Constrained Swelling of Polymer Networks: Characterization of Vapor

Article pubs.acs.org/Macromolecules

Constrained Swelling of Polymer Networks: Characterization of Vapor-Deposited Cross-Linked Polymer Thin Films Karel Dušek,†,* Andrei Choukourov,‡ Miroslava Dušková-Smrčková,†,‡ and Hynek Biederman‡ †

Institute of Macromolecular Chemistry, Academy of Sciences of the Czech Republic, Heyrovského nám. 2, Prague 6, 162 06, Czech Republic ‡ Charles University in Prague, Faculty of Mathematics and Physics, Department of Macromolecular Physics, V Holešovičkách 2, 180 00 Prague, Czech Republic ABSTRACT: In certain real situations, polymer networks cannot swell freely but are subjected to constraints by action of external forces or geometric restrictions. In this contribution, the restriction to swelling by adhesion of a cross-linked polymer film deposited on a rigid substrate is considered. Such constraints prevent swelling in plane, and the osmotic force concentrates on extension of network chains in the direction normal to the surface. We analyze existing rubber elasticity models taking into account finite extensibility of network chains and derive the elastic contribution to the chemical potential of the solvent in swollen networks. For the mixing contribution to the change of the Gibbs energy, it is more appropriate to consider the network containing cross-links, but with undeformed network chains than an un-cross-linked polymer of infinite degree of polymerization, because continuing formation of bonds within an infinite molecule further decreases the entropy. The additional term is proportional to the cycle rank of the network. The reformulation of the mixing contribution is important for consideration of cross-linking induced phase separation during network formation. The phase separation limit is due to the crosslink contribution to the mixing part of the Gibbs energy and has nothing to do with chain stretching. The swelling theories respecting finite extensibility of network chains are then applied to cross-linked poly(ethylene oxide) films obtained by plasmaassisted vapor deposition with the aim to determine their effective cross-link density.

1. INTRODUCTION Equilibrium swelling is one of the most characteristic properties of polymer gels and networks. Theories have been developed linking the equilibrium content of liquids in swollen networks to the concentration of elastically active network chains (EANC) and other network parameters and to thermodynamic interactions of the solvent molecules with segments of network chains (cf., e.g., refs 1−6). An overwhelming number of applications of this method concerns free swelling. However, in certain important applications, the gels are not allowed to swell freely and their swelling is constrained by geometric limitations or external forces. Swelling of cross-linked polymer films adhering to a substrate is one of such cases. Cross-linked polymer films exist and are prepared in various ways: such as protective organic coatings cross-linked in the course of film formation, stimuli-responsive materials for sensors and other devices, and electroactive layers.7−17 Such films are excellent candidates for smart surfaces with sensing and actuating characteristics. Plasma-assisted chemical vapor deposition is an easy way of making such films (cf., e.g., refs 17−26). The film structure and properties determine their application potential. To reveal the structural features of the network such as the concentration of EANC and other network characteristics, the film polymer must be in the rubbery state. The film © 2014 American Chemical Society

material is often in glassy state under normal conditions, but its Tg can be substantially lowered by absorption of a lowmolecular-weight liquid. The swollen state is often the working state of such adhering films and swelling and swelling transitions are used for characterization of the cross-linked film structure.11−24 Although the connection between free swelling of a polymer network and swelling of an adhering film has been recognized, much less attention was paid to the confinement preventing the adhering film from expansion in the lateral dimensions leaving the osmotic forces to increase the film thickness. On the contrary, when the film is prepared in the presence of a diluent and the diluent is evaporated, the adhesion prevents the network from shrinking which affects the vapor pressure over the film.27 Toomey et al.28−30 obtained a direct proof that a cross-linked film chemically attached to a rigid substrate swells less than a free film. They also confirmed that the scaling between the degree of swelling and cross-link density, predicted by adapting the Flory−Rehner theory, for free and confined swelling were different. Received: March 26, 2014 Revised: May 21, 2014 Published: June 19, 2014 4417

dx.doi.org/10.1021/ma5006217 | Macromolecules 2014, 47, 4417−4427

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When a cross-linked film deposited on a stiff substrate swells, (a) the film delaminates from the substrate when the swelling stresses exceed the adhesion strength, (b) only the film thickness increases because the film is prohibited from dilation in plane of the substrate, or (c) the limit of instability is exceeded and surface pattern formation like buckling or wrinkling is observed. The development of instabilities is observed for highly swollen and thicker films and the observed pattern can be used for characterization of elastic properties of the material.31−37 Here, we will be concerned with case b: swelling of a thin film (on the order of 101−102 nm) adhering to the substrate in a stable regime, when by absorption of liquid only its thickness increases. In this contribution, we are concerned with the equilibrium swelling of an adhering film and effect of adhesion on determination of cross-link density by swelling. The classical mean-field theory is used which is based on the concept of additivity of elastic and mixing Gibbs energies. Since in unidirectional swelling of the film some of the chains can be rather highly extended (much more than in free swelling) while the other are not, the finite extensibility of network chains plays a more important role than in the isotropic free swelling. Several current finite chain extensibility models are considered, and explicit relations for swelling equilibrium are derived and compared. It is shown that respecting finite extensibility of network chains is necessary for higher cross-link densities or/and high degrees of swelling. Swelling data for poly(ethylene oxide) plasma polymer films described in ref 21 are analyzed in terms of these two effects.

