J. Phys. Chem. B 2006, 110, 12721-12727
12721
Effect of Added Free Polymer on the Swelling of Neutral Microgel Particles: A Thermodynamic Approach Alexander F. Routh,*,† Alberto Fernandez-Nieves,‡ Melanie Bradley,§ and Brian Vincent§ Department of Chemical and Process Engineering, UniVersity of Sheffield, Mappin Street, Sheffield, S1 3JD, United Kingdom, Group of Complex Fluids Physics, Department of Applied Physics, UniVersity of Almeria, 04120 Almeria, Spain, and School of Chemistry, Bristol UniVersity, Cantocks Close, Bristol BS8 1TS, United Kingdom ReceiVed: October 14, 2005; In Final Form: March 3, 2006
Microgel particles based on poly (N-isopropylacrylamide) have been shown to display an initial swelling behavior, followed by a collapse, with increasing concentration of added poly(ethylene oxide), PEO, chains. This paper considers the thermodynamic reasons for the observed expansion and subsequent shrinkage of the particles. At low concentrations of PEO, the free chains permeate into the microgel particles and cause an increase in osmotic pressure, expanding the particles. At higher concentrations of PEO, the particles are saturated and an increase in osmotic pressure in the external phase causes the particles to collapse again. The calculated magnitude of swelling and the effect of PEO molecular weight are, at least qualitatively, in agreement with the experimental observations reported elsewhere.
1. Introduction Microgel particles based on poly(N-isopropylacrylamide), PNIPAM, have been studied for a number of years.1-3 They exhibit a volume phase transition around 32 °C. At higher temperatures, the particles are collapsed and rigid, and the flow of solvent in their vicinity is such that the microgel particles behave as impermeable solid particles.4 At lower temperatures, the interaction between the aqueous solvent and polymer chains becomes favorable, and the particles swell to maximize polymersolvent contact. A cross-linking agent (typically methylenebisacrylamide) is required to prevent complete dissolution and to keep the particles as distinct entities. Because of their thermal triggering, microgels are attractive for a number of smart material applications, such as adsorption of metal ions5 and drug delivery.6 The swelling of microgel particles allows controlled variation of the interparticle interaction potentials, and this has led to the use of PNIPAM particles as models for aggregation studies.7-13 The phase behavior of these particles has also been extensively studied.14-21 For a microgel particle, the chemical potential of water inside and outside the particle must balance at equilibrium. The internal chemical potential is dependent on the local polymer concentration, the interaction between the polymer and solvent, characterized by the Flory Huggins χ parameter, and the elasticity of the chains, which is dependent on the cross-linker concentration. The chemical potential outside the microgel is dependent on the external solvent conditions, e.g., salt concentration. The thermodynamics of charge-controlled swelling has been considered previously,22 and this paper follows a similar approach in that a chemical potential balance of each component is * To whom correspondence should be addressed. Email:
[email protected]. Present address: Department of Chemical Engineering and BP Institute, University of Cambridge, Madingley Rise, Madingley Road, Cambridge CB3 0EZ. † University of Sheffield. ‡ University of Almeria. § Bristol University.
