Solid-fluid mixtures and hydrogels

Transcription

1 Chapter 6 Solid-fluid mixtures and hydrogels Ahydrogelisapolymernetworkfilledwithasolvent(fluid). Thismeansthatitcanchange volume by means of absorbing (or expelling) solvent. A dry gel can change its volume by more than 100 fold is immersed in water; this phenomenon is called swelling. Typically, the swelling ratio depends on both the polymer/solvent and the elasticity of the polymer. If the polymer is too sti or the a nity is too low, then swelling will be weak. In contrast, low elasticity and high a nity will favor high swelling. In other words, one may think of swelling as the result of the competition between the force that make solvent particle mix with polymer chain (measured by the mixing energy and the force that prevents polymer chains from stretching (measured by the elastic energy). To capture this competition, we will therefore work with a combined free energy E, that possess both contributions in the form: E = E e + E mix (6.1) where the elastic potential was derived in chapter 4 for permanent polymer networks. Let us thus explore the form of the mixing energy, starting from a simple system of two particles (representing polymer and solvent) and their propensity to mix. 6.1 Free energy of mixing To derive an expression for the mixing energy, we take here a theoretical approach in which water and polymer chains are represented as a mixture of solvent (water) and solute particles (polymer chains) whose behavior is mediated by physical interactions. To simplify the approach, we make two key assumptions: The particles are circular and have the same volume. Thisassumptionwillberelaxed at the latter stage to account for the elongated shape of polymer chains. The particles are distributed on a regular lattice as shown in Fig Further assuming that the mixture is saturated with these two components, the total volume of the mixture can be related to the number M of particles by: V = M (6.2) 63

2 64 CHAPTER 6. SOLID-FLUID MIXTURES AND HYDROGELS Figure 6.1: Lattice model representing the location of M p solute molecules and M s solvent molecules on M = M p + M s lattice sites. On the right is a polymer solution, where solute molecules are connected to form long chains of N molecules. It is however more convenient to work with intensive quantities, and in particular the volume fraction of solvent : = V s V = M s M (6.3) where M s is the number of solvent particles and M is the total number of particles. The Boltzmann formula According to Boltzmann, the free energy of a system is related to the energy of all possible states. The energy of a given state (i.e., the one showed in (6.1)) depends on the interactions between each particle and its neighbor. We can identify three types of neighboring pairs: (1) solvent-solvent interaction " ss,(2)solute-solventinteraction" sp and (3) solute-solute interaction " pp. By definition, these interaction energies are negative if attractive ("<0) and positive (" >0) if repulsive. For a given state, one can count the number of each pair M i ss, M i sp and M i pp and find that the total interaction energy for this so-called macrostate i is: E i = M i ss" ss + M i sp" sp + M i pp" pp (6.4) The total energy F is usually written in the form: F (, T )= k B Tln(z) where z = X i Ei exp k B T (6.5) is the partition function. Computing all possible combinations and the associated sum for a large number of particle would be extremely challenging. To bypass this di culty, one can

3 6.1. FREE ENERGY OF MIXING 65 use another level of approximation, known as the mean field approximation that consists of evaluating an average value Ē of the above sum. In this case, the partition function reads: Ē z = Wexp (6.6) k B T where W is the number of di erent macrostates. In this case, the free energy (6.7) becomes: F (, T )= TS + Ē where S = k Bln(W ) (6.7) The free energy is therefore comprised of two terms, that are associated with ehir own physical interpretation. The entropic term S is only a function of the possible number of macrostates and should therefore depend on the size of the system and the volume fraction. The enthalpic term Ē depends on the interaction energies and will therefore be very sensitive on whether solvent and solute tend to attract one-another. Let us derive an expression for each of these two terms Enthalpic term. As discussed above, the enthalpic term involves analyzing the di erent types of possible macrostates, and for each one of them, estimating the number of p-p, s-s and s-p pairs. For this, one first need to calculate the coordination number z, which measures the number of potential number for each lattice space. For instance, on a two-dimensional square lattice, for which we consider both side and diagonal boxes as neighbors, the coordination number is z =8. Thus,foragivenparticle,thenumberofconnectionswithasolvents is, on average z while the number of connections with a solute particle is (1 )z. Next, to determine the number of pair-neighbors on a lattice of M particles, M ss = 1 2 M s z = 1 2 Mz 2 (6.8) M pp = 1 2 M p z(1 ) = 1 2 Mz(1 )2 (6.9) M pp = M p z(1 ) = Mz (1 ) (6.10) The total energy (6.4) in the system is thus: Ē = 1 2 Mz " ss 2 + " pp (1 ) 2 +2" sp (1 ) = 1 2 Mz[ " (1 )+" pp + " ss (1 )] (6.11) where " = ss + pp 2 sp is the energy associated with the recombination in which two particles of the same species become neighbors (Fig. 6.2). In other words, if this energy is positive, this recombination is not favorable and the solution will mix. If this energy is negative, the recombination is favorable and and particle of the same species will segregate. We will have demixing. ow, by omitting the linear terms in (6.11) (they are not relevant in this context), we obtain: Ē = Mk B T (1 ) (6.12)