are then used for determination of swelling equilibrium under the conditions of free swelling and swelling of an adhering layer. Of the existing models, those have been selected that are based on random flight generation of end-to-end distances of chains composed of a finite number of segments, and for which the end-to-end distance cannot exceed the contour length of the chain nl (n is number of segments and l is the segment length). The exact Rayleigh distribution of population of endto-end distances of a chain composed of n statistical segments was transformed by Kuhn and Grün38 into a functional form containing the inverse Langevin function of the deformation ratio taken relative to the maximum extension nl, λrel. Usually the chain deformation λ is taken relative to the unperturbed state (r/n1/2l), so that λrel = λn−1/2. The Kuhn−Grün function has been a basis of various models such as the three-chain model, eight-chain model, and various series expansions (cf., e.g., refs 39−46). The semiempirical function developed by Gent et al.47−49 is based on the logarithm of the first invariant of the Cauchy−Green strain vector, J1 = λx2 + λy2 + λz2 − 3, as a way of inclusion of higher powers of the first strain invariant. The invariant J1 is taken relative to its maximum value JM for which at least one of the components x, y, z reaches the value corresponding to the maximum possible extension. The E or G equilibrium moduli are interpreted in terms of the concentration of EANCs and state of network chains in the same way as for the Gaussian model. Also, ΔFel correctly converges to the Gaussian limit for large values of n and low degrees of swelling. Horgan and Saccomandi48 have shown that the functional form of the Gent function is close to that of the three-chain model if the inverse Langevin function is expressed by the Paddé approximant. It was claimed that such modification improved the agreement with experiment;48 however, when applied to highly swollen hydrogel, it was not so successful and had to be modified by a term allowing for coupling between strains along the three strain axes.50 Kovac51 proposed a modified Gaussian model based on the end-to-end distance distribution function for a chain subjected to an equilibrium force which can extend the chain up to its maximum length equal to nl. The Kovac model and Gent model are functionally similar. While a power series expansion was used in applications to the deformed state, the simple basic form is shown in Table 1 (Kovac’s51 eq 24), and it was used here. The modified tube models52−55 represent the third group of theories; however, more input parameters are required. The application of the respective finite-extensibility models focused on stress−strain behavior of dry and swollen networks; the degree of swelling and its dependence on strains was studied much less. For the current state of chain finite extensibility problems, the reader is referred to Treloar’s monograph56 and recent reviews.57−59 In Table 1, the elastic contributions to the Helmholtz energy are summarized for various models of chains of finite extensibility valid for arbitrary chain deformation in x, y, z directions. The contributions are compared with the ideal expression for the network of Gaussian chains, for which the extensibility is not limited and the end-to-end distance may exceed the contour length. The theoretical models are based mainly on the Kuhn−Grün distribution function or its amended form. An amendment was introduced by Jernigan and Flory60,61 in order to correct the Kuhn−Grün approximate distribution in the limit of deformation ratios λ → 1. The 3-chain and the 8-chain models rank currently among the most frequently used ones for description of equilibrium stress−strain properties of

2. FINITE EXTENSIBILITY MODELS FOR ELASTIC CONTRIBUTIONS TO THE GIBBS ENERGY AND THE SWELLING EQUATION The change of Gibbs energy due to absorption of a liquid by a polymer network, ΔGSW = ΔGel,n + ΔGmix is usually considered as sum of contributions due to deformation of the network and mixing of a polymer of infinite molecular weight with the liquid, respectively.1−4 The term ΔGel,n traditionally includes the contribution of deformation of elastically active network chains (EANC), grouping of EANC in f-functional (elastically active) junctions (cross-links) (contribution ΔGcr), and the effect of possible fluctuations of the junctions on the force generated by deformation of EANCs. (Sometimes, ΔGel is associated only with stretching of EANCs and the sum ΔGel + ΔGcr is called ΔGnet.) In such a connotation, ΔGmix refers to mixing of a polymer of infinite degree of polymerization, linear or branched, but of cycle rank ξ equal to zero, i.e., with such a number of terminal units that the structure remains tree-like. We think that it is more convenient to consider the mixing contribution of the network with junctions, i.e., of ξ > 0, because network-solvent systems exist that are being formed and remain in their state of ease (undeformed) (section 3). Next, we will discuss only contribution by deforming a network of Ne EANCs when their ends are displaced affinely with the macroscopic strain (cf., eq 2.44 of ref 4). Under the usual conditions, the change of ΔGel,n is equal to the change of Helmholtz energy, ΔFel, which is a function of the entropy change. For a system where the reference volume V0 changes to the volume V: ΔGel,n ≈ ΔFel,n = ΔFel(λ) − NekT ln(V /V0)

(1)

The quantity ΔFel(λ) ≡ ΔFel is derived for various models of chain end-to-end distance distribution functions. These functions 4418


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Table 1. Forms of ΔFel/kT for Various Models of Networks with Finite Extensibility of Network Chains Discussed in This Worka ΔGel ≈ ΔFel

function acronym Gauss

1−6

ΔFel /kT =

3-chain39,41

Ne 2 (λx + λy 2 + λz 2 − 3) 2

⎡ ⎛ β ⎞ ⎛ β ⎞ ΔFel Nn1/2 ⎢ y ⎟+λβ = λxβx + n1/2 ln⎜⎜ x ⎟⎟ + λyβy + n1/2 ln⎜⎜ z z kT 3 ⎢⎣ sinh βy ⎟⎠ ⎝ sinh βx ⎠ ⎝ ⎛ β ⎞⎤ + n1/2 ln⎜⎜ z ⎟⎟⎥ ⎝ sinh βz ⎠⎥⎦ ⎛ λ ⎞ ⎛ λ ⎞ 1 λ = coth β − ; β = S −1⎜ 1/2 ⎟; β = S −1⎜ 1/2 ⎟ ⎝n ⎠ ⎝n ⎠ β n1/2 λn−1/2(3 − λ 2n−1) ≈ 1 − λ 2n−1