applied. The effect of free poly(ethylene oxide), PEO, chains on microgel particles has also been explored with osmotic deswelling examined theoretically.23 Recent work by Bradley et al.24 has shown that PNIPAM microgels swell further at 25 °C in the presence on PEO. The microgel particles contained up to 10 wt % acrylic acid (AAc), which provided a driving force for the PEO chains to enter the microgel particles; this arises from PEO-AAc hydrogen bonding. The swelling was found to be dependent on the PEO molecular weight. For molecular weights of PEO up to 100 000 g mol-1, the PNIPAM particles displayed a swelling at low PEO concentrations and then a subsequent decrease in size as the PEO concentration increased further. For larger molecular weight PEO chains, the PNIPAM particles did not swell and merely displayed a collapse in size. It was observed that, for those cases that displayed a swelling, the concentration of PEO at which the maximum in particle size was observed decreased with increasing PEO molecular weight. The reason for the swelling is infusion of PEO into the microgel particles, causing an increase in osmotic pressure. Further addition of PEO results in the external phase osmotic pressure increasing and, hence, the subsequent decrease in size. Thermodynamically, at equilibrium, the chemical potential of water, inside and outside the microgel particles, must balance, as must that of the PEO chains. In this paper, the thermodynamics of swelling and the subsequent collapse of PNIPAM microgel particles in the presence of PEO chains is considered. The different contributions to the Gibbs energy changes in the system are explained, and the chemical potential balances, discussed above, provide a prediction for the swelling and collapse behavior, as seen experimentally. The model also captures the observed dependence on free polymer molecular weight. 2. Theory Consider a microgel particle in a solution containing water and PEO. The volume fractions of water, NIPAM, and PEO
10.1021/jp0558831 CCC: $33.50 © 2006 American Chemical Society Published on Web 06/02/2006
12722 J. Phys. Chem. B, Vol. 110, No. 25, 2006
Routh et al.
inside the microgel particle are φ1, φ2, and φ3, respectively. In the external phase, the corresponding volume fractions are: ext ext φext 1 , φ2 , and φ3 . At equilibrium, the chemical potential of water inside and outside the microgel and also the PEO chemical potential inside and outside will balance. Writing the total Gibbs energy of the system as G, the chemical potentials of water and PEO follow, respectively, as
µ1 )
∂G ∂n1
µ3 )
∂G ∂n3
(1)
where n1 is the total number of moles of water and n3 is the number of moles of PEO. Equating chemical potentials inside and outside the microgel is equivalent to balancing the internal and external osmotic pressure.22 In addition to n1 moles of water and n3 moles of PEO, there are n2 moles of NIPAM. If the volume of a NIPAM molecule relative to water is x and the volume of a PEO molecule relative to water is y, then the volume fractions follow as
φ1 )
n1 n1 + xn2 + yn3
(2)
φ2 )
xn2 n1 + xn2 + yn3
(3)
φ3 )
yn3 n1 + xn2 + yn3
(4)
To calculate the Gibbs energy inside the microgel, there are a number of terms to be considered, namely, the mixing of the three components, the chain elasticity, and the confinement of PEO chains inside the microgel. For the external Gibbs energy, only the mixing of PEO and water must be accounted for. Each term is considered separately. 2.1. Mixing Contribution. When water, NIPAM, and PEO are mixed, there are enthalpic terms dependent on the FloryHuggins interaction parameters χ12, χ13, and χ23 and entropic components to the change in chemical potential ∆G, such that
∆Gmix ) kT[n1 ln φ1 + n2 ln φ2 + n3 ln φ3 + xn2φ1χ12 + yn3φ1χ13 + xn2φ3χ23] (5) The mixing contributions for the chemical potentials inside the microgel follow as
µmix 1 )
mix
∂∆G ∂n1
(
) kT 1 - φ1 + ln φ1 + χ12φ2(1 - φ1) +
)
φ2 φ 3 χ13φ3(1 - φ1) - χ23φ2φ3 - (6) x y µmix 3 )
∂∆Gmix ) ∂n3 kT(1 - φ3 + ln φ3 + φ1y(χ13(1 - φ3) - 1) + φ2y(χ23(1 - φ3) - χ12φ1)) (7)
2.2. Elastic Contribution. When swelling occurs, the PNIPAM chains extend and thus lose entropy. The entropy change is dependent on the average number of NIPAM chains per particle, Nc, the molar volume of water ν, the original volume of the
particle V0, and the volume fraction of PNIPAM in this state, φi2. The chemical potential of the water associated with the chain elasticity inside a particle, with polymer volume fraction φ2, has been derived explicitly previously.22
µelas 1 )-
( ())
νNckT φ2 φ2 - i V0 2φi φ2 2
1/3
(8)
2.3. Confinement Contribution. As the PEO is forced to be inside pores, it must adopt a configuration that is smaller than its desired size. Hence, there is a loss of entropy for confining the PEO chain. Following Young and Lovell 25 and Flory 26 for a polymer with y segments of length l in a box of size L, the change in entropy due to the confinement is given by
3 ∆Gcon ) T∆Scon ) Tn3yk[λ2 - 1] 2
(9)
where the factor λ relates the size of the unperturbed chain, taken as y3/5l, to the dimensions available to the chain. The latter is taken as a swelling-dependent box size L, corrected for the volume of PEO already confined.
