I n d . E n g . C h e m . Res. 1990, 29, 554-560
554
nickel is relative to both the formation rate of the carbon deposit and the adsorption strength of hydrogen on 15% Ni/A1203catalysts prepared a t various initial pH values of the Ni(N03)2solution.
Conclusions The change of the initial controlled-pH value of the preparation of the catalysts can influence the catalytic characterization of nickel-alumina catalysts significantly. The XRD results show that the smaller nickel particle size is obtained by adjusting the initial p H values of nickel nitrate solution to pH 6 or 8.3 with N H 4 0 H . However, same sintering extent of nickel occurs for catalysts prepared a t various initial pH values during the catalytic cracking of n-hexane. The order of the initial pH values of the preparation of the catalysts by the coke formation rates of nickel catalysts in the catalytic cracking of nhexane is in good agreement with that by the peak temperature of the T P D spectra of these nickel catalysts, which is pH 6 > 8.3 > 5 > 3. These results indicate that the formation rate of carbon deposit has a strong relationship with the adsorption strength of hydrogen on the nickel catalyst. Accordingly, both the tendency of the average methane yield and the tendency of the average ratio of n-hexane conversion to the amount of carbon deposit can be reasonably expected from the tendency of the formation rate of carbon deposit for the initial pH values of the preparation of the catalysts. The order of the initial pH values of the preparation of the catalysts by the average methane yield is pH 8.3 > 6 > 5 > 3. However, that by the average ratio of hexane conversion to the amount of carbon is pH 3 > 5 > 8.376. Acknowledgment
Registry No. Ni, 7440-02-0; N i ( N 0 J 2 , 13138-45-9; hexane, 110-54-3; carbon. 7440-44-0; methane, 74-82-8.
Literature Cited Barbier, J. Deactivation of reforming catalysts by coking. Appl. Catal. 1986, 23, 225-243. Blackmond, D. G.; Williams, J. A.; Kesraoui, S.; Blazewick, D. S.The Effects of Cs Promotion of Rh/AI,O, Catalysts. J . Catal. 1986, 101, 496-504. Burch, R. The Selective Isomerization of n-Hexane over Nickel on Silica-Alumina Catalysts. J . Catal. 1979, 58, 220-229. Chen, I. W.; Shiue, D. W. Reduction of Nickel-Alumina Catalysts. Ind. Eng. Chem. Res. 1988a, 27, 429-434. Chen, I. W.; Shiue, D. W. Resistivity to Sulfur Poisoning of Nickel-Alumina Catalysts. Ind. Eng. Chem. Res. 1988b, 27,1391-1396. Corella, J.; Monzon, A. Modeling of the Deactivation Kinetics of Solid Catalysts by Two or More Simultaneous and Different Causes. Ind. Eng. Chem. Res. 1988, 27, 369-374. Cullity, B. D. Elements of X-ray Diffraction, 2nd ed.; AddisonWesley: Reading, MA, 1978. Diffenbach, R. A,; Fauth, D. J. The role of p H in the Performance of Precipitated Iron Fischer-Tropsch Catalysts. J . Catal. 1986, 100, 466-476. Hegedus, L. L.; Summers, J . C. Improving the poison Resistance of Supported Catalysts. J . Catal. 1977, 48, 345-353. Martin, G. A,; Primet, M.; Dalmon, J. A. Reactions of CO and C 0 2 on Ni/Si02 above 373K as Studied by Infrared Spectroscopic and Magnetic methods. J. Catal. 1978, 53, 321-330. Mirodatos, C.; Praliand, H.; Primet, M. Deactivation of Nickel-Based Catalysts during CO Methanation and Disproportionation. J . Catai. 1987,107,275-287. Mustard, D. G.; Bartholomew, G. H. Determination of Metal Crystalline Size and Morphology in Supported Nickel Catalysts. J . C'atal. 1981, 67, 186-206. Richardson, J. T. SNG Catalyst Technology. Hydrocarbon Process. 1973, Dec, 91-95. Richardson, J. T.; Crunip, J. G. Crystalline Size Distributions of Sintered Nickel Catalysts, J . Catal. 1979, 57, 417-425. Received for review September 21, 1988 Revised manuscript received J u n e 5, 1989 Accepted December 5, 1989
We thank the Tatung Company and the Chinese National Science Council for their financial aid.