S(β) =

Smith43

ΔFel 1 = Ne (1 − N −1)(λx 2 + λy 2 + λz 2 − 3) kT 2 3 + (λx 4 + λy 4 + λz 4 − 3) 20N 1 + (λx 2λy 2 + λx 2λz 2 + λz 2λy 2 − 3) 10N

{

}

8-chain

40,41

⎡ ⎛ β ⎞⎤ ΔFel /kT = Nen1/2⎢λchainβchain + n1/2 ln⎜⎜ chain ⎟⎟⎥ ⎢⎣ ⎥ sinh β ⎝ chain ⎠⎦ ⎞ ⎛λ ⎛1 ⎞1/2 ⎟; λchain = ⎜ (λx 2 + λy 2 + λz 2)⎟ βchain = S −1⎜ chain 1/2 ⎝3 ⎠ ⎝n ⎠

3-chain amended60,61

⎡ ⎛ β ⎞ ⎛ β ⎞ N n1/2 ⎢ ΔFel y ⎟+λβ λxβx + n1/2 ln⎜⎜ x ⎟⎟ + λyβy + n1/2 ln⎜⎜ = e z z ⎢ kT 3 ⎣ sinh βy ⎟⎠ ⎝ sinh βx ⎠ ⎝ ⎛ β ⎞⎤ 1 βx βyβz + n1/2 ln⎜⎜ z ⎟⎟⎥ − Ne ln 3 λxλyλz ⎝ sinh βz ⎠⎥⎦

8-chain amended60,61

⎡ ⎛ β ⎞⎤ ⎞ ⎛ β chain ⎟ ΔFel /kT = Nen1/2⎢λchainβchain + n1/2 ln⎜⎜ chain ⎟⎟⎥ − Ne ln⎜⎜ −1/2 ⎟ ⎢⎣ ⎥ sinh β ⎠ ⎝ λchainn ⎝ chain ⎠⎦

Gent47−49

ΔFel /kT = −

EJM ⎡ J ⎤ ln⎢1 − 1 ⎥; J1 = I1 − 3; I1 = λx 2 + λy 2 + λz 2 6 ⎢⎣ JM ⎥⎦

JM = I1M − 3; I1M = I1(max λi = n1/2) Kovac51

ΔFel /kT = −

⎤ Nen ⎡ ⎛ ⎛ ⎛ 21⎞ 21⎞ 21⎞ ⎢ln⎜1 − λx ⎟⎠ + ln⎜⎝1 − λy ⎟⎠ + ln⎜⎝1 − λz ⎟⎠ + const⎥ ⎦ 2 ⎣ ⎝ n n n

a

k Boltzmann constant, T temperature in K, Ne number of elastically active network chains (EANC), n number of statistical segments per EANC, β = S −1(λ /n1/2) inverse of the Langevin function S(β) = coth β − 1/β , λi deformation ratio of chain along the axis i with respect to the reference (unperturbed) state.

The exact solution would result from the integration over all possible directions of the chain end-to-end vectors,62,63 which is, however, difficult. The conceptual simplicity is the advantage of the semiempirical model of Gent. It is based on the impossibility for the network chains in either one of the three directions to be extended over their contour lengths. In free swelling, this condition concerns all three directions; for swelling of the adhering film, only the direction perpendicular to the surface is involved. Interesting is the functional similarity (logarithmic form of the strain invariant) of the Gent and Kovac models. Also, both theories converge to the same value for n → ∞ (Gaussian function) and (ϕ02/ϕ2) → n3/2 for Δμ1el → ∞. The limiting value is, however, different for ϕ2 = 1, ϕ02 = 1: Δμ1el/RTνeV1m = n/(n − 1) for the Kovac model and

cross-linked rubbers in the region of large strains. They differ in orientation of network chains with respect to the three main strain components x, y, z. In the three-chain model with the junction placed in the center of a cubic element, the stress components are oriented normally to the cube faces and the orientation of the chain end-to-end vectors are the same. In the eight-chain model, the end-to-end chain vectors are oriented in the direction of the body diagonals of the cube. The three- and eight-chain models give the same results for isotropic changes of volume, but the results are different for the anisotropic deformation. In the three-chain model, the effect of anisotropy is the strongest while in the eight-chain model, the anisotropy effect is smeared by the transformation λchain = ((1/3)(λx2 + λy2 + λz2))1/2 (Table 1). 4419


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Table 2. Elastic Contribution to the Chemical Potential of Solvent: Freely Swollen Network and Adhering Filma function acronym Gauss

3-chain

(Δμ1)el

type of swelling free

(Δμ1)el /RT = V1mνeϕ21/3(ϕ20)2/3

constrained

(Δμ1)el /RT = V1mνe(ϕ20)2 ϕ2−1

(Δμ1)el /RT =

⎡ ⎤ β′ 1 V1mνen1/2⎢(β + β′λ) + n1/2 (1 − β coth β)⎥ϕ22/3(ϕ20)1/3 ⎣ ⎦ β 3

(Δμ1)el /RT =

⎡ ⎤ β′ 1 V1mνen1/2⎢(βx + βx′λx) + n1/2 x (1 − βx coth βx )⎥ϕ20 ⎢ ⎥⎦ 3 βx ⎣

free

constrained

Smith approx.