λ)
y3/5l L(1 - φ3)1/3
(10)
We assume the swelling is isotropic and relate L to the polymer volume fraction by
()
φi2 L ) L0 φ2
1/3
(11)
where L0 refers to the confining box size at the polymer volume fraction φi2, corresponding to the absence of PEO. Hence the chemical potential follows as
µcon 3 )
[ (
φ2 ∂T∆Scon 3y y6/5l2 ) kT 2 i ∂n3 2 L0 φ2(1 - φ3)
) ] 2/3
-1
(12)
Different forms for the confinement Gibbs energy are considered explicitly later. 2.4. External Phase. Externally, there is only PEO and water, ext ext with φext 1 + φ3 ) 1 and φ2 ) 0. The mixing term is given by
∆Gext mix ) kT[n1 ln φ1 + n3 ln φ3 + χ13n1φ3].
(13)
The chemical potentials follow as
(
(
ext µext 1 ) kT ln φ1 + 1 -
1 ext 2 φ + χ13(φext 3 ) y 3
)
)
ext ext ext 2 µext 3 ) kT(ln φ3 - (y - 1)φ1 + yχ13(φ1 ) ).
(14) (15)
2.5. Overall Balance. The balance at equilibrium relates the chemical potentials and the total volume fractions ext φ1 + φ2 + φ3 ) 1; φext 1 + φ3 ) 1
(16)
mix elas µext 1 ) µ1 + µ1
(17)
mix con µext 3 ) µ3 + µ3
(18)
Finally, we obtain the particle size R and particle volume V
Swelling of PNIPAM Microgels by Penetrating PEO
J. Phys. Chem. B, Vol. 110, No. 25, 2006 12723
from the solution for the internal volume fraction. Hence,
() ()
R V ) R0 V0
1/3
)
φi2 φ2
1/3
(19)
Equations 11 and 19 imply adoption of the affine approximation, where the microgel particles are assumed to swell isotropically with a uniform volume fraction spatially. 2.6. Parameter Values. There are numerous parameters occurring in the above equations, values for which are considered below: The average volume fraction φ2 of PNIPAM inside the microgel particles is well-defined. However, the PNIPAM relative molar volume (x) and number of moles (n2) require some consideration. n2 is not a well-defined parameter, but it does not arise in the equations for the chemical potential. Because of the cross-linking of the PNIPAM chains, x is very large. In the mixing term (eq 6), φ2/x is therefore taken as zero, which simplifies the equation. The Flory Huggins χ parameters for the interaction between NIPAM and water (χ12) and water and PEO (χ13) are additional variables. In the experiments reported by Bradley et al.,24 the microgels were at room temperature and, therefore, in the swollen state. This implies that the interaction between NIPAM and water is favorable; hence, a value of χ12 of 0.4 is taken as reasonable. In addition, PEO freely mixes with water, implying a favorable interaction, and so χ13 is also taken as 0.4. Although the values used seem reasonable, it is important to recognize that they are merely representative and can be altered to any specific problem. Confinement Box Size and Repeat Length of PEO: L0 and l. We have assumed that the size of the pores inside the microgel particle increases as the particle swells, following the affine approximation. This is supported by data from Saunders,27 which shows a microgel mesh size to scale linearly with the overall particle size. It is not sensible to have a negative confinement chemical potential, because this would imply that the added polymer gains entropy when confined. Hence, the size of an unhindered chain y3/5l must be bigger than the confinement size, L0. The confinement size before addition of PEO, L0, is taken as 10 nm and the PEO repeat length size, l, as 1 nm. It therefore follows from solving y3/5 > 10 that y > 47. This requirement is merely to ensure that the confinement chemical potential is positive and physically reasonable. The case for y < 47 is considered separately, because the confinement term is zero (rather than negative) in these cases. Although the confinement value of 10 nm is larger than mesh sizes reported previously27 and the repeat length of 1 nm may be too large, it is important to note that the ratio of polymer to confinement size is the important parameter, rather than just the confinement. (This is discussed later in relation to Figure 7.) The other variables are the PEO/NIPAM interaction parameter, χ23, the PEO molecular weight, y, and the external PEO concentration, φext 3 . The effect of these is considered explicitly in the next sections. 3. Solution with No PEO Present When there is no PEO present, the PNIPAM particle size is only determined by the balance between polymer mixing and chain elasticity. This scenario provides the base solution with φ2 ) φi2 Equating the water chemical potential inside and outside the microgel, eqs 16 and 17, respectively, with the
Figure 1. Volume fraction of polymer in the microgel as a function of the degree of cross-linking for φ3 ) 0. Solution from eq 20.
