J. Phys. Chem. B 2001, 105, 4979-4986
4979
Polyelectrolytic Amphiphilic Model Networks in Water: A Molecular Thermodynamic Theory for Their Microphase Separation† Maria Vamvakaki and Costas S. Patrickios* Department of Chemistry, UniVersity of Cyprus, P.O. Box 20537, 1678 Nicosia, Cyprus ReceiVed: September 16, 2000; In Final Form: January 12, 2001
The aqueous aggregation behavior of networks comprising hydrophilic ionic blocks and hydrophobic nonionic blocks was studied by formulating a molecular thermodynamic theory, which considers the Gibbs free energies of the two possible states of the networks: the micelle-like state and the unimer-like state. The appropriate expressions for the elastic, mixing, and electrostatic components of the Gibbs free energy were developed for each of the two cases. For the micelle-like state, the interfacial free energy for the contact of the micellar core with the aqueous solvent was also included. For each of the two states, the total Gibbs free energy was minimized with respect to the polymer volume fraction. The lower from the two minimum Gibbs free energies corresponds to that of the more stable state. The effects of the length and degree of ionization of the hydrophilic block, the effect of the length of the hydrophobic block, the effect of the value of the Flory-Huggins interaction parameter between the hydrophobic block and water, the effect of the initial polymer volume fraction, and the effect of the number of arms per cross link were investigated. Under certain conditions, a unimer-tomicelle transition was observed, accompanied by a discontinuous change in the degree of swelling of the networks.
Introduction
Geometry and Forces
Ionic hydrogels are cross-linked polyelectrolytes possessing some unique properties,1,2 which make them promising materials for various practical applications.3 One relatively new type of ionic hydrogels is that of ionic amphiphilic hydrogels, also containing hydrophobic units in addition to the hydrophilic ionic ones.4-9 The presence of hydrophobic units can lead to microphase separation, which induces order formation and can impart new properties to the networks, ultimately resulting in new applications. The way the hydrophobic units are introduced in the network, belonging either to the side chains4-6 or to the main chain,7-9 may further affect the hydrogel structure and properties. Despite the extensive experimental work on ionic amphiphilic hydrogels, there have been no efforts to model their behavior. However, there is modeling work on a relevant system, that of noncovalent, physically associating polymer networks.10,11 Most of the modeling work has focused on nonionic associating polymer networks12-14 rather than on ionic associating networks.15,16 The aim of this study is to extend theories on the behavior of simple ionic networks17-19 to cover ionic amphiphilic networks. Our approach is also relevant to existing theories on the micellization of linear (not cross-linked) ionic amphiphilic block copolymers.20-22 Our main finding from this work is the observation of a unimer-to-micelle transition for block copolymer-based ionic amphiphilic networks under certain conditions. Moreover, we reproduce the volume phase transition already known (both experimentally and theoretically17-19) for statistical copolymer-based ionic amphiphilic networks.
We consider an amphiphilic model (precise chain lengths between cross-links and constant functionality of the crosslink23) network, based on ABA triblock copolymers24,25 comprising nonionic hydrophobic end-blocks and an ionic hydrophilic mid-block. The system can exist in one of two possible states. First, a unimer-like state in which no microphase separation takes place. And, second, a micelle-like state in which the hydrophobic blocks aggregate to form spherical hydrophobic microdomains. Figure 1 illustrates these two states, with the hydrophobic blocks drawn in black and the hydrophilic blocks shown in white. The main aim of this work was to determine, under a variety of conditions and network composition, the state, micelle-like or unimer-like, in which the amphiphilic networks exist. Aggregation into the micelle-like state would be driven by the reduction of the hydrophobic area, which would lead to the decrease of the unfavorable water-hydrophobe contact. At the same time, this aggregation would influence the values of the other energy components in the system. The preferred structure of the network will be determined by the relative magnitude of all participating forces. It must be recognized that it is also possible that a higher order of aggregation takes place in which two or more hydrophobic cores are combined, resulting in structures such as cylinders and lamellae.26 Such an assembly would be favored by the hydrophobic force because it would further reduce the interfacial area per chain (depending on composition), but it would be opposed by the elastic force because it would create local stretching and compression of the network. The exact balance of the various forces would again dictate whether this intercore assembly would take place or not. This issue will be addressed in the future. However, we feel that, for our own experimental system,9 containing many (typically 20) hydro-
† This work is dedicated to the memory of Professor Toyoichi Tanaka, formerly of the Physics Department of the Massachusetts Institute of Technology (MIT), who introduced one of the authors (C.S.P.) to the science of hydrogels. * To whom correspondence should be addressed.