MATERIALS AND INTERFACES Swelling of Ionic Gels in Electrolyte Solutions Ebrahim Vasheghani-Farahani, Juan H. Vera, David G. Cooper, and Martin E. Weber* D e p a r t m e n t of Chemical Engineering, McGill University, Montreal, Quebec, Canada H3A 2A7
T h e swelling behavior of a n anionic gel and a cationic gel in electrolyte solutions of different p H and salt concentrations was investigated. T h e cationic gel imbibed more water than the anionic gel a t p H 5 6. Addition of salt to the external solution depressed the degree of swelling, and solutions containing counterions of higher charge were more effective in shrinking of t h e gels. T h e rule of additivity for t h e osmotic pressure of polyelectrolyte-salt solutions described t h e swelling of ionic copolymers of acrylamide as a function of t h e ionic composition of t h e external solution. Cross-linked polymer gels exhibit large changes in volume in response to changes in the composition (Tanaka, 1978), temperature (Hirokawa and Tanaka, 1984), or osmotic pressure (Katchalsky and Zwick, 1955) of the external solution. This has led to the use of such gels as
* Author
t o whom correspondence should be addressed.
extraction solvents in chemical separation systems (Cusler et al., 1984; Freitas and Cussler, 1987) and as physiologically sensitive compounds in biomedical and pharmaceutical applications (Peppas, 1987). In this work, the swelling behavior of anionic and cationic copolymer gels of acrylamide in aqueous electrolyte solutions of various pH and ionic strength was studied.
0888-5885/90/2629-0554$02.50/0 C 1990 American Chemical Society
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 555 These experiments were carried out with solute concentrations typical of those in fermentation broths and in buffered enzyme systems. The ionic monomer was sodium acrylate for the anionic gel and [3-(methacrylamido)propylltrimethylammonium chloride for the cationic gel. The swelling behavior has been interpreted by using theory for ionic gels, which accounts for the non-Gaussian behavior of rubberlike elasticity and the dissociation behavior of polyelectrolytes.
Theory Early theories describing the swelling of polyelectrolyte gels have been reviewed by Helfferich (1962). Flory’s (1953) treatment for the thermodynamics of gel swelling provides the basis for the subsequent work. Many researchers (Hasa et al., 1975; Ricka and Tanaka, 1984; Gehrke et al., 1986; Sun, 1986) attempted to modify it for quantitative prediction of swelling effects. Among the most recent efforts, Brannon-Peppas and Peppas (1988) presented a derivation in which the osmotic coefficients are set equal to unity, the distribution of the polymer chains is considered Gaussian, and all the effects are accounted for by a structural term. Prange et al. (1989) proposed a model based on the quasi-chemical theory (Guggenheim, 1952; Panayiotou and Vera, 1980) suitable for the study of temperature-sensitive hydrogels which exhibit lower critical solution temperatures. Konak and Bansil (1989) modified the elasticity term by including an effect of the electrostatic persistence length. They included an explicit electrostatic term and used a simplified osmotic term. Otake et al. (1989) presented a model that considers a hydrophobic interaction for the thermally induced discontinuous shrinkage of hydrogels. In this work, we follow Flory’s treatment as discussed by Hasa et al. (1975) and apply it to the swelling of gels. The theoretical description of the swelling of the polyelectrolyte gels a t equilibrium is based on the minimization of the Gibbs free energy of the gel. The freeenergy change corresponding to the volume change during swelling of a gel, 53,is the sum of contributions due to mixing of pure solvent with an initially pure, amorphous, unstrained gel network, AG,, due to configurational changes of the gel structure, AG2, and due to mixing of ions with solvent, AG3. An ionic gel is subjected to a swelling pressure, x , which is expressed as the sum of three components corresponding to each contribution to AG:
volume fraction a t gel formation. The configurational contribution, x 2 , is evaluated from the configurational free-energy change, AG2, during swelling. Assuming isotropic swelling, by differentiating AG2 with respect to volume and expanding the inverse Langevin function in a power series (Treloar, 1958; Hasa et al., 1975), x 2 is obtained as
1
-
n
99 + -(u,/u)175
x1 =
RT
--[ln
(1 - u )
+ u + xu2]
(2)
VI
where u is the polymer volume fraction and VI is the molar volume of the solvent. The polymer-solvent interaction parameter, x, can be expressed as (Hasa et al., 1975) x = 0.44 + 0 . 0 6 ~ (3) Expanding the logarithmic term in eq 2 in a power series, neglecting terms higher than cubic, and combining the result with eq 3 give
RT
+ 0.267Uo/x) (4) Vl where X = u o / u is the swelling ratio and uo is the polymer “1
= --(~o/X)~(-0.06
1
(5)
The osmotic pressure, x 3 , attributed to the difference between the osmotic pressure of the mobile ions in the gel and in the external solution is given by (Flory, 1953) i
I
where Ciand Ci are the concentrations of mobile ions in the external solution and in the gel, respectively. The osmotic coefficients, @ and 4, are discussed in detail below. Osmotic Coefficients. Charged groups attached to the network play an essential role in swelling phenomena. To account for the nonideal behavior of the polyelectrolyte gel, an osmotic coefficient for the gel phase, a, is defined: @
=
(7)
Tp/xideal
The ideal osmotic pressure of the salt-free polyelectrolyte solution is given by the Van’t Hoff expression Tideal
= RT(n,a
+ np)
(8)
where n, is the molarity of the monomer, a is the degree of ionization, and n p is the molarity of the polymer. Since polyelectrolytes are strongly nonideal, a correction factor, 4 , called the osmotic coefficient, is introduced (Alexan&owicz, 1960): xp =
RT(n,&,
+ np)
(9)
Combination of eqs 7-9 gives
When n,a
>> n p ,eq 9 becomes irp
The equilibrium condition is obtained when x is set equal to zero. The osmotic pressure of a polymer solution, xl, is given by the Flory-Huggins theory (Flory, 1953):
1 + -513 ( u , / u ) ~ / ~ - + ... 875 n3
1
n2
= RTn,a@,
(11)
According to Katchalsky (1971), the dissociation behavior of polyelectrolyte solutions can be summarized as follows. It is often found that at higher degrees of dissociation cqbP is approximately constant or that the osmotic coefficient decreases somewhat as the degree of ionization increases. It is also observed that the dilution of a polyelectrolyte solution does not lead to stronger dissociation of counterions, and often the opposite effect is observed; i.e., 4pdecreases upon dilution, which indicates a strong binding of the counterions to the polyion with lower osmotic activity. The stronger electrostatic attraction of small polyvalent ions to the polyion should reduce the fraction of free counterions; hence, & for bivalent counterions should be about one-half of the value for the monovalent counterions. This expectation was confirmed experimentally. Measurements of +p for magnesium alginate gave a value of 0.15, compared with 4p = 0.4 for sodium alginate or 4 = 0.35 for potassium alginate (Katchalsky et al., 1961). f’or mixed salts, the results are more complex. Dolar and Peterlin (1969) extended the rodlike model for the evalu-
556
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990
ation of the osmotic coefficient of mixed polyelectrolyte salts in which the polyion was neutralized by both monoand bivalent ions. The theoretical predictions were tested qualitatively by Dolar and Kozak (1970). Both theory and experiment indicated the existence of a maximum value of $p a t a certain ratio of monovalent to bivalent ions. Alexandrowicz (1960, 1962) found that the osmotic pressure of a polyelectrolyte and a low molecular weight monomonovalent salt system could be represented by an additivity rule as the sum of the osmotic pressure of the salt-free polyelectrolyte solution, 7rp, and the polyelectrolyte-free salt solution, 7rs: 7r=7rp+7r
(12)
Following Alexandrowicz for the polyelectrolyte/monomonovalent salt system, x,
= RT(2ns&)
(13)
f = CNa+/nma
(23)
and combining eqs 21 and 22 yield
2an,cuf2
+f
- 1=0
(24)
where a = CM2+/CNa+2.The ion swelling pressure, 7r3, for CM2+< IO-* M in the presence of Na+ is then obtained through eq 6 as 7~~
= RTN[n,a(l