Figure 2. Swelling of microgel particles for different χ23 values (effectively the “driving force”) for an initial PNIPAM volume fraction of φi2 ) 0.5 and a PEO molecular weight equivalent to y ) 50.
assumption that x .1 one obtains
φi2 + ln(1 - φi2) + χ12(φi2)2 +
νNc )0 2V0
(20)
This is equivalent to stating that the amount of water in the microgel originally is dependent on the degree of cross-linking. The relation between the polymer volume fraction in the absence of PEO, φi2, and the degree of cross-linking νNc/2V0 is shown in Figure 1 for several interaction parameters χ12. As can be seen, and as is expected, an increase in νNc/2V0 (the degree of cross-linking) results in an increase in φi2. 4. Solution with PEO Present Assuming the values discussed in Section 2.6, one may obtain solutions to eqs 16-18 for a range of initial values of φi2 (which is determined by the degree of cross-linking and χ12, as we have seen), χ23, and y. Solutions were obtained using Mathematica. 4.1. The Effect of Added PEO Concentration. Figure 2 shows how the swelling ratio of a microgel particle varies with φext 3 , the added PEO volume fraction, for various values of χ23 (PEO/NIPAM interaction) and for the case where y ) 50. The
12724 J. Phys. Chem. B, Vol. 110, No. 25, 2006
Figure 3. Swelling of microgel with different initial polymer volume fractions for χ23 ) -2 and y ) 50.
main feature of these plots is the presence of a maximum. The swelling is on the order of 3% in particle radius for χ23 ) -2, and the maximum in particle size occurs around a volume fraction of PEO of 0.25. Upon increasing χ23, the maximum swelling occurs at higher PEO concentrations. ext R/R0 must be unity at φext 3 ) 0, but the low φ3 region is not well captured with the solution to the base equations described above. This is due to the failure of the basic Flory Huggins model for dilute polymer solutions. This point is discussed in detail later. After the initial swelling, there is collapse of the microgel particles at higher PEO concentrations, as observed by Bradley et al.24 This is due to the confinement term starting to dominate; the PEO does not enter the particle, causing an increase of the external osmotic pressure. This type of deswelling behavior has also been observed by other authors experimentally.23,28 4.2. Effect of Initial PNIPAM Volume Fraction (and, hence, Cross-Linker Concentration). As we have seen, the cross-linker concentration in the microgel particles, in the absence of added PEO, is the primary determinant of the (average) volume fraction of PNIPAM in unswollen microgels, φi2 (Figure 1). However, the absolute value of φi2 is difficult to obtain, as the microgel particles contain water molecules even in the collapsed state, at higher temperatures. The addition of ethanol has been shown to reduce the size of microgel particles even below this collapsed state.29 It would seem that in the hightemperature collapsed state, the volume fraction of water may be as high as 50% (φi2 ) 0.5).30 Figure 3 shows the variation in swelling ratio of a microgel particle with added PEO volume fraction (φext 3 ), for a range of initial PNIPAM volume fractions (φi2) in the microgel particles. As the initial volume fraction of PNIPAM (φi2) is increased: (i) the degree of swelling increases, for a given value ext of φext 3 , and (ii) the maximum swelling occurs at higher φ3 values. This is easily understood by recognizing that the more NIPAM segments present in the microgel particles, the more PEO molecules that can enter due to the NIPAM-PEO hydrogen bonding, as expressed through the negative value of χ23 that reflects this attraction. 4.3. Effect of PEO Molecular Weight. Increasing the size of the polymer chain, y, has dramatic effects on the swelling behavior. As discussed in section 2.6, the size of the polymer chain is limited, at this stage, to be greater than 47. Section 4.4
Routh et al.