10.1021/jp003307t CCC: $20.00 © 2001 American Chemical Society Published on Web 05/03/2001
4980 J. Phys. Chem. B, Vol. 105, No. 21, 2001
Vamvakaki and Patrickios
Figure 1. Schematic representation of the two possible states of a triblock copolymer-based ionic amphiphilic model network. The hydrophobic blocks are shown in black and the hydrophilic in white. A change in pH, temperature, solvent composition, or copolymer composition may induce a transition between the nonsegregated state and the microphase separated state.
[
∆Gunimer mixing ) kTβ(1-φ) ζχ +
Figure 2. Schematic representation of an ionic amphiphilic model network based on a statistical copolymer. The hydrophobic units are shown in black and the hydrophilic in white. The random distribution of hydrophobic and hydrophilic units precludes microphase separation in this network.
phobic blocks “frozen” in the cores via cross-linking, higher order aggregation would be difficult. Another system, for which the results of the present investigation are relevant, is an amphiphilic network based on statistical copolymers comprising nonionic hydrophobic and ionic hydrophilic units. This system, illustrated in Figure 2, can only exist in the unimer-like state because of the random distribution of the hydrophobic units along the polymer chains. The mathematical description of the statistical copolymer-based network is covered by the description of the unimer-like state of the triblock copolymer-based network. It is noteworthy that systems similar to the statistical copolymer-based network have been studied extensively in the past, both experimentally18,19 and theoretically,17 and they are known to undergo a first-order volume phase transition (discontinuous swelling) upon certain change in conditions.
We present in the following section the equations used in the model. These equations were written for a unit cell, which was assumed to be a cube with center at a cross-link and extending to the middle of the (adjacent) hydrophilic blocks. Analogous models have been presented by other workers for the micellization of ionic20-22 and nonionic27-29 linear block copolymers in selective solvents. Free Energy of Mixing. For the unimer-like gel, the free energy of mixing is given by the expression:
]
(1)
where k is Boltzmann’s constant, T is the absolute temperature, β is the number of arms per cross-link (Figures 1 and 2 show three arms per cross-link), φ is the polymer volume fraction, (1- φ) is the solvent volume fraction, ζ is the number of units in the hydrophobic block (or one-half of the hydrophobic units between cross-links; each elastic chain has two hydrophobic blocks), and η is the number of units in the hydrophilic semiblock (or one-half of the hydrophilic units between crosslinks; only one-half of each hydrophilic block is located within the unit cell), and χ is the Flory-Huggins interaction parameter between the solvent (water) and the hydrophobic units. The Flory-Huggins interaction parameter between water and the hydrophilic units was set equal to zero. Equation 1 is the classical Flory-Huggins equation for the Gibbs free energy of mixing of polymer solutions30,31 applied to a network unit cell. The expression lacks the term of the polymer translational entropy due to the infinite size of the polymer network, but the logarithmic term represents the solvent translational entropy, while the term with χ is the enthalpic interaction between the hydrophobic block of the polymer and the solvent. The factor β (ζ + η) in the solvent translational entropy term is the total number of polymer units within the unit cell, while β(ζ + η)(1 - φ)/φ is the number of solvent molecules within the unit cell. Note that φ-1 is the degree of swelling (DS) of the network, defined as the total number of segments (solvent plus polymer) in the unit cell divided by the number of polymer segments in the unit cell. It was assumed that a solvent (water) molecule and a monomer repeat unit (e.g., methylene) occupy the same volume υ ) R3, where R is the length of the side of a cube circumscribing one water molecule. For the micelle-like gel, the free energy expression is
∆Gmicelle mixing ) kTβ(ζ+η) Theory
ζ+η ln(1-φ) φ
[
]
1-φ (η+ζ)(1-φ) ln φ η + ζ(1-φ)
(2)