4 66 CHAPTER 6. SOLID-FLUID MIXTURES AND HYDROGELS where the polymer solvent interaction parameter is: We note the following. = 1 z " 2 k B T (6.13) if >0, the term is positive and the solution will tend to mix. if <0, the term is negative and the solution will tend to demix. Phase separation should be observed between the two species. Figure 6.2: Change of enthalpic energy from the motion of two solute particles getting to adjacent lattice spaces Entropic term. To derive an expression for the entropic term, we use combinatorial to estimate the number of state W for a lattice with M spots, M s being occupied by the solvent (and thus M M s being occupied by the solute): W = M! M s!(m M s )! (6.14) This can be simplified for large M by using the Stirling formula ln(m!) = Mln(M) M, yielding the following expression for the entropy: S = k B M [ ln( )+(1 )ln(1 )] (6.15) When polymer chains are considered, instead of single particle, a similar approach can be found based on the schematic of Fig. XX. By introducing the number N of monomer in a chain, it can be shown that the corresponding entropic term becomes: apple S = k B M ln( )+ 1 (1 )ln(1 ) (6.16) N It can be seen here that since long polymer chains can take less configuration, their entropy is reduced by a factor 1/N. In the case where the chain is made of a single monomer, this formula reduces to (6.15). In the other end of the spectrum, when polymer chain are very long (N!1), equation (6.16) only contains the contribution of the solvent (terms in ).

5 6.1. FREE ENERGY OF MIXING Mixing energy The total energy of mixing for a system of M particles (with solvent volume fraction )is found by adding up the entropic and enthalpic contributions. It is however more convenient to express this energy in terms of a volume density, i.e. we divide it by the total volume V of the solution (or gel) to obtain: mix = k BT apple ln( )+ 1 (1 )ln(1 )+ (1 ) (6.17) N where = N/V is the number density. Let us known explore how the above form of the mixing energy predicts the behavior of the polymer-solvent solution. We are particularly interested in identifying when the solution is prone to mixing or demixing as this is what dictates the swelling behavior of hydrogels. For this, let us start by plotting (Fig. 6.4) the free energy of mixing for polymers with small chains (N! 1andincreasingvaluesofthepolymer-solventinteractionparameter. For small values of (for instance =1.7), we see that the energy has a convex shape, while when increases to a value above 2, we see that part of this curve becomes concave. We show below that this may, in certain cases, lead to demixing. When the chain increases (see N = 30 in Fig. 6.4), we see that the minimum energy is shifted to lower solvent volume fractions. We still however observe the evolution of the cruve from convex to non-convex as the polymer-solvent parameter increases. Figure 6.3: Mixing energy in terms of the solvent fraction when N =1andN =30andfor three di erent values of the polymer-solvent interaction parameter ( =1.7, 2and2.3).

6 68 CHAPTER 6. SOLID-FLUID MIXTURES AND HYDROGELS 6.2 Elastic and mixing forces Free energy of a hydrogel To write the total free energy for a hydrogel, let us now consider a volume V 0 of polymer in its dry state, which after being infiltrated by the solvent deforms to a new volume V = QV 0, where Q is the selling ratio. Referring to (6.18), the elastic energy may is then related to the elastic potential by E e = V 0 (note here that the elastic potential is defined by reference (or dry) volume V 0 ). In contrast, the mixing energy density mix is defined by current volume VandthetotalmixingenergyisthusE mix = V mix. This means that the free energy for this volume is: E = V 0 ( e + Q mix ) (6.18) To obtain a closed form of the final fre energy density for hydrogels, we then make the following observations: The free energy density is measured per unit reference volume V 0,i.e. = E/V 0. The swelling ratio can be directly related to the solvent volume fraction by the fact that the polymer volume V p = V 0 =(1 )V does not change during swelling. This means that Q = V V 0 = 1 1 (6.19) The polymer chains are typically long in a connected hydrogel network and N!1. In this case, the mixing energy density (6.17) can be approximated as: mix = k BT [ ln( )+ (1 )] (6.20) Finally, the total free energy density (per dry volume) for the hydrogel becomes: = G (tr(c) 3) + 1 k B T [ ln( )+ (1 )] 2 {z } 1 {z } e m (6.21) where the elastic energy is that derived from rubber elasticity and G = ck B T is the shear modulus Swelling equilibrium Let us consider a hydrogel specimen of volume V 0 = a 3 and and submerge it in water for a long time. Because of the mixing force, the specimen swells until it reaches a steady state volume V = QV 0. The swelling equilibrium, characterized by the ratio Q is such that the hydrogel s free energy is minimized. We =0 (6.22)