8-chain

free

⎛ ⎞ 1 1 (Δμ1)el /RT = V1mνe(ϕ20)2/3 ϕ21/3⎜1 − + (ϕ 0)2/3 ϕ2−2/3⎟ ⎝ ⎠ n 2n 2

constrained

⎛⎛ ⎞ 3 ⎞⎟ 0 2 ‐1 3 (Δμ1)el /RT = V1mνe⎜⎜1 + (ϕ ) ϕ2 + (ϕ 0)4 ϕ2−3⎟ ⎝⎝ ⎠ 10n ⎠ 2 5n 2

free constrained

=3-chain, because λchain = λ ⎡ 1 ′ λchain) (Δμ1)el /RT = V1mνen1/2⎢(βchain + βchain ⎢⎣ 3

+ n1/2

⎤ (1 − βchain coth βchain )⎥ϕ20 ⎥⎦ βchain

′ βchain

with λchain = 3-chain enhanced

⎛1 2 ⎞1/2 ⎜ (λ + 2)⎟ ⎝3 x ⎠

free

the same as 3‐chain model plus term

8-chain enhanced

Gent

constrained

⎡ β′ 1 1⎤ the same as 3‐chain model constrained plus term V1mνe⎢ x − ⎥ϕ20 ⎢⎣ βx 3 λx ⎥⎦

free constrained

the same as 3-chain and 8-chain models enhanced

⎡ β′ 1 1 ⎤ 0 ⎥ϕ the same as 8‐chain model constrained plus term V1mνe⎢ chain − ⎢⎣ βchain 3 λchain ⎥⎦ 2

free

(Δμ1)el /RT = νeV1m constrained

(Δμ1)el /RT = V1mνe Kovac

free

(Δμ1)el /RT = νeV1m constrained

(Δμ1)el /RT = νeV1m Lucht−Peppas

⎡ β′ 1 1⎤ V1mνe⎢ − ⎥ϕ2 2/3(ϕ20)1/3 ⎣β 3 λ⎦

free

1−

(ϕ20)2/3 ϕ21/3 0 [(ϕ2 /ϕ2)2/3 − 1](n

1 − ((ϕ20 /ϕ2)2 − 1)/(n − 1) ϕ21/3(ϕ20)2/3 1 − (ϕ20 /ϕ2)2/3 /n

ϕ2−1(ϕ20)2 1 − (ϕ20 /ϕ2)2 /n 1/3⎞ 1⎛ ϕ ⎞ + ⎜⎜ 20 ⎟⎟ ⎟⎟ n ⎝ ϕ2 ⎠ ⎠ ⎝

⎛

(Δμ1)el /RT =

− 1)

(ϕ20)2 ϕ2−1

νeV1ϕ21/3(ϕ20)2/3 ⎜⎜1

2

2/3⎞3 ⎛ 1 ⎛ ϕ2 ⎞ ⎟ ⎜ /⎜1 − ⎜⎜ 0 ⎟⎟ ⎟ n ⎝ ϕ2 ⎠ ⎠ ⎝

a For free swelling: λx = λy = λz = λ = (ϕ2/ϕ02)1/3; for swelling of an adhering film: λy = λz = 1; λx = λ = (ϕ2/ϕ02); (Δμ1)el = (∂ΔGel(λ))/(∂λ) (∂λ)/ (∂ϕ2) (∂ϕ2)/(∂N1). R gas constant, N1 number of moles of solvent, λy = λz = 1; λx = λ = (ϕ2/ϕ02); ϕ2volume fraction of polymer in the swollen network, ϕ02 volume fraction of nonextractable material at network formation, V1m molar volume of the solvent, νe concentration of EANCs in moles per dry volume of the network, β inverse Langevin function of λ/n1/2, β′ derivative of β with respect to λ; β is usually expressed in the form of Paddé approximant.

of ΔGel and the Lucht−Peppas swelling function64,65 for which the parent ΔGel was not available. For short chains (higher cross-link density), it was suggested to use elastic Gibbs energy based on the Wall−White end-to-end distribution function.69 However, this function gives a good approximation for the excluded volume effect in the region of compact conformations and does not consider the finite extensibility limitations. In our calculations, the inverse Langevin function β = S −1(λ /n1/2) was approximated by the Paddé approximant

Δμ1el/RTνeV1m = 1 for the Gent model. A relation differing from those discussed above was offered by Lucht and Peppas.64,65 The effect of finite extensibility for free swelling was also considered by Oppermann et al.66,67 and Okay et al.68 The value of n chosen for comparison of (Δμ1)el varies between >400 and 4; the lower limit (4) is already beyond that for which random walk approximation is permissible. Table 2 contains contributions to the change of chemical potential of the solvent (Δμ1)el associated with the change 4420


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Figure 1. Plot of elastic contribution to the change of chemical potential of the solvent (Δμ1)el/RT by free swelling listed in Table 1 in dependence on the volume fraction of the polymer in the swollen network ϕ2. Concentration of EANCs, νe, and average number of statistical segments per EANC, n: (a) νe = 10−5 mol/cm3, n = 400; (b) νe = 10−4 mol/cm3, n = 40; (c) νe = 10−3 mol/cm3, n = 4; values of other parameters: ϕ02 = 1, molar volume of solvent V1m = 100 cm3, specific gravity of polymer ρ2 = 1.2 g/cm3, molecular mass of a statistical segment Ms = 300 g/mol. Identification of the curves (see Table 1): black, Gauss; blue, Smith; red, 3-chain and 8-chain; red dashed, 3-chain and 8-chain amended; green, Gent; magenta, Kovac; cyan, Lucht−Peppas. In part a, all curves (colors) practically merge.