whose logarithmic term is more complicated than that of the unimer-like gel due to the more complex structure of the micellelike gel in which mixing with the solvent is allowed only in the shell region (but not in the core region). Equation 2 lacks the enthalpic term of eq 1 because all interactions arising from the contact between water and the hydrophobic segments are described by the interfacial free energy expression, discussed next. Interfacial Free Energy. The micellar interfacial free energy for the contact of the hydrophobic core with water is
Polyelectrolytic Amphiphilic Model Networks micelle ∆Ginterfacial ) kT(4π)1/3(3βζ)2/3
J. Phys. Chem. B, Vol. 105, No. 21, 2001 4981
(6χ)
1/2
(3)
The derivation of this equation is explained below. The interfacial free energy is equal to the product of the interfacial tension between water and the hydrophobic units γ with the hydrophobic interfacial area Sinterfacial: micelle ∆Ginterfacial
) γSinterfacial
kT χ 1/2 R2 6
()
(5)
where R2 is the area of interaction between a water molecule and a hydrophobic monomer repeat unit. The hydrophobic interfacial area of the spherical micellar core Sinterfacial is calculated from the volume balance of the hydrophobic blocks at a cross-link: the volume of the micellar core Vcore is equal to the sum of the volumes of the constituting hydrophobic blocks:
4π 3 R ) βζR3 Vcore ) 3
(6)
where R is the radius of the core, which leads to the calculation of the interfacial area Sinterfacial:
Sinterfacial ) 4πR2 ) (4π)1/3(3βζ)2/3R2
(7)
Combining eqs 5 and 7 with eq 4 yields eq 3. Elastic Free Energy. The unimer-like elastic (deformation) free energy is given by the equation:
[( )
φ0 3 ∆Gunimer elastic ) kTβ 4 φ
2/3
( )]
1 φ0 - 1 - ln 3 φ
(8)
where φ0 is the polymer volume fraction in the unperturbed state, taken as that during synthesis. A typical value for φ0 is 0.2. 9 It must be pointed out that the numerical coefficient in this equation [3β/4 ) (β/2)(3/2)] originates from the fact that β chains emanate from each cross-link point in each unit cell, but only half of each effective chain lies within the unit cell. The original Flory coefficient is 3/2 for one elastic chain. The micelle-like network elastic free energy is expressed as
{[ (
)]
2/3 φ0 3 ζ ∆Gmicelle 1 + (1-φ) elastic ) kTβ 4 φ η ζ 1 φ0 1 + (1-φ) 1 - ln 3 φ η
[ (
unimer ∆Gelectrostatic ) kTβηιln
(ηφηι+ ζ)
(10)
(4)
The interfacial tension between water and the hydrophobic units γ can be related to their Flory-Huggins interaction parameter χ as follows:
γ)
numerical constant of 3β/4 in front of the elastic term for the micellar shell can be rationalized in the same way as that in eq 8. Electrostatic Free Energy. The electrostatic free energy expression for the unimer-like gel is
)]} (9)
which is the result of applying eq 8 only in the micellar shell. The elastic contribution from the core is small and difficult to calculate, and it is therefore neglected. Initial calculations were performed by estimating the core elastic energy using eq 8 and setting the equilibrium volume fraction of the core equal to 1 due to exclusion of water from the core. The results (equilibrium Gibbs free energies and DSs) obtained in that case were very similar to those obtained in this approach with some slight differences in the position of the volume transitions. The term in square brackets in eq 9 represents the ratio of φ0 divided by the polymer volume fraction in the shell. Note that φ represents the overall polymer volume fraction in the network. The
where ι is the degree of ionization of the ionizable groups in the network, and the argument of the logarithm is the volume fraction of counterions.32,33 Equation 10 represents the Gibbs free energy contribution from the translational entropy of the counterions to the network and ignores any electrostatic interaction arising from the repulsion between the polymeric charges. Such a simple approach is more accurate at low degrees of ionization,34 but it still gives a qualitative description of highly ionized systems where more complete approaches also break down.35,36 The electrostatic free energy for the micelle-like gel is micelle ) kTβηιln ∆Gelectrostatic
[
φηι η + ζ(1-φ)
]
(11)
