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A lattice model for thermally-sensitive core–shell hydrogels
Journal of Colloid and Interface Science 406 (2013) 148–153
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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
A lattice model for thermally-sensitive core–shell hydrogels Cheng Lian, Dongyan Zhi, Shouhong Xu ⇑, Honglai Liu, Ying Hu Key Laboratory for Advanced Materials and Department of Chemistry, East China University of Science and Technology, Shanghai 200237, China
a r t i c l e
i n f o
Article history: Received 30 January 2013 Accepted 3 June 2013 Available online 18 June 2013 Keywords: Molecular thermodynamic model Core–shell hydrogel Swelling behavior Thermally-sensitive
a b s t r a c t A lattice molecular thermodynamic model for describing the swelling behavior of thermally-sensitive core–shell hydrogels was developed by integrating a close-packed lattice model for mixing free energy and a Flory Gaussian chain model for the elastic free energy. The thermodynamic model is characterized with two parameters: the temperature-dependent exchange energy parameter e and the Topologydependent size parameter V. The input values of both parameters can be obtained by experimental results of pure polymer hydrogels. With the help of proposed model, swelling behaviors of two kinds of hydrogel systems were analyzed: one is a doubly thermally-sensitive core–shell hydrogels (with two LCSTs) and the other comprises a thermally-sensitive hydrogel shell with a hard internal core. We show that the calculated results are in good agreement with the experimental data. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Smart hydrogels are soft materials which can swell through absorbing solvent and deswell through ejecting the solvent as a response to external stimuli (such as temperature, pH, ionic strength, and composition of solvent). These hydrogels have been widely applied in sensors [1,2], immobilized enzymes [3], material extraction [4,5], drug delivery devices [6–8], catalysts [9], and so on. In recent years, novel architectures (such as core–shell structures [10–18] and interpenetrating network structures [19,20]) have been designed to obtain materials with superior properties and functions. The properties of these particles, such as stability, biocompatibility, mechanical strength, and even functionalities, have been improved. In the related theoretical filed, as early as 1953, Flory [21] established the swelling theory of such special network structures based on the polymer solution theory and the rubber theory. Until 1987, Tanaka and his coworker [22] established a thermodynamic model for hydrogels. They considered the chemical potential inside the hydrogel was equal to that of outside hydrogel at swelling equilibrium. From then on, researchers have paid more and more attention to establishing molecular thermodynamic models for hydrogels. For the homogeneous hydrogels, corresponding molecular thermodynamic models for describing their swelling behaviors have been widely reported. Considering the influence of hydrogen bonding, Shenoy et al. [23] combined an association model with a Flory–Rehner approach to develop a new model. Starting from a quasi-chemical partition function [24], which con-
⇑ Corresponding author. Fax: +86 21 64252921. E-mail address: [email protected] (S. Xu). 0021-9797/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jcis.2013.06.005
sidered both the rubber elasticity theories and competition between hydrogen bonding and dispersion forces, Oliveira et al. [25] presented an oriented quasi-chemical thermodynamic model to explain the swelling behavior of hydrogels. Huang et al. [26] developed a molecular thermodynamic model based on a lattice model [27–29] for thermally-sensitive hydrogels. Then, Zhi et al. [30,31] established molecular thermodynamic models for thermal and solvent sensitive hydrogels and also multi-sensitive hydrogels. Jung et al. [32] combined modified double lattice model (MDL) theory with Flory–Erman theory to describe the reentrant swelling behavior of thermally-sensitive hydrogels. Nevertheless, the above models are essentially proposed for homogeneous hydrogels, and the properties of the core–shell hydrogels are fundamentally different. In core–shell hydrogel system, the core and shell are constrained by and correlated with each other, and thus, the swelling process is an inhomogeneous deformation. The thermodynamic study on phase transitions of such hydrogels is rather complicated. With a general yet complex formalism of field theory, Sekimoto and Kawasaki [33] have studied the phase coexistence in hydrogels, and using the Legendre transformation, the inhomogeneous swelling behavior of covalently cross-linked polymers is investigated by Marcombe et al. [34]. Furthermore, the thermodynamic theory of Gibbs and the Flory–Huggins statistical-mechanical model were combined to establish a field theory for describing the phase transition result from a mechanical constraint [35]. Base on the field theory, Gernandt et al. [36] presented a general thermodynamic model together with a conventional thermodynamics theory of hydrogel swelling for calculating the internal structure of core–shell hydrogels. Despite the success of those models, they are generally too complex for application.
C. Lian et al. / Journal of Colloid and Interface Science 406 (2013) 148–153
In present work, we propose a simple yet meaning thermodynamic model to address the swelling behavior of two kinds of hydrogels. One is composed of hydrogels in both core and shell. In other words, the swelling of the core and the shell is affected by thermodynamic and solution properties and shows very complicated swelling behavior. The other is composed of non-hydrogel core and an external hydrogen shell, i.e., a hydrogel is coated onto a hard core surface (e.g., gold, polystyrene PS, silica, etc.). The swelling behavior of these core–shell hydrogels is dominated by two different contributions: the interaction between polymer chains and solvent and the elasticity of polymer networks. Here, we combine the close-packed lattice model [27–29] for mixing free energy and Flory’s Gaussian chain model [21] for the elastic free energy. With proposed thermodynamic model, two sorts of core–shell hydrogels have been studied. One is composed of hydrogels in both the core and the shell with different LCSTs and the other is composed of a hard core and a thermally-sensitive gel shell. In particular, they are employed to represent poly(N-isopropylacrylamide)-poly(Nisopropylmethacrylamide) (PNIPAM–PNIPMAM) and polystyrenepoly(N-isopropylacrylamide) (PS-PNIPAM) core–shell hydrogels, respectively. 2. Molecular thermodynamic model The swelling behaviors of hydrogels are decided by two separate and additive contributions. One is the mixing interaction between polymers and solvent and the other is the elasticity of polymer hydrogel network, which describes the cross-linking degree and other constraint. The total changes of Gibbs free energy DG are composed of the mixing contribution and the elasticity contribution:
DG ¼ DGmix þ DGelas
ð1Þ
where the DGmix and DGelas are the changes of mixing Gibbs free energy and elastic Gibbs free energy, respectively. The hydrogel can be regarded as a close-packed lattice mixture composed of polymer network and solvent. If ignore the effect of pressure, DGmix value of hydrogels can be calculated by using the closepacked lattice model proposed by Yang and his coworkers [27–29]:
DGmix u z hS uP qP hP ¼ uS ln uS þ P ln uP þ uS ln þ ln 2 Nr kT rP uS rP uP z z z 2 2 2 2 2 uS uP ðuS þ u2P Þ þ uS uP 2 uS uP 2T 4T 12T 2 u ðrP 1 þ fP Þ 1 þ uS ðexpð1=T Þ 1Þ ln P rP 1 þ uS uP ðexpð1=T Þ 1Þ
ð2Þ
In above equation, the energy parameter T = kT/e (e is the exchange energy), k is the Boltzmann constant, z is the coordination number z = 6, the subscripts ‘‘s’’ and ‘‘p’’ represent solvent and polymer network, and rs and rp are the chain lengths of solvent and of polymer, respectively. In our study, the value of rs is set as a unit. Nr is the total number of chains, i.e., Nr = NS + NP. Here, NS and NP stand for the chain numbers of the solvent and polymer. ui and hi are, respectively, the volume and surface fractions of component i with i = s, p. They can be calculated by
ui ¼ Ni ri =ðNS rS þ NP rP Þ
ð3Þ
hi ¼ Ni qi =ðNS qS þ NP qP Þ
ð4Þ
where qi is the surface area parameters of solvent or polymer defined by
qi ¼ ½r i ðz 2Þ þ 2=z
ð5Þ
149
The last term of Eq. (2) denotes the contribution of chain connectivity of polymer, and the involved parameter fP characterizes the long-range correlations between segments on the same polymer chain, which can be written as:
fP ¼
ðr P 1Þðr P 2Þ ð0:1321rP þ 0:5918Þ r 2P
ð6Þ
Since the elasticity of polymer network is entropic, it is reasonable to suppose that the macroscopic deformation is proportional to the microscopic deformation of polymer chains. The free energy difference during the structural change of polymer network can be calculated by the Gaussian model [21],
3 DGelas ¼ NC kTðk2 1 ln kÞ 2
ð7Þ
where the NC is the number of polymer chains in hydrogel network and the k is the swelling ratio of hydrogel, which is defined as
k3 ¼
uP;0 V ¼ V0 uP
ð8Þ
The subscript ‘‘0’’ is the initial state of hydrogel network. Combining Eqs. (1)–(7), the Gibbs free energy difference, i.e., DG, during the swelling (or deswelling) can be expressed as:
DG DGmix DGelas ¼ þ Nr kT Nr kT Nr kT
z h uq h uS ln S þ P P ln P 2 rP uS rP uP z z z u2S u2P ðu2S þ u2P Þ þ uS uP 2 u2S u2P 2T 4T 12T 2 u ðrP 1 þ kP Þ 1 þ uS ðexpð1=T Þ 1Þ ln P rP 1 þ uS uP ðexpð1=T Þ 1Þ
¼ uS ln uS þ
þ
uP
ln uP þ
3NC 2 ðk 1 ln kÞ 2Nr
ð9Þ
The chemical potential of solvent in the core and shell can be derived from Eq. (9) as follows:
DlS 3 2uP 2u2P ¼ ln uS þ uP þ 3 ln þ uP þ 2 þ uS 3 uP kT 6 þ 3uS 2 1 1 þ 3u2P 1:5uS u2P ð2uP uS Þ T T 3 1 0:5uS u2P ð4u2S uP u3S þ 2u3P 3uS u2P Þ T 1 uP uS 1:1312u2P C 1 C2 C3 2 !1=3 3 v S NC 4 uP uP 5 NA V 0 2uP;0 uP;0
ð10Þ
where C1 = exp (1/T), C2 = 1 + uSC1, C3 = 1 + uSuPCC1. For the chemical potential outside the core–shell hydrogel, only the mixing contribution should be accounted for. Since the solvent is pure in our discussion, Dlsolv is considered as zero. S 3. Swelling behaviors of doubly thermally-sensitive core–shell hydrogels 3.1. Swelling equilibrium In the initial state (or equivalently reference state), a polymer network contains no solvent water, and the core is of radius Rcore which is also called the inner radius. We use Rshell to denote the total radius of the core–shell hydrogel. The schematic illustration of such state is shown in Fig. 1a. In the swelling equilibrium state, the
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Dlshell s ¼ ln uSS þ uPS kT 3 2uPS 2u2PS þ 3 ln þ uPS 2 þ uSS 3 uPS 6 þ 3uSS 2 1 1 þ 3u2PS 1:5uSS u2PS ð2uPS uSS Þ T shell T shell 0:5uSS u2PS ð4u2SS uPS u3SS þ 2u3PS 3 1 3uSS u2PS Þ T shell 1 uPS uSS 1:1312u2PS D1 D2 D3 2 !1=3 3 v S Nshell u u PS PS C 4 5 NA V 0 2uPS;0 uPS;0
Fig. 1. Schematic illustration of deformation of a core–shell hydrogel: (a) initial state and (b) swelling equilibrium state.
corresponding radius of core and the whole core–shell are rcore and rshell, respectively. Because the core and shell parts are chemical linked, the interaction between core and shell may result in complicated thermally-sensitive behavior of the hydrogels. In addition, both the core and the shell gel swell by the same stretch in all directions, so that the swelling ratio of the core and shell gel can be expressed, respectively, as
3
k3core ¼
uPC;0 4p V core ¼ ¼ V core;0 uPC 4p
k3shell ¼
uPS;0 4pr3shell =3 4pr3core =3 V shell ¼ ¼ V shell;0 uPS 4pR3shell =3 4pR3core =3
¼
r3core =3 R3core =3
¼
rcore Rcore
ð11Þ
r 3shell r 3core
ð12Þ
R3shell R3core
where Vcore and Vshell are the volumes of the core and shell in swelling equilibrium state, respectively. uPC denotes the volume fraction of polymer in the core, while uPS denotes the volume fraction of polymer in the shell. From Eqs. (8), (11), and (12), the ratio of total equilibrium volume and total reference volume V/V0 can be expressed as:
3 4pr 3shell =3 V rshell ¼ ¼ 3 V 0 4pRshell =3 Rshell 3 ! 3 uPS;0 Rcore Rcore uPC;0 ¼ 1 þ Rshell uPS Rshell uPC
ð13Þ
Dlcore s ¼ ln uSC þ uPC kT 3 2uPC 2u2PC þ 3 ln þ uPC 2 þ uS 3 uPC 6 þ 3uSC 2 1 1 1:5uSC u2PC ð2uPC uSC Þ þ 3u2PC T core T core 0:5uSC u
2 SC
3 SC
Dlcore ¼ Dlshell ¼ Dlsolv S S S
ð16Þ
Here, Dl and Dl denote the chemical potential of solvent in core and in shell gel, respectively, and Dlslov denotes that in solvent s phase. Combining Eqs. (13)–(16), the swelling equilibrium of core– shell hydrogel can be described with three parameters. The first is the relative thickness which was defined as the ratio of the core radius to the whole core–shell particle radius Rcore/Rshell. The second is T of core or shell hydrogel which depends on temperature. The last is the dimensionless volume parameters V of core or shell hydrogels, which can be expressed as: core s
1 V core 1 V shell
¼
¼
shell s
v s Ncore C NA V 0
v s Nshell C NA V 0
ð17Þ
ð18Þ
1 core e1C e2C e3C ¼ ¼ þ þ T core kT kT ðkTÞ2 ðkTÞ3
ð19Þ
1 shell e1S e2S e3S ¼ ¼ þ þ T shell kT kT ðkTÞ2 ðkTÞ3
ð20Þ
Coefficients in Eqs. (19) and (20) can be obtained by fitting the experimental swelling data of the pure core and pure shell hydrogel. 3.2. Correlation
3 PC
u uPC u þ 2u 3 1 3uSC u2PC Þ T core 1 uPC uSC 1:1312u2PC C 1 C2 C3 2 !1=3 3 v S Ncore u u PC PC C 4 5 NA V 0 2uPC;0 uPC;0
where the uSC denotes the volume fraction of solvent in the core and uSS denotes the volume fraction of solvent in the shell. C 1 ¼ expð1=T core Þ, C2 = 1 + uSC1, C3 = 1 + uSuPCC1. D1 ¼ expð1=T shell Þ, D2 = 1 + uSD1, D3 = 1 + uSuPSu1. Ncore and N shell are the numbers of C C polymer chains in core and shell, which are determined by the cross-linking degree of hydrogel. The swelling equilibrium equation of hydrogel system can be expressed as:
where V is dependent on the cross-linking degree of hydrogel. The higher the degree of cross-linking is, the lower the value of V⁄. We adopt a quadratic of inverse temperature to express T[28].