Figure 4. Effect of PEO molecular weight (as expressed by y) on microgel particle swelling for φi2 ) 0.5 and χ23 ) -2.
considers the case with very large molecular weight, and in this section, the effect of intermediate values is examined. Figure 4 shows the size of a microgel particle, as a function of φext 3 , for an initial PNIPAM volume fraction of φi2 ) 0.5, an interaction parameter χ23 ) -2, and a range of y values, reflecting the MW of the added PEO chains. As the PEO size is increased, the degree of swelling decreases and the position of this maximum moves to lower PEO concentrations. This result should be compared with Figure 10 from the paper by Bradley et al.24 Exploring the maximum in the swelling behavior further, Figure 5a,b shows the PEO concentration at which the maximum swelling is observed and the maximum degree of swelling, both as a function of PEO size. As intuitively expected, as y increases, the free polymer volume fraction needed to reach maximum swelling decreases; moreover, this decrease is linear within the y range probed. Perhaps less intuitive is the decrease in the maximum swelling degree as y increases (Figure 5b). This arises as a result of the larger Gibbs energy confinement contribution for a larger PEO molecule; it restricts the number of PEO molecules that actually enter the microgel particle, and thus, the microgel particles begin to deswell earlier from a smaller degree of swelling. 4.4. Collapse of Microgel Particles with Very Large Molecular Weight PEO. If the limit of large free polymer molecular weight is considered such that the chains cannot fit inside the microgel particles, the base equations may be simplified with the assumption that y . 1. The PEO balance cannot be satisfied, implying that no PEO exists inside the microgel particles and φ3 ) 0. For the water balance, one obtains
φ2 + ln(1 - φ2) + χ12φ22 -
[ ()]
νNc φ2 φ2 - i i V0 2φ φ2 2
1/3
)
ext ext 2 φext 3 + ln(1 - φ3 ) + χ13(φ3 ) (21)
Using the previous expression linking the cross-linker concentration to the initial PNIPAM volume fraction, eq 20, one obtains an algebraic expression for the PNIPAM volume fraction as a function of the external PEO volume fraction. The result is shown in Figure 6 for χ12 ) χ13 ) 0.4 and an initial polymer concentration of φi2 ) 0.5. As expected, as more PEO is introduced into the system, the microgel particles contract because the osmotic pressure in the external phase increases.
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J. Phys. Chem. B, Vol. 110, No. 25, 2006 12725
Figure 7. Swelling of microgel particles for φi2 ) 0.5, χ23 ) -2, y ) 50, L0 ) 10 nm, and various values for l.
Figure 5. (a) Value of the external PEO concentration at the maximum degree of swelling, as a function of y (in effect, the PEO molecular weight), for φi2 ) 0.5 and χ23 ) -2. (b) maximum swelling as a function of y (in effect, the molecular weight) for χ23 ) -2 and φi2 ) 0.5.