which is again the Gibbs free energy contribution from the translational entropy of the counterions which are now restricted within the micellar shell. Thus, the more complex geometry of the micelle-like gel gives rise to a more complicated argument of the logarithm in eq 11 than in eq 10, similar to eqs 2 and 1. Total Free Energy. The total free energy of the networks is equal to the sum of all free energy components. For the unimerlike networks, the total energy is unimer unimer unimer ∆Gunimer total ) ∆Gmixing + ∆Gelastic + ∆Gelectrostatic
(12)
while for the micelle-like networks the interfacial free energy component must also be included:
) ∆Gmicelle total micelle micelle micelle ∆Gmicelle mixing + ∆Gelastic + ∆Gelectrostatic + ∆Ginterfacial (13)
Equilibrium Degree of Swelling (DS). Each one of the two total free energies in eqs 12 and 13 is minimized with respect to φ. The lowest of the two minimum total free energies is the most favorable and corresponds to the preferred network state, micelle-like or unimer-like. The equilibrium polymer volume fraction is that corresponding to the lowest minimum total free energy, while the equilibrium DS is calculated as its inverse. For statistical copolymer-based networks, existing only in the unimer-like state, the equilibrium DS is that corresponding to the minimum of the unimer-like total free energy. Minimization Procedure. The total free energies were minimized numerically using Newton’s method, programmed in GWBASIC. To this end, the first and second derivatives of the total free energies with respect to φ were calculated. The total free energy is minimized when the first derivative is equal to zero and the second derivative is positive. Maximum Degrees of Swelling. The limiting (maximum) DS of unimer-like networks and micelle-like networks was calculated and compared against the equilibrium DS to ensure that the latter is always lower than the former. The maximum DS is that corresponding to the case when the chains attain their maximum possible extension. For the unimer-like networks, this implies fully extended hydrophilic and hydrophobic blocks, and the maximum DS is given by the formula
4982 J. Phys. Chem. B, Vol. 105, No. 21, 2001
DSunimers maximum )
8 (η + ζ)2 β
Vamvakaki and Patrickios
(14)
TABLE 1: Values of the Parameters Investigated in the Calculations parameter’s values
For the micelle-like networks, the maximum DS is calculated for fully extended hydrophilic blocks and collapsed hydrophobic blocks, according to the equation
[η + (3βζ 4π ) ]
1/3 3
8 DSmicelles maximum ) β
η+ζ
(15)
Assumptions Made and Range of Variables Investigated. The system was assumed to consist of model networks, comprising linear chains of exact length and composition and a precise number of arms at the cross-link. The ABA triblock copolymer chains were considered to have hydrophobic endblocks and a hydrophilic ionizable midblock, as shown in Figure 1. The system was assumed to be salt-free. The effect of the following variables on the equilibrium total Gibbs free energy and the equilibrium DS was investigated: the initial polymer volume fraction, the number of arms per cross-link, the number of units and degree of ionization of the hydrophilic block, the number of units in the hydrophobic block, and the FloryHuggins interaction parameter. Table 1 lists the names, symbols, range of values, and central values of these variables explored in this study. In each set of calculations, a certain variable was changed, with the rest of the variables kept constant at their central value. Central values were chosen to represent the most educated values of the variables, as those would occur in a typical experimental system. For example, the central value for the initial polymer volume fraction was chosen to be 0.2, equal to that in our recent experimental system.9 The central value for the solvent-hydrophobe interaction parameter was fixed at 2.0, typical for the value of the water-propylene oxide pair.29 Results and Discussion Figures 3-8 illustrate the results of our calculations. The micelle-like state exists in all Figures: the micelle-like state dominates in Figures 3, 4, and 6, while both the unimer-like state and the micelle-like state appear in Figures 5, 7, and 8. All these are relevant to block copolymer-based