β ≈ ((λn−1/2(3 − λ2n−1)/(1 − λ2n−1)) which is claimed to be a good approximation in the relevant range of λ and n.70 The (Δμ1)el functions were obtained by differentiation of ΔGel with respect to the number of moles of the solvent (component 1), N1. Figure 1a−c and 2a−c show the values of (Δμ1)el for three levels of concentration of EANCs. For a network, Δμ1,el,n RT

=

Δμ1,el RT

up the expected upturn to the finite extensibility limits and convergence to the Gaussian limit for long EANCs. However, only the Kuhn−Grün function amended by Jernigan and Flory and the Gent function satisfy the Gaussian limit for ϕ2 = 1 and ϕ02 = 1. However, both the 3-chain and the Gent models assume that one-third of EANCs are oriented parallel to x-direction and equal parts parallel to y and z directions. Despite of functional similarity, there is a relatively large difference in the onset of upturn between the 3-chain and the 8-chain models for which the directions of the EANCs are oriented along the body diagonals. The 8-chain model is softer. Experiments on swelling of adhering films can further contribute to elucidation of the chain orientation problem. In the considerations above, we have been interested in the functional forms of the Gibbs energy functions and the chemical potential functions and we tacitly worked with the direct proportionality of the elastic contribution to the number (concentration) of elastically active network chains (EANC), νe, which is a proportionality characteristic of the affine model. The subject of the phantom and affine network behavior has been widely discussed in the literature and is well explained in the Erman−Mark monograph.4 It is obvious that the elastic modulus of a Gaussian phantom network is not proportional to Ne, but to the cycle rank ξ = [(fe − 2)/fe]Ne, where fe is the average functionality of elastically active junctions; fe is equal to the chemical functionality f (f ≥ 3) for a perfect network. However, the finite chain extensibility reduces the junction

− V1mνeϕ2

Several conclusions can be drawn from the comparison of elastic contribution to the chemical potential of solvent in the swollen gel. The most important one is that the constraints by adhesion drastically affect the degree of swelling and its maximum attainable value. For the free three-dimensional swelling, the maximum volume expansion is close to the third power of the maximum limit for the one-dimensional swelling. Also, this is a warning not to relay on applicability of the Gaussian theory for constrained swelling even if it is a good approximation for free swelling. Concerning the choice of the Gibbs energy function: a closed form is always preferred over the power series expansion. The expansion − Taylor series, Lagrange expansion, Hermite polynomials, etc. are usually good for particular deformation geometry but they are less successful for another geometry. From the choice in Table 2, we can see that those based on the Kuhn−Grün function (3-chain and 8-chain models) and the Kovac and the Gent models show 4421


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Figure 2. Plot of elastic contribution to the change of chemical potential of the solvent Δμ1,el/RT by constrained swelling (listed in Table 2) in dependence on the volume fraction of polymer in the swollen network. Concentration of EANCs, νe, and average number of statistical segments per EANC, n: (a) νe = 10−5 mol/cm3, n = 400; (b) νe = 10−4 mol/cm3, n = 40; (c) νe = 10−3 mol/cm3, n = 4; values of other parameters are as in Figure 1. Identification of the curves (see Table 2): black, Gauss; blue, Smith; red, 3-chain; red dashed, 3-chain amended; violet, 8-chain; violet dashed, 8-chain amended; green, Gent; magenta, Kovac. In part a, all colors practically merge.

fluctuation and a passage to affine behavior is likely. Also, at this point, we do not go deeper into the problem of finite volume of chains and trapped entanglement contribution, as discussed recently.71

as strongly associating contact points (cf. also refs 74 and 75). Thus, the contribution to ΔG by mixing of a perfect network with the solvent should read

3. CONTRIBUTION TO ΔG BY MIXING WITH THE SOLVENT MOLECULES Usually, this contribution is expressed by the Flory−Huggins lattice model for a polymer of infinite molecular weight because the gel can be considered only one giant molecule (N2 = 1). Indeed, each bond formed between two finite molecules reduces the freedom of chain segments to occupy lattice sites and the combinatorial entropy. Equally, each bond between segments of a giant moleculean infinitely long chain or a branched crosslinked structurecancels the former independence of their placement on the lattice and reduces the combinatorial entropy as well. Therefore, the change of combinatorial entropy does not stop as soon as the molecule gets infinite but continues decreasing when more cross-links are formed. At the same time, the formed network chains as generally assumed are in their states of ease. The contribution by cross-links additional to those forming an acyclic molecule is proportional to the cycle rank of the network, ξ. The contribution is concentration dependent. This conclusion also follows from application of the Guggenheim−-Tompa quasichemical equilibrium theory of polymer solutions72,73 when the cross-linked units are considered

where N1 is the number of moles of solvent molecules and g is the interaction function which can be concentration dependent.76 Considering first formation of cross-links between segments of an infinite chain, each cross-link connects two segments (tetrafunctional junction) and ξ = Nν/2, where Nν is the number of moles of cross-linked segments. For an imperfect network, for instance such that develops during the network formation process or under nonstoichiometric conditions with dangling chains (and, possibly, small, elastically inactive loops), the cycle rank should be considered as effective cycle rank where the chains in cycles span elastically active junctions:

ΔGmix = RT (N1 ln ϕ1 + ξ ln ϕ2 + gN1ϕ2)

ξe =

(fe − 2) fe

νe

(2)

(3)

The elastically effective junction is a junction participating in at least 3 bonds with infinite continuation. The effective functionality, fe, is equal to the number of EANCs issuing from an elastically active junction (it is the number-average number). fe increases from 3 near the gel point possibly reaching the chemical functionality of the cross-link depending on the network 4422