swelling curves for a range of repeat chain lengths. As the size increases, the confinement term becomes larger and the swelling decreases. For sufficiently large values of l, no swelling is observed and the particle merely contracts with PEO addition. 4.6. Solutions when the PEO Concentration is Very Small (φext 3 , 1). The Flory Huggins theory itself is a mean field theory that only strictly applies for concentrated polymer solutions, where there is an assumed uniform segment concentration throughout the volume space of the solution. Not surprising, it breaks down in the limit of small φext 3 . Despite this, we obtain a solution for the swelling in the limit of very small φext 3 and deduce that the concentration range over which this applies is so small that a more interesting case is where the solvent polymer interactions are considered. This is captured by the governing equations, without the need for any asymptotics. The governing equations are singular when the PEO concentration is small. The solution to the governing equations fails to adequately cover this regime because the PEO balance is divergent due to the logarithm term in the mixing both inside and outside the microgel. For sufficiently small values of φ3 and φext 3 , the divergent terms must balance, providing the solution: φ3 ) φext for φ3ext , 1, and simplifying the 3 expressions considerably. The water balance remains the same with φ3 replaced by φext 3 . This solution is only valid when the logarithm term in eq 15 is dominant over the terms containing y. Hence,
ln φ3ext < y(χ13 - 1)
Figure 6. The collapse of microgel particles for the case of y . 1 for χ12 ) χ13 ) 0.4 and φi2 ) 0.5.
4.5. Effect of Repeat Chain Length. The size of the polymer chain needs to be compared to the confinement size. This is done through the dimensionless group y3/5l/L0. Figure 7 shows
(22)
- (1-χ13)y. Numerically this is φext < which provides φext 3 < e 3 -13 10 for χ13 ) 0.4 and y ) 50. Of more interest is when the terms of order ln φext 3 and y balance. Figure 8 shows a solution to eq 16-18, although only considering very small external PEO concentrations. Figure 8 should be compared with Figures 2 and 4, with the important point being that in these figures, the swelling is actually zero (R/R0 ) 1) with no PEO in the system. 4.7. Different Confinement Terms. The confinement term used to generate the predictions of swelling is based on the affine approximation, i.e., a cage that expands as the microgel particle expands. This results in a confinement term that decreases as more PEO enters the particle. In addition, the volume of the microgel particle already occupied by PEO is used to calculate
12726 J. Phys. Chem. B, Vol. 110, No. 25, 2006
Figure 8. Swelling of microgel particles for very small PEO concentrations for χ23 ) -2, y ) 48, and φi2 ) 0.5.
the available volume for additional free polymer. Without the correction for PEO occupation, no maximum is observed in the particle size with PEO concentration. In the following discussion, a number of different possible expressions for the confinement are considered. If the cage does not increase with swelling, then the term λ in eq 10 is simply dependent on a constant box size L0. Similarly, it is possible not to include the reduction in available volume due to PEO concentration, (i.e., remove the (1 - φ3)1/3 term). There are four possibilities to explore
Case 1: λ )
y3/5l Constant box size L0
Case 2: λ )
y3/5l Variable box size L
Case 3: λ )
Case 4: λ )
y3/5l L0(1 - φ3)1/3 Constant box size, accounts for PEO exclusion y3/5l L(1 - φ3)1/3 Variable box size, accounts for PEO exclusion
In the previous sections, case 4 has been used to generate the swelling curves. Figure 9 shows the swelling curves obtained assuming the different forms for the confinement term. In this example, the original polymer volume fraction is taken as φi2 ) 0.5, L0 ) 10 nm, χ23 ) -2, and l ) 1 nm. The sensitivity of the swelling to the functional form of the confinement is evident. The only cases that display a maximum in the particle size are the cases that include a term for the volume of PEO inside the microgel hindering further ingression. The swelling is more pronounced in case 4 than in case 2 because the box size mimics the added polymer size behavior; with increased box size, more PEO molecules are allowed to accommodate inside the microgel, further increasing the internal osmotic pressure. 5. Discussion Equations 16-18 are decidedly nonlinear and, therefore, difficult to solve. The balance in the confinement Gibbs energy is crucial. If this term is too small, then there is no restriction on the PEO entering the microgel and unphysically large
Routh et al.
Figure 9. Swelling of microgel particles, for different forms for the confinement chemical potential for L0 ) 10 nm, l ) 1 nm, φi2 ) 0.5, χ23 ) -2, and y ) 48.