From Eq. (10), the chemical potential of solvent in the core and shell hydrogel can be obtained.
2 PC ð4
ð15Þ
ð14Þ
Using our molecular thermodynamic model of core–shell hydrogel, we can analyze the correlation with the experimental swelling data of the PNIPAM–PNIPMAM core–shell hydrogels [10]. The energy parameters between polymers and the solvent can be obtained by fitting the experimental data of pure PNIPAM and pure PNIPMAM hydrogels. Fig. 2 shows the comparisons between experimental data of the pure hydrogels and the calculated curves by using our model. Adjustable interaction parameters are set as e1C/k = 81.1 102/K, e2C/k2 = 81.5 104/K2, and e3C/
C. Lian et al. / Journal of Colloid and Interface Science 406 (2013) 148–153
Fig. 2. Swelling curves of pure PNIPAM and PNIPMAM hydrogel. The circles are experimental data of pure PNIPMAM hydrogels; the triangles are experimental data of pure PNIPAM hydrogels [10]. Solid lines exhibit theoretical prediction obtained by our model.
k3 = 507 106/K3 for pure PNIPAM hydrogel, and e1S/ k = 8.6 102/K, e2S/k2 = 8.5 104/K2, and e3S/k3 = 50.0 106/K3 for pure PNIPMAM hydrogel. In addition, the V is related with the cross-linking degree, so that their values are different under various experimental conditions. Here, the V of pure PNIPAM and PNIPMAM are 58 and 33. As shown in Fig. 2, the maximum swelling ratio of PNIPAM is much larger than that of PNIPMAM hydrogel. The transition temperatures of PNIPAM and PNIPMAM are 307 K and 317 K, respectively. When calculating the swelling ratio of the core–shell hydrogel, these energy parameters are used and the V of core and shell is adjusted according to the cross-linking degrees. Fig. 3 shows the effect of shell thickness on the swelling behavior of PNIPAM–PNIPMAM core–shell hydrogel with the same crosslinking degree. The thicker the shell is, the smaller the value of Rcore/Rshell. The solid lines obtained by calculation are correlated with the corresponding experimental results. The energy parameters used here were obtained from correlating pure polymer hydrogels as shown in Fig. 2. V of core and shell hydrogel are 23 and 27, which are different with the values of pure hydrogels. This
151
is because the cross-linking degree of PNIPAM or PNIPMAM in the pure gel is different from that in core–shell hydrogels. The semiquantitative agreement of the predicted swelling equilibrium with the experimental results is very encouraging. Comparing with the swelling ratio of pure PNIPAM, those of core–shell hydrogels decreased in different degrees. When the shell thickness increases, i.e., Rcore/Rshell decreases, the swelling ratio decreases and two phase transition points appear gradually on swelling curves. When the value of Rcore/Rshell is less than 0.57, the phase transition at 307 K disappeared. The core–shell hydrogels with Rcore/Rshell = 0.57 behave very similarly to the pure PNIPMAM gel (see Fig. 2). Obviously, a thicker PNIPMAM shell gives a stronger restriction to the swelling behavior of the PNIPAM core and even to the properties of the whole core–shell hydrogel. Fig. 4 plots the swelling ratios of core–shell hydrogels (Rcore/ Rshell = 0.79) with varied cross-linker contents of the shell. It shows that the swelling ratio increases as cross-linker content decreases. Berndt and Richtering [10] drew the same conclusion from experimental data by prepared core–shell hydrogels using different cross-linker contents in the shell. The more cross-linker contains in the shell, the higher cross-linking degree of the hydrogels. Then, due to the less porous ratio in a higher cross-linking degree network, the swelling ratio becomes lower. Here, all energy parameters are used as the same as mentioned in Fig. 3. The values of V are 27, 99, and 220 for the various cross-linker contents (9%, 5%, and 3%), respectively. Comparing with LCSTs of pure PNIPMAM and PNIPAM hydrogel, there is a slight deviation in those of core–shell hydrogels. The mutual constriction between the core and the shell should cause the change in LCSTs of both the core and the shell hydrogels. The results calculated by using our model are quantitatively in good agreement with the experimental data [10]. The results demonstrate that the thickness and the cross-linking density of shell can be used to control the thermally-sensitive behavior of the whole core–shell structural hydrogel. 4. Swelling behaviors of hardcore hydrogels 4.1. Swelling equilibrium Hardcore hydrogel is a particle composed of a hardcore inside and a hydrogel shell. In the reference state, the polymer network does not contain any water. Similarly, the core is of radius Rcore, and the whole core–shell particle is of radius Rshell. Since the hard
6
5
V/V0
4
Rcore/Rshell
3
0.79 0.74 0.66 0.57
2
1 300
305
310
315
320
325
T/K Fig. 3. Swelling curves of PNIPAM–PNIPMAM core–shell hydrogels with different Rcore/Rshell. Solid lines exhibit theoretical prediction by our model. The points are experimental data [10]. Inset is three-dimensional calculated results.