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The fluctuations of cross-links of a phantom network affect both the entropy change connected with stretching of EANCs as well as that determined by positional probability of crosslinks in the amount proportional to Ne − ξe (eq 2.48 of ref 4), so that for the phantom network:

perfectness. Thus, the change of the chemical potential of the solvent derived from ΔGmix reads Δμ1,mix = RT[ln ϕ1 + ϕ2 + ξeϕ2 + (g − g ′ϕ1)ϕ22]

(4a)

where g′ = ∂g′/∂ϕ2. For concentration independent interaction function (g′ = 0), g = χ - the Flory−Huggins interaction parameter. Often, the concentration dependence of χ in Δμ1,mix(eq 4b) is given as Δμ1,mix = RT[ln ϕ1 + ϕ2 + ξeϕ2 + χ (ϕ2)ϕ2 2]

⎛ 3ξ ΔGsw,ph = RT ⎜N1 ln ϕ1 + ξe ln(Vd /V ) + gN1ϕ2 + e (λ 2 − 1) ⎝ 2 ⎛ ⎞ 3ξe 2 + ξe ln(V /V0)⎟ = RT ⎜N1 ln ϕ1 + gN1ϕ2 + (λ − 1) ⎝ ⎠ 2 ⎞ + ξe ln(Vd /V0)⎟ ; λ 2 = (V /V0)2/3 ⎠ (8)

(4b)

but then the functional form of g in ΔGmix is different from the concentration dependence of χ. A typical process in which the network chains are assumed to remain in their state of ease is the process of network formation in solution. The volume of the system remains constant (or nearly so) and only the number of junctions and EANCs increases. The thermodynamic changes are fully covered by ΔGmix (eq 2) and by the corresponding change of the chemical potential Δμ1,mix. As ξe and νe increase, Δμ1,mix increases which means that the vapor pressure of the solvent increases. Eventually (if the sol fraction is negligible at this point), Δμ1,mix = 0 and the system gets into equilibrium with pure solvent. At that point, also ϕ2 = ϕ02 and one gets the condition for the incipience of phase separation: ln(1 − ϕ20) + ϕ20 +

fe − 2 fe

The term ξe ln(Vd/V0) does not depend on volume of the swollen network (Vd is the volume of dry network), i.e., on N1, and does not affect Δμ1,mixwhich is now equal: ⎡ Δμ1,ph = RT ⎢ln(1 − ϕ2) + ϕ2 + (g − g ′ϕ1)ϕ2 2 ⎢⎣ +

a relation which already was obtained earlier75 based on somewhat different reasoning. Thus, the loss of the thermodynamic stability at incipient phase separation has nothing to do with deformation of EANCs and the corresponding change of the Gibbs energy. A prosaic explanation of why the system must phase separate is simple: when cross-linking continues, it is not possible in the given space to connect all junctions by EANCs while keeping their average end-to-end dimensions constant (at their state of ease). To avoid phase separation, further formation of bonds should not increase ξe and νe; which means that the newly formed are only allowed closing loops. The issue of phantom and af f ine Gaussian networks is very well-known. It is comprehensively settled in many publications by Erman and Flory. The results are summarized in the Erman−Mark monograph4 (especially, chapter 2). How does this relation look like in light of redefinition of ΔGmix? For isotropic swelling and affine behavior, ΔGel,n (eq 1) is just added to ΔGmix which results in

+ (g − g ′ϕ10)(ϕ20)2 +

Δμ1 RT

(10)

= ln a1 =

Δμ1,mix

+

Δμ1,el,n

RT RT ⎡ 2 ⎤ = ⎢ln(1 − ϕ2) + ϕ2 + (g − g ′ϕ1)ϕ2 2 + Δμ1,el − νeV1m ϕ2 ⎥ ⎢⎣ fe ⎥⎦ (11)

(6)

where for Δμ1,el a function selected from Table 2 either for free swelling or constrained swelling is substituted. The Gent function is taken as an example, because it closely resembles the most frequently used three-chain model in its amended form:

Δμ1,aff = RT (ln(1 − ϕ2) + ϕ2 + (g − g ′ϕ1)ϕ2 2) + Δμ1,el (7)

For the affine Gaussian network, as before

ln a1 = ln(1 − ϕ2) + ϕ2 + (g − g ′ϕ1)ϕ2 2

Δμ1,aff = RT[ln(1 − ϕ2) + ϕ2 + (g − g ′ϕ1)ϕ2 2 − (2/fe )ϕ2)]

fe

⎤ νeV1mϕ20 ⎥ ⎥⎦

4. CONCENTRATION OF EANCS AND ADHESION EFFECTS ON SWELLING For equilibrium of the swollen network phase with liquid solvent or solvent vapor, the equality of chemical potentials holds:

which gives

+

fe − 2

It is interesting to notice that eq 9 turns into eq 10 for ϕ2 = ϕ02 although their last terms come from terms of ΔGSW of different origin. For chains of finite extensibility for which the end-to-end distance distribution differs from Gaussian, the free fluctuations of cross-links cannot be expected. The phantom network behavior can be reached only asymptotically for network chain dimensions not deviating too much from those of their state of ease.

