volume]) were calculated from measured mass 21 ACS Paragon Plus Environment


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Page 22 of 44

swelling ratios by assuming additivity of polymer and solvent volumes. For pure solvent swelling, the volume fractions of solvent at equilibrium are related to the mass and volume swelling ratios by eqs. 23 and 24. 1

′ 1 − #

1

"

1 − #

= 7 = = , =

− 1 X890,< + X7,< 23 X7,

Swelling of Random Copolymer Networks in Pure ... - ACS Publications

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