Fig. 4. Swelling ratios of PNIPAM–PNIPMAM core–shell hydrogels with the shell prepared by using different cross-linker contents (w/w). The points are experimental data [10]. Solid lines exhibit theoretical prediction obtained by our model.
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core does not swell, the value of Rcore remains constant in reference or equilibrium state. rshell denotes the whole core–shell radius in the equilibrium state. Due to the core and shell are chemically linked, the interaction between core and shell should bring about more complicated thermally-sensitive behavior. Since the shell swells in the same stretch degree in all directions, the swelling ratio of shell, which is constricted by the core, is considered to be
k3shell ¼ ¼
uPS;0 4pr3shell =3 4pR3core =3 V shell ¼ ¼ V shell;0 uPS 4pR3shell =3 4pR3core =3 r3shell R3core R3shell R3core
ð21Þ
The subscript ‘‘0’’ means the reference state. The V/V0 can be expressed as follows:
3 4pr 3shell =3 V rshell ¼ ¼ V 0 4pR3shell =3 Rshell 3 ! 3 uPS;0 Rcore Rcore ¼ 1 þ Rshell uPS Rshell
Fig. 6. The calculated swelling ratios of PS-PNIPAM hydrogels are plotted against values of Rcore/Rshell. Inset is three-dimensional calculated results.
ð22Þ
From Section 2, the chemical potential of solvent in the shell is obtained by Eq. (15). The swelling equilibrium equation of hydrogel system can be expressed as
Dlshell ¼ Dlsolv S S
ð23Þ
Combining Eqs. (15), (21), (22), and (23), the swelling equilibrium of core–shell hydrogel can be described by three kinds of parameters as mentioned in Section 3. 4.2. Correlation In Section 3, the energy parameters were obtained to be e1/ k = 81.1 102/K, e2/k2 = 81.5 104/K2, and e3/k3 = 507 106/ K3 for pure PNIPAM hydrogel. The experimental swelling data of the PS-PNIPAM core–shell hydrogels [37] were correlated with the calculated results using the above parameters. Fig. 5 shows the effect of cross-linker content on swelling ratio of the PS-PNIPAM hydrogels with Rcore/Rshell = 0.79. Smaller V corresponds to the higher cross-linking degree, which induced an increase in network density and then stiffness, so that the swelling ratio decreases. On the other hand, the LCST is mainly related to the incompatibility between the solvent and polymer network, i.e.,
the hydrophilicity of the polymer chains, so that V has no influence on the point of phase transition. As shown in Fig. 5, the LCST of this hydrogel remains unchanged comparing with that of pure PNIPAM gel. The calculated data by our model are quantitatively agreed with the experimental results. For this kind of hydrogels, no experimental data are yet available for exploring the influence of the Rcore/Rshell value on the swelling behavior. Fig. 6 shows the calculated results only by our model. Using the same energy parameters and volume parameter as mentioned above, the effect of Rcore/Rshell on the swelling behavior of PS-PNIPAM core–shell hydrogels can be predicted. Rcore/ Rshell = 0 means that the hydrogel is pure PNIPAM without PS core. When Rcore/Rshell value increases from 0.25 to 0.90, the swelling ratios of the hardcore hydrogels decrease from 10.8 to 3.6. In principle, swelling of a hydrogel with hard core can be considered as a restricted process. Generally, the swelling ratio should decrease under the constraint. When discussing the influence of Rcore/Rshell value on the swelling ratio, two factors should be considered. They are the contact area of core/shell interface Ac and the value of Rshell. As for the increase of Rcore/Rshell value, there are three possible cases: (1) Rcore and Ac are constant and Rshell decrease. In this case, the swelling ratios are inhomogeneous along the direction of radius, i.e., swelling of the hydrogel near to the hardcore must be restricted more severely resulting in a relative low swelling ratio. Then, a thinner shell shows a smaller macroscopic swelling ratio. (2) Rshell is constant and Rcore and Ac increase. In this case, the swelling ratio is mainly decided by Ac. It is easy to understand the swelling ratio decrease with the increase of Ac. (3) Rshell is constant and Rcore (increase) and shell thickness (decrease) changed simultaneously. This case seems to possess both the (1) and (2) at the same time. Consequently, all these cases show that the swelling ratios decrease with the increase of Rcore/Rshell value. The swelling behavior under a restricted condition is very important for understanding more physiological phenomenon and processes. More experimental data and theoretical models are needed for further research. 5. Conclusions
Fig. 5. The swelling curves of PS-PNIPAM hardcore hydrogels with different contents of cross-linker (w/w). Solid lines exhibit theoretical prediction obtained by our model. The points are experimental data.
In this work, we have proposed a new molecular thermodynamic model for describing the swelling behavior of thermallysensitive core–shell hydrogels. The thermodynamic model combines the close-packed lattice model [27–29] for the mixing contribution of free energy and the Gaussian chain model [21] for the elastic free energy. The model involves two parameters:
C. Lian et al. / Journal of Colloid and Interface Science 406 (2013) 148–153
temperature-dependent exchange energy parameter e and the Topology-dependent size parameter V, and both of them are obtained by experimental results of pure polymer hydrogels. We showed that, with this simple model, we can analyze not only the doubly thermally-sensitive core–shell hydrogels, but also the hydrogels with hard core. The calculated results in both cases are overall in very good quantitative agreement with the experimental data [10,37]. Besides, considering the correlated constraints between core and shell parts during the process of swelling, the geometrical relationships of the two kinds of core–shell hydrogels have been considered to relate to Ac. Both experimental data and calculated results indicate that the swelling ratios in two hydrogels systems were determined by the cross-linking degree and the Rcore/Rshell value. The thermodynamic model developed here can in principle be extended to address, e.g., the pH sensitive swelling behavior, etc. We believe that the current thermodynamic model can shed some lights to understand the complex yet interesting properties in the challenging areas of core–shell hydrogel science. Acknowledgments Financial support for this work is provided by the National Natural Science Foundation of China (Nos. 21076071, 20990224, and 21276074) and the 111 Project (No. B08021) of China and the Fundamental Research Funds for the central Universities of China. References [1] G. Gerlach, M. Guenther, J. Sorber, G. Suchaneck, K.F. Arndt, A. Richter, Sens. Actuat. B 111–112 (2005) 555–561. [2] S.M.M. Quintero, R.V. Ponce F, M. Cremona, A.L.C. Triques, A.R. d’Almeida, A.M.B. Braga, Polymer 51 (2010) 953–958. [3] K. Podual, F.J. Doylelll, N.A. Peppas, Polymer 41 (2000) 3975–3983. [4] J. Hendri, A. Hiroki, Y. Maekawa, M. Yoshida, R. Katakai, Radiat. Phys. Chem. 60 (2001) 617–624.
