Multi-Effect-Coupling pH-Stimulus (MECpH) Model for pH-Sensitive Hydrogel
2.5 Parameter Studies by Steady-State Simulation for Equilibrium of Hydrogelfor Equilibrium of Hydrogel
2.5.2 Influence of Young’s Modulus of Hydrogel
counterions, e.g. Na+in the present case, diffuse into the interior hydrogel to com- pensate the surplus charge. In contrast, the mobile anions are repulsed from entering the interior hydrogel. It is thus evident that the concentration of the Na+ion is higher whereas that of Cl−ion is lower in the case of the alkaline solution, compared with the acidic solution. As a consequence, the concentration differences between the interior hydrogel and the exterior solution increase tremendously, leading to higher osmotic pressure which effectively drives higher degree of swelling.
Figure 2.8a, b shows the theoretically predicted dependence of the swelling of the pH-sensitive hydrogel response to the changes in the initially fixed charge den- sity for an ideal solution at different pH levels. It is obvious that the change of the fixed charge concentration csm0at dry state strongly influences the equilibrium swelling of the hydrogels at high pH values, whereby the decrease of csm0dramat- ically reduces the degree of swelling at high pH values. The initial concentration of fixed charge csm0is a function of molar ratio of the comonomers during prepa- ration (Chu et al., 1995). As the molar ratio of carboxylic acid to 2-hydroxyethyl methacrylate decreases, the initially fixed charge density decreases dramatically.
The concentration difference thus decreases between the interior hydrogel and the exterior solution. As a result, this in turn mitigates the osmotic pressure and gener- ates smaller degree of hydrogel swelling. These observations are in agreement with the experimental trend reported by Siegel (1990). Figure 2.8b characterizes well the experimental phenomena, where a monotonic swelling is predicted with increas- ing the total molar concentration of ionizable groups per volume of solid network polymer.
Figure 2.9 exhibits the relation between the equilibrium swelling of the hydrogel and the concentration of fixed charge group csm0for three different buffer solutions, the ideal solution, the phosphate buffer and the Briton–Robinson buffer. The larger csm0is, the greater degree of swelling the hydrogel performs for both the buffer solu- tions at higher pH. At low pH level, however, the degree of swelling keeps almost constant for both the buffer systems. The figure evidently shows that the swelling equilibrium achievable in the Britton–Robinson system is always higher than that in the ideal solution and the phosphate buffer, especially as the initial concentra- tion of fixed charge group csm0increases highly. Swelling of the hydrogel is almost the same if bathed with either the phosphate buffer or the ideal solution, e.g. only NaCl and/or HCl in solution. However, the phosphate buffer shows greater degree of swelling with the increase of initially fixed charge group concentration csm0.
0 2 4 6 8 10 12 450
500 550 600 650 700 750 800 850 900 950 1000 1050 1100 1150
Hydrogel diameter (μm)
pH
Cs =2400mM
Cs =1800mM Cs =1200mM Cs =600mM
(a)
Hydrogel diameter (μm)
0 500 1000 1500 2000 2500 3000
500 600 700 800 900 1000 1100 1200 1300
Initial fix-charged concentration, Cs (mM) pH3
pH7 pH12
(b)
mo mo mo mo
mo
Fig. 2.8 Dependence of swelling on (a) bathing pH as the function of ionizable fixed charge concentration csmoand (b) varying ionizable fixed charge concentrations csmoin acidic, neutral and basic solutions
and basic solution (pH 12). The cationic concentrations, e.g. H+and Na+within the hydrogel are higher than those in the bath solution. In contrast, the anion concen- tration in the interior hydrogel is at a lower level than that in the external solution.
Electroneutrality is conserved everywhere.