networks for which the micelle-like state is possible. Statistical copolymerbased networks are described by the unimer-like state, for which Figures 5-8 predict first-order volume phase transitions. The results in each figure are analyzed in detail below. Figure 3 shows the effect of the initial polymer volume fraction on the (equilibrium) total Gibbs free energy ∆Gtotal and the (equilibrium) degree of swelling DS, while the other variables are kept constant at their central values. ∆Gtotal for the micelle-like state is always lower than that for the unimerlike state by about 1400 kT. Thus, all networks described in this Figure should exist in the micelle-like state. Examination of the components of the total free energies reveals that the prevalence of the micelle-like state is due mainly to the lower value of the combined mixing-interfacial energy for the micelle-like state than the mixing energy in the unimer-like state. The DSs are shown in the lower part of the figure. The DSs of the energetically favored micelle-like state are lower by a factor of 2 than those of the unimer-like state due to the shorter effective chains in the former case (the hydrophobic block is “lost” in the micellar core and is not available for stretching). Both micellar and unimer DSs decrease with φ0, as expected: the denser the initial state of the network the less it swells. It should be noted that, although in the case of block copolymerbased networks the micelle-like state prevails under all the
parameter’s name and symbol initial polymer vol fraction φ0 no. of arms per cross link β no. of units in hydrophilic semiblock η degree of ionization of hydrophilic block ι no. of units in hydrophobic block ζ Flory-Huggins interaction parameter between hydrophobic block and water χ
range of values 0.1-1.0 3-50 1-2000 0.001-0.5 1-12000 0.0-5.0
central results value in Figure 0.2 20 120 0.3 120
3 4 5 6 7
2.0
8
Figure 3. Effect of the initial polymer volume fraction φ0 on (a) the equilibrium total Gibbs free energy and (b) the equilibrium degree of swelling for the micelle-like and unimer-like states of the ionic amphiphilic model networks. The other variables are kept constant at their central values: β ) 20, η ) 120, ι ) 0.3, ζ ) 120, and χ ) 2.0.
conditions covered by this figure, the unimer-like state should be encountered in the case of statistical copolymer-based networks. In qualitative agreement with the results of the present theory, experimental measurements on ionized amphiphilic networks indicated that the DSs of statistical copolymer-based hydrogels were about 50% higher than those of their block copolymer-based counterparts.9 However, the DSs in the experimental system had significantly lower values, order of 10, than the model predictions, order of 1000, despite the molecular weight and composition similarities in the two systems. This may be due to chain entanglements in the experimental system, which can greatly restrict the swelling of the network. Figure 4 presents the effect of the number of arms per crosslink. Similar to Figure 3, ∆Gtotal for the micelle-like state is lower than that for the unimer-like state, for the same reason described for Figure 3. Figure 4a shows that the energy difference between the micelle-like state and the unimer-like state increases with the number of arms. This is due to the smaller hydrophobic surface area per arm for larger micellar cores compared to smaller ones, which allows for greater savings in hydrophobic contact for higher values of β in the case of micelles. Figure 4b presents the DSs for the two states and, similar to Figure 3, the DSs for the micelle-like state are always lower than those for the unimer-like state due to shorter effective chains after micellization. Interestingly, the DSs of both states are independent of β. This can be explained from the fact that
Polyelectrolytic Amphiphilic Model Networks
Figure 4. Effect of the functionality of the cross-links β on (a) the equilibrium total Gibbs free energy and (b) the equilibrium degree of swelling for the micelle-like and unimer-like states of the ionic amphiphilic model networks. The other variables are kept constant at their central values: φ0 ) 0.2, η ) 120, ι ) 0.3, ζ ) 120, and χ ) 2.0.