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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
A lattice model for thermally-sensitive core–shell hydrogels Cheng Lian, Dongyan Zhi, Shouhong Xu ⇑, Honglai Liu, Ying Hu Key Laboratory for Advanced Materials and Department of Chemistry, East China University of Science and Technology, Shanghai 200237, China
a r t i c l e
i n f o
Article history: Received 30 January 2013 Accepted 3 June 2013 Available online 18 June 2013 Keywords: Molecular thermodynamic model Core–shell hydrogel Swelling behavior Thermally-sensitive
a b s t r a c t A lattice molecular thermodynamic model for describing the swelling behavior of thermally-sensitive core–shell hydrogels was developed by integrating a close-packed lattice model for mixing free energy and a Flory Gaussian chain model for the elastic free energy. The thermodynamic model is characterized with two parameters: the temperature-dependent exchange energy parameter e and the Topologydependent size parameter V. The input values of both parameters can be obtained by experimental results of pure polymer hydrogels. With the help of proposed model, swelling behaviors of two kinds of hydrogel systems were analyzed: one is a doubly thermally-sensitive core–shell hydrogels (with two LCSTs) and the other comprises a thermally-sensitive hydrogel shell with a hard internal core. We show that the calculated results are in good agreement with the experimental data. Ó 2013 Elsevier Inc. All rights reserved.
1. Introduction Smart hydrogels are soft materials which can swell through absorbing solvent and deswell through ejecting the solvent as a response to external stimuli (such as temperature, pH, ionic strength, and composition of solvent). These hydrogels have been widely applied in sensors [1,2], immobilized enzymes [3], material extraction [4,5], drug delivery devices [6–8], catalysts [9], and so on. In recent years, novel architectures (such as core–shell structures [10–18] and interpenetrating network structures [19,20]) have been designed to obtain materials with superior properties and functions. The properties of these particles, such as stability, biocompatibility, mechanical strength, and even functionalities, have been improved. In the related theoretical filed, as early as 1953, Flory [21] established the swelling theory of such special network structures based on the polymer solution theory and the rubber theory. Until 1987, Tanaka and his coworker [22] established a thermodynamic model for hydrogels. They considered the chemical potential inside the hydrogel was equal to that of outside hydrogel at swelling equilibrium. From then on, researchers have paid more and more attention to establishing molecular thermodynamic models for hydrogels. For the homogeneous hydrogels, corresponding molecular thermodynamic models for describing their swelling behaviors have been widely reported. Considering the influence of hydrogen bonding, Shenoy et al. [23] combined an association model with a Flory–Rehner approach to develop a new model. Starting from a quasi-chemical partition function [24], which con-
⇑ Corresponding author. Fax: +86 21 64252921. E-mail address: [email protected] (S. Xu). 0021-9797/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jcis.2013.06.005
sidered both the rubber elasticity theories and competition between hydrogen bonding and dispersion forces, Oliveira et al. [25] presented an oriented quasi-chemical thermodynamic model to explain the swelling behavior of hydrogels. Huang et al. [26] developed a molecular thermodynamic model based on a lattice model [27–29] for thermally-sensitive hydrogels. Then, Zhi et al. [30,31] established molecular thermodynamic models for thermal and solvent sensitive hydrogels and also multi-sensitive hydrogels. Jung et al. [32] combined modified double lattice model (MDL) theory with Flory–Erman theory to describe the reentrant swelling behavior of thermally-sensitive hydrogels. Nevertheless, the above models are essentially proposed for homogeneous hydrogels, and the properties of the core–shell hydrogels are fundamentally different. In core–shell hydrogel system, the core and shell are constrained by and correlated with each other, and thus, the swelling process is an inhomogeneous deformation. The thermodynamic study on phase transitions of such hydrogels is rather complicated. With a general yet complex formalism of field theory, Sekimoto and Kawasaki [33] have studied the phase coexistence in hydrogels, and using the Legendre transformation, the inhomogeneous swelling behavior of covalently cross-linked polymers is investigated by Marcombe et al. [34]. Furthermore, the thermodynamic theory of Gibbs and the Flory–Huggins statistical-mechanical model were combined to establish a field theory for describing the phase transition result from a mechanical constraint [35]. Base on the field theory, Gernandt et al. [36] presented a general thermodynamic model together with a conventional thermodynamics theory of hydrogel swelling for calculating the internal structure of core–shell hydrogels. Despite the success of those models, they are generally too complex for application.
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In present work, we propose a simple yet meaning thermodynamic model to address the swelling behavior of two kinds of hydrogels. One is composed of hydrogels in both core and shell. In other words, the swelling of the core and the shell is affected by thermodynamic and solution properties and shows very complicated swelling behavior. The other is composed of non-hydrogel core and an external hydrogen shell, i.e., a hydrogel is coated onto a hard core surface (e.g., gold, polystyrene PS, silica, etc.). The swelling behavior of these core–shell hydrogels is dominated by two different contributions: the interaction between polymer chains and solvent and the elasticity of polymer networks. Here, we combine the close-packed lattice model [27–29] for mixing free energy and Flory’s Gaussian chain model [21] for the elastic free energy. With proposed thermodynamic model, two sorts of core–shell hydrogels have been studied. One is composed of hydrogels in both the core and the shell with different LCSTs and the other is composed of a hard core and a thermally-sensitive gel shell. In particular, they are employed to represent poly(N-isopropylacrylamide)-poly(Nisopropylmethacrylamide) (PNIPAM–PNIPMAM) and polystyrenepoly(N-isopropylacrylamide) (PS-PNIPAM) core–shell hydrogels, respectively. 2. Molecular thermodynamic model The swelling behaviors of hydrogels are decided by two separate and additive contributions. One is the mixing interaction between polymers and solvent and the other is the elasticity of polymer hydrogel network, which describes the cross-linking degree and other constraint. The total changes of Gibbs free energy DG are composed of the mixing contribution and the elasticity contribution:
DG ¼ DGmix þ DGelas