Figure 10. Swelling of microgel particles with no confinement term for φi2 ) 0.5.
swelling degrees are observed. The opposite case, when the confinement is too large, results in unphysical solutions, and considering the limit of large molecular weight, as in section 4.4, is the way to obtain solutions. Solutions for very low concentrations of PEO have been obtained separately because of the divergent behavior of the logarithmic terms in the mixing contributions to the Gibbs energy. Although easy to handle as a separate case, the nonconvergence of the base equations in this limit is numerically inconvenient. Some of the physical values of the parameters in the model are not known accurately, and we have simply taken representative numbers to demonstrate the swelling and subsequent collapse of the microgel. For simplicity, the number of equivalent chains is set to be equal to the size of a PEO molecule relative to water. This has implications for the length of each chain, and the estimate of l ≈ 1 nm along with L0 ) 10 nm are merely representative numbers. The most crucial parameter values relate to the confinement term. It is difficult to imagine the physical case where y3/5l < L0, with no confinement term operating, but an example of this case is shown in Figure 10. As expected, the interaction between the PEO and PNIPAM (captured in χ23) is the dominant parameter and the PEO molecular weight plays only a small
Swelling of PNIPAM Microgels by Penetrating PEO role. It is also noticeable that no maximum is observed in the microgel swelling because there is no “cost” in inserting the PEO into the microgel, and thus, the swelling continues over the entire PEO range. At the other end of the scale, when considering confinement, it is surprising that the molecular weight window over which the PEO can fit into the microgel is so small, and this indicates there is something unphysical with our confinement term. The derivation of eq 8 assumes that there are sufficient segments between each cross-link for the PNIPAM chains to adopt random statistics. There are two entropy terms: (i) one associated with the PNIPAM chains (between cross-links); as soon as solvent or PEO enters the particles, the PNIPAM chains stretch away from their assumed random coil conformations; equation 8 does not account for these deviations, which are thus not accounted for in our model; (ii) the confinement of the PEO chains, i.e., they are not free coils as in solution. We put this in as a polymer molecule confined within a volume L in eq 9. This is one limit (and gives rise to the “cage” or “pore” concept), but a problem, which we have not considered, then arises: does the Flory Huggins mean field mixing term hold (eq 7)? That is, are the PNIPAM and PEO chains sufficiently intermixed for mean field conditions to apply? The other limit is to ignore the confinement term per se and just let the PEO chains enter and freely mix with the PNIPAM chains, but then, as we see from Figure 10, thermodynamically there is no upper MW limit on PEO chains entering. Only the elastic term from the crosslinking defines the maximum swelling under these conditions. Thus, “reality” lies somewhere between these two limits. A partially “open” cage or pore, rather than a closed one, would be more realistic, but we do not currently have the correct freeenergy term to handle this situation.31 It is, however, pleasing that the physical reason for the microgel swelling and collapse can be explained. The question remains as to which form the confinement term should take. The assumption of isotropic compression and expansion of the PEO to squeeze the chain inside the particle is needed to provide a tractable confinement term, although this, as discussed above, may be physically unsound. In practice, a stretched-out chain will be able to fit inside a particle of very low porosity and, although entropically very unfavorable, for sufficiently high enthalpic interactions (large and negative χ23) may be the method of PEO induced swelling over the wider molecular weight range observed by Bradley et al.24 An example of this is shown in Figure 7, where the size of the repeat unit l is varied from 1 to 1.05 nm. The swelling curves are all similar in shape although the magnitude of the swelling varies wildly. This is because for lower values of l, the confinement chemical potential is negligible and there is no hindrance to the PEO entering the microgel with the correspondingly large enthalpy gain. For larger l values, the drop off in swelling is very rapid. As argued above, the confinement term rapidly becomes dominant, and thus, the ingress of PEO is unfavorable. It seems likely that our simplistic view of a confinement box overestimates the PEO confinement and, hence, the predicted rapid decrease in swelling that we observe. 6. Conclusions The experimental swelling behavior of poly(N-isopropylacrylamide) microgels by penetrating poly(ethylene oxide)
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