ΔGsw,aff = RT (N1 ln ϕ1 + ξe ln ϕ2 + gN1ϕ2) + ΔFel

νeV1m((ϕ20)2/3 ϕ21/3

(9)

⎡ Δμ1,ph = Δμ1,aff = RT ⎢ln(1 − ϕ20) + ϕ20 ⎢⎣

(5)

− RTνeV1m(2/fe )ϕ2)

fe

⎤ νeV1m(ϕ20)2/3 ϕ21/3⎥ ⎥⎦

However, for thermodynamics of systems in which network chains are formed or remain in their states of ease (ϕ2 = ϕ02), λ2 − 1 = 0, and the term ξe ln(Vd/V0) becomes relevant. For such system characterized by λ2 − 1 = 0:

νeϕ20 + (g − g ′(1 − ϕ20))(ϕ20)2 = 0

− RTNekT ln(V /V0))

fe − 2

⎛ (ϕ20)2 ϕ2−1 2 ⎞ + V1mνe⎜⎜ − ϕ2⎟⎟ 0 2 fe ⎠ ⎝ 1 − [(ϕ2 /ϕ2) − 1]/(n − 1)

(7a) 4423

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Table 3. Changes of thickness of the PEO plasma polymers films by swelling in water (cf., ref20) sample power, W 1 2 3 4 5 6 7

1 2 3 4 5 10 30

initial thickness, d0, nm

equilibrium swelling thickness, d1, nm

final dry thickness, d2, nm

polymer volume fraction in swollen gel, ϕ2 = d2/d1

volume fraction of gel in dry polymer, ϕ02 = d2/d0

150 122 110 132 135 137 132

320 161 139 158 160 150 139

90 61 76 113 120 132 129

0.28 0.38 0.55 0.72 0.75 0.88 0.93

0.60 0.50 0.69 0.86 0.89 0.96 0.98

temperature to form a soft, wax-like film. Alternatively, the flux of the PEO oligomers can be let to pass through the glow discharge. The resultant films swell when put in contact with water. In Table 3, typical data are shown on swelling of PEO films prepared at different power of discharge. The swelling degree was calculated from the change of film thickness measured by ellipsometry with liquid cell. Also, the sol fraction was measured by comparing the thickness of dry films before and after swelling. The values of the Flory−Huggins interaction parameter for PEG(PEO)−water system given in the literature vary widely between 0.4 and 0.8 depending on their molecular weight, character of the end groups and on the method used for their determination. From several studies,75,78−81 it is evident that χ increases with increasing polymer concentration. This dependence is expressed as linear or quadratic dependence on ϕ2 or in a closed form. We have used the latter form,80 which results from a detailed study:

For equilibrium with pure liquid solvent (a1 = 1), the determination of νe was straightforward because νe stood only in front of the last term of eq 12. However, also the number of statistical segments in an EANC, n, is a function of νe: ρ νe = 2 nMs (13) Here ρ2 is the specific gravity of the polymer and Ms is the molecular weight of the statistical segment. After substitution for n from eq 13, eq 12 and analogous equations with different functional forms of Δμ1,el should be solved numerically. Here, we do not consider the effect of exclusion of a part of the surface of polymer chains by cross-links from interaction with solvent on g or χ as is done in the quasichemical equilibrium approach (cf., refs 76,77). If g or χ are calculated from equilibrium swelling degree, the effect of cross-links is usually hidden in the concentration dependences of g or χ.

5. APPLICATION TO CROSS-LINKED POLY(ETHYLENE OXIDE) PLASMA POLYMER FILMS In plasma-assisted vapor deposition, soft materials are formed as a result of passage of organic vapors through a glow discharge. The precursor molecules undergo interactions with energetic electrons (and UV photons) in plasma and may dissociate with formation of free radicals. The latter participate in recombination reactions to create a variety of chemical bonds. Under pressure typically used for plasma polymerization (several Pa), the probability of the homogeneous gas phase radical recombination is much lower than that of the heterogeneous recombination, and solid materials usually grow on surfaces adjacent to plasma in the form of thin films. Furthermore, the surface being in contact with plasma acquires a floating (negative) potential and it is subject to bombardment by positive ions that may also contribute to the chemical and structural changes in the growing film. The resultant materials are in general highly crosslinked and often chemically more diverse than the precursor. (Nevertheless, these polymers are widely referred to as “plasma polymers” within the plasma polymerization community). Crosslink density of plasma polymers can be controlled both by the power delivered to the discharge (related to the concentration and energetic spectrum of the electrons) and by the flow rate of the precursor (related to the concentration of the monomer molecules). Usually, low molar mass monomers are used for plasma polymerization to ensure the sufficient volatility of the precursor but heavier species can be supplied into the gas phase as well. The flux of oligomeric PEO species with Mn = 1050 g/mol was generated by thermal vacuum depolymerization of conventional PEO as it was shown in detail in ref 20. Under experimental conditions, the vapors of the PEO oligomers are supersaturated, and they readily condense on substrates of room

χ (T , ϕ2) = D(T )B(ϕ2); 1 B(ϕ2) = 1 − bϕ2

D(T ) = d0 + d1/T + d 2 ln T ; (14)

using values of the constants D(T = 298) = 0.44, b = 0.517. The molecular weight of the statistical segment, Ms = 88 g/mol, was estimated from the characteristic ratio of PEO chain equal to 5.182 (close to 2 PEO units). The value of calculated concentrations of EANCs using eq 12 and eq 13 are shown in Table 4. They are compared with values calculated using equations for free swelling both in the Gaussian limit and representative relations taking into account the finite extensibility of network chains (Table 4). Both the finite extensibility and constraints by adhesion (unidirectional swelling) affect the calculated values in the same waythey make them smaller compared to values calculated for an unconstrained Gaussian network. Using the example above, we can compare two effects: (a) isotropic swelling of free network with unidirectional swelling of an adhering film respecting finite extensibility of network chains (adhesion effect), and (b) unlimited extensibility model (Gaussian distribution function) with finite extensibility limit. The Gent model was chosen as representing the finite extensibility behavior mainly because its Δμ1,el/RT contribution (Figure 2) almost copies the contribution of the (amended) tree-chain model which has been most frequently used. In the intermediate range of cross-link densities, the eight-chain model is always softer than the threechain model. This difference demonstrates in fact a special case of cross-coupling between coordinate axes without using any additional adjustable parameter (for more exact solution, cf. ref 63). The difference gets smaller when the concentrationdependent interaction parameter is determined by swelling using 4424


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Table 4. Calculated Concentration of Elastically Active Network Chains (mol/cm3) Using Eqs 11−13 for Various Forms of the Function Δμ1,el(Constrained and Unconstrained Case, Table 2)a sample no.