ð1Þ
where the DGmix and DGelas are the changes of mixing Gibbs free energy and elastic Gibbs free energy, respectively. The hydrogel can be regarded as a close-packed lattice mixture composed of polymer network and solvent. If ignore the effect of pressure, DGmix value of hydrogels can be calculated by using the closepacked lattice model proposed by Yang and his coworkers [27–29]:
DGmix u z hS uP qP hP ¼ uS ln uS þ P ln uP þ uS ln þ ln 2 Nr kT rP uS rP uP z z z 2 2 2 2 2 uS uP ðuS þ u2P Þ þ uS uP 2 uS uP 2T 4T 12T 2 u ðrP 1 þ fP Þ 1 þ uS ðexpð1=T Þ 1Þ ln P rP 1 þ uS uP ðexpð1=T Þ 1Þ
ð2Þ
In above equation, the energy parameter T = kT/e (e is the exchange energy), k is the Boltzmann constant, z is the coordination number z = 6, the subscripts ‘‘s’’ and ‘‘p’’ represent solvent and polymer network, and rs and rp are the chain lengths of solvent and of polymer, respectively. In our study, the value of rs is set as a unit. Nr is the total number of chains, i.e., Nr = NS + NP. Here, NS and NP stand for the chain numbers of the solvent and polymer. ui and hi are, respectively, the volume and surface fractions of component i with i = s, p. They can be calculated by
ui ¼ Ni ri =ðNS rS þ NP rP Þ
ð3Þ
hi ¼ Ni qi =ðNS qS þ NP qP Þ
ð4Þ
where qi is the surface area parameters of solvent or polymer defined by
qi ¼ ½r i ðz 2Þ þ 2=z
ð5Þ
149
The last term of Eq. (2) denotes the contribution of chain connectivity of polymer, and the involved parameter fP characterizes the long-range correlations between segments on the same polymer chain, which can be written as:
fP ¼
ðr P 1Þðr P 2Þ ð0:1321rP þ 0:5918Þ r 2P
ð6Þ
Since the elasticity of polymer network is entropic, it is reasonable to suppose that the macroscopic deformation is proportional to the microscopic deformation of polymer chains. The free energy difference during the structural change of polymer network can be calculated by the Gaussian model [21],
3 DGelas ¼ NC kTðk2 1 ln kÞ 2
ð7Þ
where the NC is the number of polymer chains in hydrogel network and the k is the swelling ratio of hydrogel, which is defined as
k3 ¼
uP;0 V ¼ V0 uP
ð8Þ
The subscript ‘‘0’’ is the initial state of hydrogel network. Combining Eqs. (1)–(7), the Gibbs free energy difference, i.e., DG, during the swelling (or deswelling) can be expressed as:
DG DGmix DGelas ¼ þ Nr kT Nr kT Nr kT
z h uq h uS ln S þ P P ln P 2 rP uS rP uP z z z u2S u2P ðu2S þ u2P Þ þ uS uP 2 u2S u2P 2T 4T 12T 2 u ðrP 1 þ kP Þ 1 þ uS ðexpð1=T Þ 1Þ ln P rP 1 þ uS uP ðexpð1=T Þ 1Þ
¼ uS ln uS þ
þ
uP
ln uP þ
3NC 2 ðk 1 ln kÞ 2Nr
ð9Þ
The chemical potential of solvent in the core and shell can be derived from Eq. (9) as follows:
DlS 3 2uP 2u2P ¼ ln uS þ uP þ 3 ln þ uP þ 2 þ uS 3 uP kT 6 þ 3uS 2 1 1 þ 3u2P 1:5uS u2P ð2uP uS Þ T T 3 1 0:5uS u2P ð4u2S uP u3S þ 2u3P 3uS u2P Þ T 1 uP uS 1:1312u2P C 1 C2 C3 2 !1=3 3 v S NC 4 uP uP 5 NA V 0 2uP;0 uP;0
ð10Þ
where C1 = exp (1/T), C2 = 1 + uSC1, C3 = 1 + uSuPCC1. For the chemical potential outside the core–shell hydrogel, only the mixing contribution should be accounted for. Since the solvent is pure in our discussion, Dlsolv is considered as zero. S 3. Swelling behaviors of doubly thermally-sensitive core–shell hydrogels 3.1. Swelling equilibrium In the initial state (or equivalently reference state), a polymer network contains no solvent water, and the core is of radius Rcore which is also called the inner radius. We use Rshell to denote the total radius of the core–shell hydrogel. The schematic illustration of such state is shown in Fig. 1a. In the swelling equilibrium state, the
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Dlshell s ¼ ln uSS þ uPS kT 3 2uPS 2u2PS þ 3 ln þ uPS 2 þ uSS 3 uPS 6 þ 3uSS 2 1 1 þ 3u2PS 1:5uSS u2PS ð2uPS uSS Þ T shell T shell 0:5uSS u2PS ð4u2SS uPS u3SS þ 2u3PS 3 1 3uSS u2PS Þ T shell 1 uPS uSS 1:1312u2PS D1 D2 D3 2 !1=3 3 v S Nshell u u PS PS C 4 5 NA V 0 2uPS;0 uPS;0
Fig. 1. Schematic illustration of deformation of a core–shell hydrogel: (a) initial state and (b) swelling equilibrium state.
corresponding radius of core and the whole core–shell are rcore and rshell, respectively. Because the core and shell parts are chemical linked, the interaction between core and shell may result in complicated thermally-sensitive behavior of the hydrogels. In addition, both the core and the shell gel swell by the same stretch in all directions, so that the swelling ratio of the core and shell gel can be expressed, respectively, as
3
k3core ¼
uPC;0 4p V core ¼ ¼ V core;0 uPC 4p
k3shell ¼
uPS;0 4pr3shell =3 4pr3core =3 V shell ¼ ¼ V shell;0 uPS 4pR3shell =3 4pR3core =3
¼
r3core =3 R3core =3
¼
rcore Rcore
ð11Þ
r 3shell r 3core
ð12Þ
R3shell R3core
where Vcore and Vshell are the volumes of the core and shell in swelling equilibrium state, respectively. uPC denotes the volume fraction of polymer in the core, while uPS denotes the volume fraction of polymer in the shell. From Eqs. (8), (11), and (12), the ratio of total equilibrium volume and total reference volume V/V0 can be expressed as:
3 4pr 3shell =3 V rshell ¼ ¼ 3 V 0 4pRshell =3 Rshell 3 ! 3 uPS;0 Rcore Rcore uPC;0 ¼ 1 þ Rshell uPS Rshell uPC
ð13Þ
Dlcore s ¼ ln uSC þ uPC kT 3 2uPC 2u2PC þ 3 ln þ uPC 2 þ uS 3 uPC 6 þ 3uSC 2 1 1 1:5uSC u2PC ð2uPC uSC Þ þ 3u2PC T core T core 0:5uSC u
2 SC
3 SC
Dlcore ¼ Dlshell ¼ Dlsolv S S S
ð16Þ
Here, Dl and Dl denote the chemical potential of solvent in core and in shell gel, respectively, and Dlslov denotes that in solvent s phase. Combining Eqs. (13)–(16), the swelling equilibrium of core– shell hydrogel can be described with three parameters. The first is the relative thickness which was defined as the ratio of the core radius to the whole core–shell particle radius Rcore/Rshell. The second is T of core or shell hydrogel which depends on temperature. The last is the dimensionless volume parameters V of core or shell hydrogels, which can be expressed as: core s
1 V core 1 V shell
¼
¼
shell s
v s Ncore C NA V 0
v s Nshell C NA V 0
ð17Þ
ð18Þ
1 core e1C e2C e3C ¼ ¼ þ þ T core kT kT ðkTÞ2 ðkTÞ3
ð19Þ
1 shell e1S e2S e3S ¼ ¼ þ þ T shell kT kT ðkTÞ2 ðkTÞ3
ð20Þ
Coefficients in Eqs. (19) and (20) can be obtained by fitting the experimental swelling data of the pure core and pure shell hydrogel. 3.2. Correlation
3 PC
u uPC u þ 2u 3 1 3uSC u2PC Þ T core 1 uPC uSC 1:1312u2PC C 1 C2 C3 2 !1=3 3 v S Ncore u u PC PC C 4 5 NA V 0 2uPC;0 uPC;0
where the uSC denotes the volume fraction of solvent in the core and uSS denotes the volume fraction of solvent in the shell. C 1 ¼ expð1=T core Þ, C2 = 1 + uSC1, C3 = 1 + uSuPCC1. D1 ¼ expð1=T shell Þ, D2 = 1 + uSD1, D3 = 1 + uSuPSu1. Ncore and N shell are the numbers of C C polymer chains in core and shell, which are determined by the cross-linking degree of hydrogel. The swelling equilibrium equation of hydrogel system can be expressed as:
where V is dependent on the cross-linking degree of hydrogel. The higher the degree of cross-linking is, the lower the value of V⁄. We adopt a quadratic of inverse temperature to express T[28].