Figure 5. Effect of the number of units in the hydrophilic semiblock η on (a) the equilibrium total Gibbs free energy and (b) the equilibrium degree of swelling for the micelle-like and unimer-like states of the ionic amphiphilic model networks. The other variables are kept constant at their central values: φ0 ) 0.2, β ) 20, ι ) 0.3, ζ ) 120, and χ ) 2.0. The inset in Figure 5a blows up the region where the unimer-tomicelle transition occurs; note that the x-axis is linear rather than logarithmic.
each energy component is linearly proportional to β, with the exception of the interfacial free energy, which is proportional to β2/3 (almost linearly proportional to β). Thus, in the minimization of ∆Gtotal the βs practically cancel out and the resulting optimum φ and corresponding equilibrium DS are almost independent of β. Note, however, that the minimum value of ∆Gtotal is a function of β. Figure 5 illustrates the effect of the length of the hydrophilic semi-block (only half of the hydrophilic block lies within the unit cell) with degree of polymerization η. Although the total
J. Phys. Chem. B, Vol. 105, No. 21, 2001 4983
Figure 6. Effect of the degree of ionization of the hydrophilic block ι on (a) the equilibrium total Gibbs free energy and (b) the equilibrium degree of swelling for the micelle-like and unimer-like states of the ionic amphiphilic model networks. The other variables are kept constant at their central values: φ0 ) 0.2, β ) 20, η ) 120, ζ ) 120, and χ ) 2.0.
Figure 7. Effect of the number of units in the hydrophobic block ζ on (a) the equilibrium total Gibbs free energy and (b) the equilibrium degree of swelling for the micelle-like and unimer-like states of the ionic amphiphilic model networks. The other variables are kept constant at their central values: φ0 ) 0.2, β ) 20, η ) 120, ι ) 0.3, and χ ) 2.0. The inset in Figure 7a blows up the region where the unimer-tomicelle transition occurs; note that the x-axis is linear rather than logarithmic.
free energies for the two states appear to coincide in Figure 5a, the inset in Figure shows that the energy of the micelle-like state is lower than that of the unimer-like state when η > 22. Thus, the system is in the unimer-like state when η < 22, and it goes to the micelle-like state when η > 22, with a unimerto-micelle transition at η ) 22, indicated by the arrow in Figure 5a. Examination of the free energy components reveals that the prevalence of the unimer-like state for small values of η ( 22, the micelle-like state wins out due to the greater decrease in its mixing energy with increasing η, conferred mainly by the higher DSs of this state than those of the unimer-like state. In both states, the total free energy decreases approximately linearly (the curvature in Figure 5a is due to the logarithmic x-axis; note the linearity in the inset to this Figure) with η due to the almost linear dependence of the electrostatic energy on η, especially at high values of η where the total energy is dominated by its electrostatic component (which is balanced by the elastic component). Figure 5b displays the dependence of the DSs on the length of the hydrophilic semiblock η. The DSs of both states increase with η, as expected: a longer hydrophilic block introduces into the network more counterions, which increase the electrostatic free energy and osmotic pressure contributions, leading to higher DSs. The arrow on the dashed line at η ) 22 indicates in Figure 5b the unimer-to-micelle transition dictated by the findings in Figure 5a. This implies a discontinuous increase in the DS from 1.1 to 45. There is also a discontinuous increase in the DS for the unimer-like state, relevant to statistical copolymer-based networks. This occurs at η ) 55 and involves a very large increase in the DS from 1.3 to 982. This volume phase transition corresponds to the break in the free energy curve of the unimerlike state in the inset of Figure 5a and it arises from the change in the balance of the free energy components. More specifically, when η < 55 (collapsed state), the equilibrium free energy is
Vamvakaki and Patrickios determined by its mixing (both the enthalpic and entropic constituents) and electrostatic components, and when η > 55 (expanded state), the equilibrium free energy is controlled by the balance between the electrostatic and elastic components. This discontinuous change in the DS for the unimer-like state has been described before both experimentally and theoretically by Tanaka18,19 and Dusˇek.17 Note that for high values of η (>55) the DSs in the unimer-like state are higher than those in the micelle-like state due to the longer effective chains in the former state. Figure 5b has some interesting experimental implications. First, for a series of amphiphilic networks based on block copolymers, those with η < 22 will be in the unimer-like state (no microphase separation) and collapsed with DS around 1.1, while those with η > 22 will be in the micelle-like state and expanded with DSs which increase with the value of η. Second, for a series of amphiphilic networks based on statistical copolymers, isomeric to those of the previous