+ f 1/21 + 4CCJ
(25)
To obtain the osmotic coefficient of the gel phase, the concentration of mobile co-ions in the gel phase must be determined. In the absence of data, these concentrations may be estimated from Donnan equilibrium (Kitchener, 1957). For a gel with a concentration of fixed charges, rima, in an ideal solution of cation C with a charge of Zc and anion A with a charge of ZA,Donnan theory gives (Gehrke, 1986)
where n, is the molarity of the salt and the factor 2 accounts for the dissociation of the salt into two ionic species. Combination of eqs 11-13 yields 7~
= RT(n,a&
+ 2ns$,)
(14)
For a cationic gel, the following result is obtained:
From eq 7 the osmotic pressure, x , can be written as 7r
= RT(n,cu
+ 2nJ@
(15)
Combination of eqs 14 and 15 yields
a=
x4p + 20, x + 2
(16)
where x = n,cu/n,. For 0.5 I x I 8, the deviation of @ from experimental results can be as large as 15% (Alexandrowicz, 1960). In the presence of a bimonovalent salt, the concentration of counterions in a solution containing a negatively charged polyion is
n, = n,
1 + -rima 2
(17)
while the concentration of co-ions is Using the additivity rule for this system gives x 4 p + 64s
@ =
x + 6
(19)
Equations 16 and 19 also hold for a mixture of positively charged polyions with monomonovalent and monobivalent salts, respectively. When the external solution contains both mono- and bivalent counterions, ideal Donnan theory is used to determine the concentration of counterions in the gel phase. According to this theory, (C/CJ1/'~ = constant
(20)
When the polyion is neutralized by both Na+ and M2+ions, eq 20 can be written: (C,,+/CN,+)2 = CMZ+/CM2+
(21)
When the concentration of salt within the gel is negligible compared to the concentration of fixed charges, rime, the electroneutrality of the gel phase implies that C N a + + 2 c M 2 + = n,(Y (22) Defining f as the fraction of the ionic sites on the polyion neutralized by Na+
Katchalsky and Michaeli (1955) concluded that the concentrations of co-ions in the gel calculated from ideal Donnan equilibrium were close to experimental values when 01 5 0.1. For LY > 0.1, there was an increasing discrepancy between theory and experiment as the degree of ionization increased.
Gel Preparation and Experimental Procedure Gels were prepared in aqueous solution a t 23 " C by free-radical copolymerization of acrylamide (632 mM), N,N'-methylenebis(acrylamide1 (8.6 mM), and either sodium acrylate (71.2 mM) for the anionic gel or [3-(methacrylamido)propyl]trimethylammonium chloride (71.2 mM) for the cationic gel. Ammonium persulfate (1.75 mM) and sodium metabisulfite (2.1 mM) or N,N,N',N'tetraethylmethylenediamine (18.6 mM, cationic gel) were used as initiator and accelerator, respectively. These compounds were added to nitrogen-sparged distilled water to make 50 mL of solution. The solution was flushed with nitrogen for 10 min and transferred to a large test tube which contained a number of smaller glass tubes of 2.5" i.d. The test tube was then sealed. After 24 h, the gels were forced from the 2.5-mm tubes and cut into cylinders having lengths roughly equal to their diameters. The original mass, M,, of each piece was determined and then it was dialyzed for 48 h against a large amount of distilled water to remove minute quantities of impurities and unreacted monomers or oligomers trapped in the network. The degree of swelling a t 23 "C was measured by immersing a piece of dialyzed gel (original mass, M,, of approximately 0.01 g) in 2 L of a solution of known concentration and pH. Every 6 h the solution was replaced with fresh solution. After equilibration for a t least 48 h, the pH was measured, and swollen gel was removed from the solution and weighed. The swelling ratio was defined as the ratio of swollen mass, M , to the original mass, M,. Experimental Results Figure 1 shows the swelling behavior of the anionic and cationic gels as the pH was varied by addition of either NaOH or HN03. It should be noted that the mass of polymer in the gel a t preparation was about 5% of M,;
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 557
\ " 0
2
4
6
8
10
PH
Figure 1. Swelling ratio as a function of pH: (A)cationic gel, ( 0 ) anionic gel.