From Eq. (10), the chemical potential of solvent in the core and shell hydrogel can be obtained.
2 PC ð4
ð15Þ
ð14Þ
Using our molecular thermodynamic model of core–shell hydrogel, we can analyze the correlation with the experimental swelling data of the PNIPAM–PNIPMAM core–shell hydrogels [10]. The energy parameters between polymers and the solvent can be obtained by fitting the experimental data of pure PNIPAM and pure PNIPMAM hydrogels. Fig. 2 shows the comparisons between experimental data of the pure hydrogels and the calculated curves by using our model. Adjustable interaction parameters are set as e1C/k = 81.1 102/K, e2C/k2 = 81.5 104/K2, and e3C/
C. Lian et al. / Journal of Colloid and Interface Science 406 (2013) 148–153
Fig. 2. Swelling curves of pure PNIPAM and PNIPMAM hydrogel. The circles are experimental data of pure PNIPMAM hydrogels; the triangles are experimental data of pure PNIPAM hydrogels [10]. Solid lines exhibit theoretical prediction obtained by our model.
k3 = 507 106/K3 for pure PNIPAM hydrogel, and e1S/ k = 8.6 102/K, e2S/k2 = 8.5 104/K2, and e3S/k3 = 50.0 106/K3 for pure PNIPMAM hydrogel. In addition, the V is related with the cross-linking degree, so that their values are different under various experimental conditions. Here, the V of pure PNIPAM and PNIPMAM are 58 and 33. As shown in Fig. 2, the maximum swelling ratio of PNIPAM is much larger than that of PNIPMAM hydrogel. The transition temperatures of PNIPAM and PNIPMAM are 307 K and 317 K, respectively. When calculating the swelling ratio of the core–shell hydrogel, these energy parameters are used and the V of core and shell is adjusted according to the cross-linking degrees. Fig. 3 shows the effect of shell thickness on the swelling behavior of PNIPAM–PNIPMAM core–shell hydrogel with the same crosslinking degree. The thicker the shell is, the smaller the value of Rcore/Rshell. The solid lines obtained by calculation are correlated with the corresponding experimental results. The energy parameters used here were obtained from correlating pure polymer hydrogels as shown in Fig. 2. V of core and shell hydrogel are 23 and 27, which are different with the values of pure hydrogels. This
151
is because the cross-linking degree of PNIPAM or PNIPMAM in the pure gel is different from that in core–shell hydrogels. The semiquantitative agreement of the predicted swelling equilibrium with the experimental results is very encouraging. Comparing with the swelling ratio of pure PNIPAM, those of core–shell hydrogels decreased in different degrees. When the shell thickness increases, i.e., Rcore/Rshell decreases, the swelling ratio decreases and two phase transition points appear gradually on swelling curves. When the value of Rcore/Rshell is less than 0.57, the phase transition at 307 K disappeared. The core–shell hydrogels with Rcore/Rshell = 0.57 behave very similarly to the pure PNIPMAM gel (see Fig. 2). Obviously, a thicker PNIPMAM shell gives a stronger restriction to the swelling behavior of the PNIPAM core and even to the properties of the whole core–shell hydrogel. Fig. 4 plots the swelling ratios of core–shell hydrogels (Rcore/ Rshell = 0.79) with varied cross-linker contents of the shell. It shows that the swelling ratio increases as cross-linker content decreases. Berndt and Richtering [10] drew the same conclusion from experimental data by prepared core–shell hydrogels using different cross-linker contents in the shell. The more cross-linker contains in the shell, the higher cross-linking degree of the hydrogels. Then, due to the less porous ratio in a higher cross-linking degree network, the swelling ratio becomes lower. Here, all energy parameters are used as the same as mentioned in Fig. 3. The values of V are 27, 99, and 220 for the various cross-linker contents (9%, 5%, and 3%), respectively. Comparing with LCSTs of pure PNIPMAM and PNIPAM hydrogel, there is a slight deviation in those of core–shell hydrogels. The mutual constriction between the core and the shell should cause the change in LCSTs of both the core and the shell hydrogels. The results calculated by using our model are quantitatively in good agreement with the experimental data [10]. The results demonstrate that the thickness and the cross-linking density of shell can be used to control the thermally-sensitive behavior of the whole core–shell structural hydrogel. 4. Swelling behaviors of hardcore hydrogels 4.1. Swelling equilibrium Hardcore hydrogel is a particle composed of a hardcore inside and a hydrogel shell. In the reference state, the polymer network does not contain any water. Similarly, the core is of radius Rcore, and the whole core–shell particle is of radius Rshell. Since the hard
6
5
V/V0
4
Rcore/Rshell
3
0.79 0.74 0.66 0.57
2
1 300
305
310
315
320
325
T/K Fig. 3. Swelling curves of PNIPAM–PNIPMAM core–shell hydrogels with different Rcore/Rshell. Solid lines exhibit theoretical prediction by our model. The points are experimental data [10]. Inset is three-dimensional calculated results.