Gauss-ad

3c-ad

3cJF-ad

8c-ad

8cJF-ad

Gent-ad

Kovac-ad

Gauss-free

Gent-free

1 2 3 4 5 6 7

3.96e-4 2.25e-3 5.88e-3 1.58e-2 1.89e-2 5.63e-2 9.75e-2

3.65e-4 1.81e-3 3.62e-3 6.05e-3 6.48e-3 9.39e-2 1.07e-2

3.67e-4 1.97e-3 4.30e-3 7.43e-3 7.90e-3 1.10e-2 1.21e-2

3.79e-4 1.91e-3 4.01e-3 6.92e-3 7.45e-3 1.03e-2 1.14e-2

3.84e-4 2.10e-3 5.00e-3 9.59e-2 1.02e-2 1.26e-2 1.31e-2

3.54e-4 1.89e-3 4.02e-3 6.99e-3 7.48e-3 1.08e-2 1.20e-2

3.45e-4 1.64e-3 3.18e-3 5.34e-3 5.77e-3 8.70e-3 1.01e-2

1.39e-3 3.95e-3 9.52e-3 2.34e-2 2.77e-2 6.93e-2 1.11e-1

1.28e-3 3.58e-3 7.38e-3 1.17e-2 1.23e-2 1.46e-2 1.52e-2

Concentration dependent interaction parameter χ(T,ϕ2) (eq 12) for T = 298 K, (D(T = 298) = 0.44, b = 0.517), V1m = 18 cm3/mol, ρ2 = 1.4 g/cm3, Ms = 88 g/mol; values of ϕ2 and ϕ02 given in Table 3 for films adhering to rigid substrate (ad). a

Figure 3. Effect of adhesion constraints on the concentration of EANCs calculated from swelling using data of Tables 3 and 4. (a) Ratio of values of νe calculated using the free-swelling model and adhesion constraint model, both using the finite-extensibility function of Gent. (b) Ratio of values of νe for free swelling using the Gaussian affine model and the Gent finite extensibility model. In both, the same values of ϕ2 and ϕ02 given in Table 3 are used.

and some other models. The differences are not very large; the largest difference was found for the three-chain and eight-chain models which differ in orientation of chains with respect to the vector of acting force. The measurement of equilibrium swelling of adhering films can be used for determination of concentration of EANC. It was applied to a series of vapor-deposited crosslinked PEO plasma polymer films. The analysis of the results shows that neglecting of swelling anisotropy and finite extensibility of network chain may result in an overestimate of concentration of EANCs reaching up to an order of magnitude. In a broader context, the contribution offers swelling equations respecting finite extensibility of network chains for free swelling and constrained swelling. The extension for other geometries of constrains is straightforward. Also, the nature of mixing contribution to the Gibbs energy change on swelling is discussed. It is more appropriate to consider mixing of solvent with the network containing cross-links but with undeformed network chains than mixing of an un-cross-linked polymer of infinite degree of polymerization, because continuing formation of bonds within an infinite molecule further decreases the entropy. The additional term is proportional to the cycle rank of the network. The reformulation of the mixing contribution is important for consideration of cross-linking induced phase separation during formation of cross-links. The limit of thermodynamic stability (incipience of phase separation) is reached as a result of increase of the cycle rank (concentration of EANCs) while network chains are formed and exist in their state of ease. Phase separation is due to the cross-link contribution to the

the same form of elastic contribution to the chemical potential of the solvent. The effect of adhesion and finite-extensibility is illustrated by parts a and b of Figure 3. From Figure 3 and Table 4, it can be seen that the effect of adhesion constraints is larger for weaker networks (for our samples up to a factor of 4) where the finite extensibility plays only a small or no role. For highly cross-linked networks, the finite extensibility of network chains dominates; of course it is questionable whether in these extremes, the entropy-based rubber-elasticity theories are applicable at all.

6. CONCLUSIONS In this contribution, we have analyzed swelling of cross-linked polymers in a broader context of current rubber elasticity models of polymer networks with finite extensibility of network chains and the mixing contribution to the Gibbs energy change on swelling. The attention was focused on the effect of constraints caused by adhesion of the cross-linked film to a rigid substrate. In such cases, swelling causes only an increase of the thickness and the lateral dimensions of the film remain unchanged. The extension of network chains in the direction normal to the film plane can be much higher than the extension of network chains by isotropic swelling of a free film. Therefore, the issue of finite extensibility of network chains is more important when adhering films are concerned. Various current rubber elasticity theories respecting the finite extensibility of elastically active network chains (EANC) were examined−the three-chain model and eightchain model and their variants, the Gent semiempirical model, 4425


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mixing part of the chemical potential and has nothing to do with chain stretching (elastic contribution).

■

AUTHOR INFORMATION

Corresponding Author

*(K.D.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was performed in part under the COST Action MP1101 and was supported by Grant LD 12066 from the program COST CZ financed by the Ministry of Education, Youth, and Sports of the Czech Republic. Also, financial support by Czech Science Foundation Grant P101/12/1306 is acknowledged.

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