7 6.2. ELASTIC AND MIXING FORCES 69 We see here that the swellling ration results from a competition between elastic forces f e e /@Q and mixing forces f m m /@Q. Theforcebalancecanbesimplywritten: f e (Q) =f m (Q) (6.23) Free swelling Let us consider a small cube of dry hydrogel, with side length a and volume V 0 = a 3 and imagine that we submerge it in water for a long time. Because of the mixing force, we expect that specimen to swell until it reaches a steady state volume V = QV 0. We here attempt to determine the steady-state value of the swelling ratio Q, or swelling equilibrium. As the hydrogel swells, its shape remains cubic, but the side length becomes a where is the stretch ratio from swelling. We here aim to determine the value of that minimizes the free energy of the gel. For this volume change, the deformation gradient F is diagonal with diagonal values therefore immediately obtain. This means that the volume change is:. We Q = V V 0 = det(f )= 3 (6.24) Now using (6.19), the stretch ratio can be related to the solvent volume fraction by: 1 = Q 1/3 = 1 1/3 (6.25) From this equation, one sees that if =0(nosolvent),thecubedoesnotswell( =1). In contrast, if the the solvent completely invades the polymer (! 1, the stretch takes very large values. From the value of,onecandeductthegreen-lagrangestraintensorand obtain the elastic energy as: e = 3G = 3G 2 Q 2/3 1 (6.26) while the mixing energy takes the same form as that shown in (6.23). m = k BT apple Q 1 (Q 1) ln Q From these two expression, we can find the elastic and mixing force as: + (6.27) Q f e = GQ 1/3 (6.28) apple k B T Q 1 f m = ln + 1 Q Q + (6.29) Q 2

8 70 CHAPTER 6. SOLID-FLUID MIXTURES AND HYDROGELS Figure 6.4: Elastic and mixing energies in terms of the swelling ratio Q for gels with di erent values of the polymer-solvent parameter. Equilibrium swelling is determined by the intersection between the two curves. 6.3 Phase separation and hydrogel instability Poly(N-isopropylacrylamide) (PNIPAm) is a well-studied hydrogel that undergoes a volume transition at a lower critical swelling temperature (LCST) of 32C (305 K). At a low temperature, it is hydrophilic, imbibes water to achieve the swollen state. When the temperature is increased above the LCST, the gel becomes hydrophobic, expels more than 90 % of its water and reaches the unswollen state. This is because the polymer-solvent interaction parameter for NIPAm is not a constant, but depends on both the temperature and the swelling ratio of polymer. The value of has been experimentally measured and fitted by Afroze et al. with the form: = 0 (T )+ 1 (T ) Q (6.30) where 0 = A 0 + B 0 T, 1 = A 1 + B 1 T and the fitting parameters take the values A 0 = , B 0 = K ( 1), A 1 =17.92 and B 1 = K ( 1). The variation of leads to the discontinuous phase transition of the NIPAm at the LCST In other words, past a critical temperature (T c 305K), the gel suddenly shrinks to about 1/10 of its initial volume. To understand this phase transition, we plot in Fig. 6.5 the mixing and elastic forces as a function of the swelling ration Q for three di erent temperatures. We observe that if T<T c, the curves intersect at only one location, which indicate a single solution at a swelling ratio Q 10. As the temperature increases to a value near T c however, we see that the curves intersect at 3 distinct points. The hydrogel therefore possess three equilibrium points at

9 6.3. PHASE SEPARATION AND HYDROGEL INSTABILITY 71 Figure 6.5: Left: Mixing and Elastic forces for a NiPam hydrogel in terms of its swelling ratio for three temperatures (below, at and above the critical temperature. Right: Corresponding free energy of a, showing a minimum when elastic and mixing forces are equal. high, intermediate and low swelling ratios. We will see that one of these solutions is usually realized, depending on conditions and temperature history. Finally, once the temperature is high, enough, we again observe a single solution but at a low swelling ratio (Q 1). Similar information can be obtained by looking at local minima of the total free energy (Fig. (6.5), right), where we see the curve displaying one minimum, 2 minima and 1 minimum, respectively as the temperature is increased around its critical value. We however see that the three solutions obtained from the balance of mixing and elastic force corresponds to two minima and one maximum. Since a local maximum is usually associated with an unstable solution, the solution at intermediate swelling will usually not be realized. To explore how these solution depend on the gel s history, we plot in Fig. (6.6), the value of the swelling ratin Q = V/V 0 corresponding to minima of the free energy in terms temperature. We obtain an inverted S curve that is characteristic of a bifurcation. When the hydrogel is heated from a low temperature (T <T c ) to a high temperature, the solution follows the upper branch of the swelling curve as shown in Fig. (6.6). In this case, the transition occurs at a temperature around T 306K. Bycontrast,ofthegeliscooleddown from a temperature T>T c,thesolutionfollowsadi erentbranchoftheswellingcurveuntil it reaches an instability at T 304.5K. This illustrate the hysotry dependence of the gel response around its transition temperature.

10 72 CHAPTER 6. SOLID-FLUID MIXTURES AND HYDROGELS Figure 6.6: Heating and cooling path for a NiPam hydrogel around the critical temperature.