Fig. 4. Swelling ratios of PNIPAM–PNIPMAM core–shell hydrogels with the shell prepared by using different cross-linker contents (w/w). The points are experimental data [10]. Solid lines exhibit theoretical prediction obtained by our model.
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core does not swell, the value of Rcore remains constant in reference or equilibrium state. rshell denotes the whole core–shell radius in the equilibrium state. Due to the core and shell are chemically linked, the interaction between core and shell should bring about more complicated thermally-sensitive behavior. Since the shell swells in the same stretch degree in all directions, the swelling ratio of shell, which is constricted by the core, is considered to be
k3shell ¼ ¼
uPS;0 4pr3shell =3 4pR3core =3 V shell ¼ ¼ V shell;0 uPS 4pR3shell =3 4pR3core =3 r3shell R3core R3shell R3core
ð21Þ
The subscript ‘‘0’’ means the reference state. The V/V0 can be expressed as follows:
3 4pr 3shell =3 V rshell ¼ ¼ V 0 4pR3shell =3 Rshell 3 ! 3 uPS;0 Rcore Rcore ¼ 1 þ Rshell uPS Rshell
Fig. 6. The calculated swelling ratios of PS-PNIPAM hydrogels are plotted against values of Rcore/Rshell. Inset is three-dimensional calculated results.
ð22Þ
From Section 2, the chemical potential of solvent in the shell is obtained by Eq. (15). The swelling equilibrium equation of hydrogel system can be expressed as
Dlshell ¼ Dlsolv S S
ð23Þ
Combining Eqs. (15), (21), (22), and (23), the swelling equilibrium of core–shell hydrogel can be described by three kinds of parameters as mentioned in Section 3. 4.2. Correlation In Section 3, the energy parameters were obtained to be e1/ k = 81.1 102/K, e2/k2 = 81.5 104/K2, and e3/k3 = 507 106/ K3 for pure PNIPAM hydrogel. The experimental swelling data of the PS-PNIPAM core–shell hydrogels [37] were correlated with the calculated results using the above parameters. Fig. 5 shows the effect of cross-linker content on swelling ratio of the PS-PNIPAM hydrogels with Rcore/Rshell = 0.79. Smaller V corresponds to the higher cross-linking degree, which induced an increase in network density and then stiffness, so that the swelling ratio decreases. On the other hand, the LCST is mainly related to the incompatibility between the solvent and polymer network, i.e.,
the hydrophilicity of the polymer chains, so that V has no influence on the point of phase transition. As shown in Fig. 5, the LCST of this hydrogel remains unchanged comparing with that of pure PNIPAM gel. The calculated data by our model are quantitatively agreed with the experimental results. For this kind of hydrogels, no experimental data are yet available for exploring the influence of the Rcore/Rshell value on the swelling behavior. Fig. 6 shows the calculated results only by our model. Using the same energy parameters and volume parameter as mentioned above, the effect of Rcore/Rshell on the swelling behavior of PS-PNIPAM core–shell hydrogels can be predicted. Rcore/ Rshell = 0 means that the hydrogel is pure PNIPAM without PS core. When Rcore/Rshell value increases from 0.25 to 0.90, the swelling ratios of the hardcore hydrogels decrease from 10.8 to 3.6. In principle, swelling of a hydrogel with hard core can be considered as a restricted process. Generally, the swelling ratio should decrease under the constraint. When discussing the influence of Rcore/Rshell value on the swelling ratio, two factors should be considered. They are the contact area of core/shell interface Ac and the value of Rshell. As for the increase of Rcore/Rshell value, there are three possible cases: (1) Rcore and Ac are constant and Rshell decrease. In this case, the swelling ratios are inhomogeneous along the direction of radius, i.e., swelling of the hydrogel near to the hardcore must be restricted more severely resulting in a relative low swelling ratio. Then, a thinner shell shows a smaller macroscopic swelling ratio. (2) Rshell is constant and Rcore and Ac increase. In this case, the swelling ratio is mainly decided by Ac. It is easy to understand the swelling ratio decrease with the increase of Ac. (3) Rshell is constant and Rcore (increase) and shell thickness (decrease) changed simultaneously. This case seems to possess both the (1) and (2) at the same time. Consequently, all these cases show that the swelling ratios decrease with the increase of Rcore/Rshell value. The swelling behavior under a restricted condition is very important for understanding more physiological phenomenon and processes. More experimental data and theoretical models are needed for further research. 5. Conclusions
Fig. 5. The swelling curves of PS-PNIPAM hardcore hydrogels with different contents of cross-linker (w/w). Solid lines exhibit theoretical prediction obtained by our model. The points are experimental data.
In this work, we have proposed a new molecular thermodynamic model for describing the swelling behavior of thermallysensitive core–shell hydrogels. The thermodynamic model combines the close-packed lattice model [27–29] for the mixing contribution of free energy and the Gaussian chain model [21] for the elastic free energy. The model involves two parameters:
C. Lian et al. / Journal of Colloid and Interface Science 406 (2013) 148–153
temperature-dependent exchange energy parameter e and the Topology-dependent size parameter V, and both of them are obtained by experimental results of pure polymer hydrogels. We showed that, with this simple model, we can analyze not only the doubly thermally-sensitive core–shell hydrogels, but also the hydrogels with hard core. The calculated results in both cases are overall in very good quantitative agreement with the experimental data [10,37]. Besides, considering the correlated constraints between core and shell parts during the process of swelling, the geometrical relationships of the two kinds of core–shell hydrogels have been considered to relate to Ac. Both experimental data and calculated results indicate that the swelling ratios in two hydrogels systems were determined by the cross-linking degree and the Rcore/Rshell value. The thermodynamic model developed here can in principle be extended to address, e.g., the pH sensitive swelling behavior, etc. We believe that the current thermodynamic model can shed some lights to understand the complex yet interesting properties in the challenging areas of core–shell hydrogel science. Acknowledgments Financial support for this work is provided by the National Natural Science Foundation of China (Nos. 21076071, 20990224, and 21276074) and the 111 Project (No. B08021) of China and the Fundamental Research Funds for the central Universities of China. References [1] G. Gerlach, M. Guenther, J. Sorber, G. Suchaneck, K.F. Arndt, A. Richter, Sens. Actuat. B 111–112 (2005) 555–561. [2] S.M.M. Quintero, R.V. Ponce F, M. Cremona, A.L.C. Triques, A.R. d’Almeida, A.M.B. Braga, Polymer 51 (2010) 953–958. [3] K. Podual, F.J. Doylelll, N.A. Peppas, Polymer 41 (2000) 3975–3983. [4] J. Hendri, A. Hiroki, Y. Maekawa, M. Yoshida, R. Katakai, Radiat. Phys. Chem. 60 (2001) 617–624.
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