499 500 501 502 503 504 505 506 507 508 509
Hydrogel diameter (μm)
HCl/NaCl solution Phosphate buffer Britton-Robinson buffer
(a)
0 500 1000 1500 2000 2500 3000
(b)
0 500 1000 1500 2000 2500 3000
500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600
Hydrogel diameter (μm)
HCl/NaCl solution Phosphate buffer Britton-Robinson buffer
Initial fixed charge concentration, Cmos (mM)
Initial fixed charge concentration, Cmos (mM)
Fig. 2.9 Influences of buffer systems on swelling equilibrium as the function of ionizable fixed charge concentration in (a) acidic medium of pH 3 and (b) basic medium of pH 9
The dissimilarity of the swelling response at lower and higher pH levels dis- plays two different conditions. As discerned from Fig. 2.10, the changes in Young’s modulus values of the hydrogel seem to have no significant effect on the degree of swelling at low pH. Probably as the hydrogel is still in compact state at low pH, the effect of changing Young’s modulus is very tiny on the swelling equilibrium. In contrast, the degree of swelling is controlled greatly by changing Young’s modulus
0 1 2 3 4 1.00
1.01 1.02 1.03 1.04 1.05
Hydrogen ion concentration (mM)
Distance across hydrogel diameter (mm) Normalized Young
modulus:
(E/E0) = 1.0 (E/E0) = 2.0 (E/E0) = 4.0
(a) Hydrogen ion (cH+)
(E/E0) = 1.0 (E/E0) = 2.0 (E/E0) = 4.0
0 1 2 3 4
300 302 304 306 308 310 312 314 316
Sodium ion concentration (mM)
Distance across hydrogel diameter (mm) Normalized Young modulus:
(b) Sodium ion (cNa+)
(E/E0) = 1.0 (E/E0) = 2.0 (E/E0) = 4.0
(c) Chloride ion (cCl–)
0 1 2 3 4
288 290 292 294 296 298 300 302
Chloride ion concentration (mM)
Distance across hydrogel diameter (mm) Normalized Young
modulus:
(E/E0) = 1.0 (E/E0) = 2.0 (E/E0) = 4.0 Normalized Young modulus:
0 1 2 3 4
0 5 10 15 20 25 30
Fixed-charge concentration (mM)
Distance across hydrogel diameter (mm)
(d) Fixed charge group (cf)
(E/E0) = 1.0 (E/E0) = 2.0 (E/E0) = 4.0 Normalized Young modulus:
0 1 2 3 4
–1.2 –1.0 –0.8 –0.6 –0.4 –0.2 0.0
Electric Potential (mV)
Distance across hydrogel diameter (mm)
(e) Electric potential (ψ)
(E/E0) = 1.0 (E/E0) = 2.0 (E/E0) = 4.0 Normalized Young modulus:
0.0 0.1 0.2 0.3 0.4 0.5
0.00 0.05 0.10 0.15 0.20 0.25
Displacement (μm)
Distance across hydrogel diameter (mm)
(f) Displacement (u)
Fig. 2.10 Distributive profiles of cH+,cNa+,cCl−,cf,ψand u as the function of normalized Young’s modulus (E/E0), where the PHEMA hydrogel is equilibrated in acidic medium of pH 3 with NaCl added to control the ionic strength
if the environmental pH level is high, as observed from Figs. 2.11 and 2.12. The phenomena mentioned occur owing to the fact that the more fixed charge groups are ionized as pH increases and thus the degree of swelling increases. However, the swelling is constrained as Young’s modulus increases. The interaction between expanding and retracting forces lasts until new equilibrium is reached.
Figure 2.13a shows the dependence of swelling of the hydrogel on the changes of environmental pH as function of Young’s modulus of the pH-sensitive hydrogel.
0 1 2 3 4 0.10
0.15 0.20 0.25 0.30 0.35
Hydrogen ion concentration (mM)
Distance across hydrogel diameter (mm) X10–3
(a) Hydrogen ion (cH+)
0 1 2 3 4
300 400 500 600 700 800 900 1000
Sodium ion concentration (mM)
Distance across hydrogel diameter (mm)
(b) Sodium ion (cNa+)
0 1 2 3 4
100 120 140 160 180 200 220 240 260 280 300 320 340
Chloride ion concentration (mM)
Distance across hydrogel diameter (mm)
(c) Chloride ion (cCl–)
0 1 2 3 4
0 100 200 300 400 500 600 700 800
Fixed-charge concentration (mM)
Distance across hydrogel diameter (mm)
(d) Fixed charge group (cf)
0 1 2 3 4
–25 –20 –15 –10 –5 0
Electric potential (mV)
Distance across hydrogel diameter (mm)
(e) Electric potential (ψ)
0.0 0.2 0.4 0.6 0.8 1.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Displacement (mm)
Distance across hydrogel diameter (mm)
(f) Displacement (u)
Normalized Young modulus:
(E/E0) = 1.0 (E/E0) = 2.0 (E/E0) = 4.0
Normalized Young modulus:
(E/E0) = 1.0 (E/E0) = 2.0 (E/E0) = 4.0
Normalized Young modulus:
(E/E0) = 1.0 (E/E0) = 2.0 (E/E0) = 4.0
Normalized Young modulus:
(E/E0) = 1.0 (E/E0) = 2.0 (E/E0) = 4.0 Normalized Young
modulus:
(E/E0) = 1.0 (E/E0) = 2.0 (E/E0) = 4.0
Normalized Young modulus:
(E/E0) = 1.0 (E/E0) = 2.0 (E/E0) = 4.0
Fig. 2.11 Distributive profiles of c+H,c+Na,c−Cl,cf,ψand u as the function of normalized Young’s modulus(E/E0), where the PHEMA hydrogel is equilibrated in neutral medium with NaCl added to control the ionic strength
The MECpH model theoretically predicts that, for the hydrogels with larger Young’s modulus, the degree of swelling decreases at higher solution pH. The characteris- tics become more visible in Fig. 2.13b when the normalized Young’s modulus is plotted against the diameters of hydrogels at equilibrium state. The magnitude of swelling reduces exponentially with the increase of Young’s modulus. Usually it is known that Young’s modulus of the hydrogel is strongly dependent on prepara- tion process, where the modulus is primarily determined by the volume per molar
0 1 2 3 4 1.0
1.5 2.0 2.5 3.0
Hydrogen ion concentration (mM)
Distance across hydrogel diameter (mm) Normalized Young modulus:
(E/E0) = 1.0 (E/E0) = 2.0 (E/E0) = 4.0 X10–9
(a) Hydrogen ion (cH+)
Normalized Young modulus:
(E/E0) = 1.0 (E/E0) = 2.0 (E/E0) = 4.0
0 1 2 3 4
300 400 500 600 700 800 900
Sodium ion concentration (mM)
Distance across hydrogel diameter (mm)
(b) Sodium ion (cNa+)
(E/E0) = 1.0 (E/E0) = 2.0 (E/E0) = 4.0
0 1 2 3 4
100 120 140 160 180 200 220 240 260 280 300 320 340
Chloride ion concentration (mM)
Distance across hydrogel diameter (mm) Normalized Young
modulus:
(c) Chloride ion (cCl–)
Normalized Young modulus:
(E/E0) = 1.0 (E/E0) = 2.0 (E/E0) = 4.0
0 1 2 3 4
0 100 200 300 400 500 600 700 800
Fixed-charge concentration (mM)
Distance across hydrogel diameter (mm)
(d) Fixed charge group (cf)
(E/E0) = 1.0 (E/E0) = 2.0 (E/E0) = 4.0 Normalized Young modulus:
0 1 2 3 4
–25 –20 –15 –10 –5 0
Electric potential (mV)
Distance across hydrogel diameter (mm)
(e) Electric potential (ψ)
Normalized Young modulus:
(E/E0) = 1.0 (E/E0) = 2.0 (E/E0) = 4.0
0.0 0.2 0.4 0.6 0.8 1.0
0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Displacement (mm)
Distance across hydrogel diameter (mm)
(f) Displacement (u)
Fig. 2.12 Distributive profiles of cH+,cNa+,cCl−,cf,ψand u as the function of normalized Young’s modulus(E/E0), where the PHEMA hydrogel is equilibrated in basic medium of pH 12 with NaCl added to control the ionic strength
ratio of copolymer mixture which directly quantifies the density of entanglement strands or crosslinking ratio. As the crosslinking content increases in the polymer network, the hydrogel enhances larger retraction force and thus develops higher Young’s modulus. The phenomenon always exists regardless of buffer contents as depicted in Fig. 2.14. The diameters of the swollen hydrogels are plotted against the normalized Young’s modulus for the buffer solutions of pH 3 and 9. The increase of Young’s modulus reduces exponentially the swelling of the hydrogels for the three
0 2 4 6 8 10 12 400
500 600 700 800 900 1000 1100 1200
Hdrogel diameter (μm)
pH Normalized Young modulus:
E/E0=0.7 E/E0=1.5 E/E0=3.0 E/E0=6.0
(a)
0 2 4 6 8 10 12 14 16 18 20
500 600 700 800 900 1000 1100 1200
Hydrogel diameter (μm)
Normalized Young modulus, E/E0 pH3 pH7 pH12
(b)
Fig. 2.13 Dependence of swelling on (a) bathing pH as the function of normalized Young’s modulus(E/E0)and (b) varying normalized Young’s modulus(E/E0)in acidic, neutral and basic solutions
different buffer systems as illustrated in Fig. 2.14. Influence of buffer contents on the swelling equilibrium at higher pH is more significant than that at lower pH. When the pH of buffer solutions is low, the degree of swelling of the hydrogel is almost insignificant even in different buffer systems and Young’s moduli. If the pH is high, the obvious differences in the degree of swelling are observed for different buffer systems. Further, the Britton–Robinson is the unchanging leader for providing the better buffer solution when large swelling scale is required. It should be pointed out that the influence of the buffer contents vanishes if Young’s modulus is high enough.