series, no microphase separation is possible and all networks will exist in the unimer-like state. Those networks with η < 55 will be collapsed with DSs lower than 1.3, while those with η > 55 will be expanded with DSs which increase with the value of η. Thus, block copolymer-based networks will expand for lower values of η (η > 22) compared to their statistical counterparts (η > 55), although the DSs of the latter type of expanded networks will be greater than those of the former (also in the expanded state) for the same values of η. Figure 6 depicts the effect of the degree of ionization of the hydrophilic block ι. Figure 6a indicates that the micelle-like state is the energetically favored one. Again, this is mainly due to the efficient hiding of the hydrophobic blocks from water in the micellar core, leading to a lower combined mixinginterfacial energy in the case of the micelle-like state than the mixing energy of the unimer-like state. The DS of the micellelike state, shown in Figure 6b, increases smoothly with the degree of ionization due to the concomitant increase in the magnitude of the electrostatic free energy component (which is of negative sign), without any transitions from or to the unimer-like state, and without any volume phase transitions. On the other hand, the DS of the unimer-like state, relevant to a network based on a statistical copolymer, will increase discontinuously from 1.6 to 547 with the increase of the degree of ionization above 0.095. This is again the same volume phase transition observed by Tanaka18,19 and encountered above in Figure 5b. Similar to Figure 5, this discontinuity corresponds to the break in the free energy curve of the unimer-like state in Figure 6a at ι ) 0.095 and it again arises from the change in the balance of the free energy components. In this case, the equilibrium value of the total free energy in the collapsed state (ι < 0.095) is controlled by the balance between the mixing energy and the electrostatic energy, whereas the equilibrium value of the total free energy in the expanded state (ι > 0.095) is defined by the balance between the electrostatic energy and the elastic energy. The important experimental implication of Figure 6 is that an amphiphilic network based on a statistical copolymer will undergo discontinuous swelling upon a certain increase in its degree of ionization. In contrast, an amphiphilic network based on a block copolymer, isomeric to the above-mentioned statistical copolymer, will swell continuously upon any increase in its degree of ionization. Therefore, when discontinuous swelling is undesired, for instance, in certain biomedical applications of
Polyelectrolytic Amphiphilic Model Networks the networks, such as for their use as implants,37 the block copolymer-based networks should be preferred over the statistical ones. Figure 7 examines the effect of the length of the hydrophobic block, ζ. The findings in this Figure are analogous to those in Figure 5, in which the effect of the length of the hydrophilic block was studied, although there is an extra transition in Figure 7. Thus, the micelle-like state prevails for ζ values between 3 and 1567, while the unimer-like state dominates outside this range of values of ζ. There is a critical value of ζ ) 3 (indicated by an arrow in Figure 7a and its inset) for aggregation to occur, above which enough savings in the contact energy between the hydrophobic units and water have been made and the network will exist in the micelle-like state. This unimer-to-micelle transition is accompanied by a small but discontinuous change in the DS, from 3092 to 3017 (indicated by an arrow in Figure 7b). For ζ values greater than 1567, the unimer-like state, which is now collapsed (see Figure 7b) due to the great hydrophobicity of the network, prevails again. The following discussion is similar to that given in Figure 5a for small values of η. At high values of ζ, the free energy of mixing in the unimer-like state is dominated by its entropic component and acquires a very negative value, despite the large value of ζ. At the same time, the interfacial free energy of the micelle-like state acquires a large positive value due to its square root dependence on a large ζ, leading to a high positive value for the combined interfacialmixing energy of this state. Thus, for ζ > 1567 the total energy of the unimer-like state is more favorable than that of the micelle-like state. This micelle-to-unimer transition also appears in Figure 7b, indicated by the dashed vertical line, and represents the abrupt change in the DS from a value of 221 to a value of 1.1. Figure 7 also predicts a volume phase transition for networks existing always in the unimer-like state. Thus, statistical copolymer-based amphiphilic networks will abruptly shrink and their DS will change discontinuously from 3078 to 1.2 when the value of ζ exceeds 300. The collapse in the unimerlike network is again due to the interplay of the various components of the total free energy. Before the collapse (ζ < 300), the minimum of the total free energy is controlled by the balance between electrostatic and elastic energy. After the collapse (ζ > 300), the equilibrium total free energy is defined by the mixing free energy and the electrostatic energy. Examination of Figure 7b implies that the collapse of the block copolymer-based networks is more difficult (ζ > 1567) than that of the statistical copolymer-based ones (ζ > 300); although, when expanded, the latter type of networks displays greater DSs than those of the former for equal values of ζ. Similar conclusions were derived from Figure 5b. Finally, Figure 8 depicts the effect of the value of the FloryHuggins interaction parameter between the hydrophobic block and water χ. The value of χ can be experimentally varied by adding to the aqueous solvent other substances. For example, addition of organic solvents, such as acetone, will lower the value of χ, depending on the composition of the mixture. The values of the equilibrium total free energies, plotted in Figure 8a, suggest that a block copolymer-based amphiphilic network will exist in the unimer-like state when χ < 1.38, but it will undergo a transition to the micelle-like state when χ > 1.38. This is so because it is only above a minimum value of χ that the combined mixing-interfacial energy components in the micelle-like state become more negative than the mixing energy component in the unimer-like state. This unimer-to-micelle transition, indicated in Figure 8b by the vertical dashed line, is accompanied by a discontinuous shrinkage of the network,
J. Phys. Chem. B, Vol. 105, No. 21, 2001 4985 whose DS abruptly changes from 3092 (unimer-like state, χ < 1.38) to 1545 (micelle-like state, χ > 1.38). Figure 8b also indicates a volume phase transition for the unimer-like state. This is a network collapse for χ values greater than 3.82, involving a change in the DS from 3079 to 1.11. Again, this is the same volume phase transition described by Tanaka.18,19 It corresponds to the change in the slope of the total free energy vs χ curve shown in Figure 8a, and it arises from the change in the relative importance of the various free energy components. In particular, the equilibrium total free energy of the unimerlike network in the swollen state (χ < 3.82) is controlled by the balance between the electrostatic and the elastic energies, whereas the minimum total free energy of the collapsed unimerlike network (χ > 3.82) is determined by the balance between the electrostatic energy and the mixing energy. It is noteworthy that all the DSs shown in Figure 8b, except those of the collapsed unimer-like network for χ > 3.82, are independent of χ. This is because the equilibrium total free energy, the equilibrium polymer volume fraction and the equilibrium DS are all determined from the competition between the electrostatic energy and the elastic energy, without involving the mixing energy which is χ-dependent. The unimer-to-micelle transition illustrated in Figure 8 is a very interesting one because, unlike the unimer-to-micelle transitions observed in the previous Figures (with the exception of that at ζ ) 3 in Figure 7), this transition involves two systems, which are both swollen. In other words, neither of the states involved is collapsed. Volume phase transitions between two swollen states have been reported for random polyampholyte gels.38 Analogous to the results of this work on coValent ionic amphiphilic networks are those of Khokhlov and co-workers15,16 on noncoValent ionic amphiphilic networks consisting of ionic associating polymers with hydrophobic end blocks. When the electrostatic forces become gradually stronger than the hydrophobic forces, the covalent networks expand either continuously or discontinuously, while the noncovalent networks first expand and then dissolve to give polymer clusters of finite size which are finally converted to a homogeneous polymer solution. Thus, the balance between electrostatic and hydrophobic forces is crucial both for covalent and noncovalent networks. Conclusions The present theory predicts interesting transitions between a unimer-like state (disordered) and a micelle-like state (ordered) for model networks based on ionic-hydrophobic block copolymers upon the variation of the degree of ionization, the solvent quality and the copolymer composition. These transitions are accompanied by discontinuous changes in the degree of swelling of the network. Careful experiments are required to verify these predictions. This theory is also capable of reproducing the results of previous models predicting the discontinuous collapse of networks based on charged statistical copolymers. Future theoretical work will involve the extension of the model by relaxing several of the assumptions made. For instance, the interaction parameter between the solvent and the hydrophilic block can be assigned a positive (nonzero) value. Another possible extension is the consideration of nonspherical morphologies for the microphase-separated networks, including cylinders, lamellae, bicontinuous, and reversed structures.26,39,40 Acknowledgment. This work was supported by the University of Cyprus Research Committee (grant 2000-2003).
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