therefore, the mass of water absorbed per unit mass of dry polymer was approximately 20 times larger than the swelling ratio, M /M,. The swelling ratio of the anionic gel increased with increasing pH of the external solution up to a maximum when ionization of the charged network was complete. A further increase of pH increased the osmotic pressure of the external solution, hence decreasing the ion swelling pressure and reducing the degree of swelling (see eq 6). The same phenomena occurred for the cationic gel as the pH decreased. Between them, these gels cover pH values that are encountered in the separation and recovery of many enzymes. According to the theory, salts affect the swelling behavior of polyelectrolyte gels. Ricka and Tanaka (1984, 1985) studied the salt effect on the swelling behavior of anionic gels. Kou et al. (1988) presented results for the effect of pH and gel composition on the swelling of anionic hydrogels a t 37 " C . Siege1 and Firestone (1988) studied the effect of pH and of ionic strength on the swelling of hydrophobic cationic gels. They showed that for these gels both factors are important. In this work, we have studied the swelling behavior of anionic and cationic gels in the presence of the salts that are frequently encountered in fermentation broths. Anionic Gel. Figure 2 shows the swelling of the anionic gel a t equilibrium in aqueous solutions, each of which contained a single salt. All salts depressed the degree of swelling if the concentrations were sufficiently high, but some were more effective than others. The monovalent sodium, potassium, and silver ions had the same influence on the gel a t the same molarity. The effect of bivalent calcium and cobalt was more pronounced than that of the monovalent cations. The trivalent lanthanum ion had the largest effect on the swelling of the anionic gel. It is known that some ions can form complexes with charged macromolecules containing carboxylate groups. Such ions are expected to have larger effects on swelling than noncomplexing ions of the same charge. Cupric ions decreased swelling more than other bivalent ions-see Figure 2 and Ricka and Tanaka (1985). Early work with this gel demonstrated that it formed a strong complex with Cu2+but not with Ni2+,Co2+,or Zn2+ (Wall and Gill, 1954). This anomaly is not a problem for biological application because copper is toxic to microorganisms and thus it is not a component of fermentation broths. Silver ions are capable of complex formation with some negatively charged macromolecules (Katchalsky et al., 19611, but we observed no difference between Ag+ and other monovalent cations.
lor-
!I
l!IP
lo-'
I(]-'
I(]-'
I(]-?
Molarity of Cations
Figure 2. Swelling of anionic gel in salt solutions (pH 7): (A) NaN03, (W &NO3, (v) KMn04, (VI KzS04,(0)Ca(N03)*, ( 0 ) CaC12,(0) Cocl,, (*I C U ( N O ~ )(A) ~ , La(N0J3.
I
'
I
I
I
I
I
I
i
22
b 9,
t -.
I)
]!I-'
*3
Ill',
lit'
I O '
1
\
Ill'
111:
I(!
Molarity of Anions
Figure 3. Swelling of cationic gel in salt solutions (pH 7): (v)NaC1, (0) NaN03, ( 0 ) CaC12, (*I Ca(N03),, (A)Na2S04, (*) NaN03 + NazS04 (1, 75% SO:-; 2, 50% SO,2-; 3, 25% SO:-), (solid line) theory.
Cationic Gel. Figure 3 shows the swelling behavior of the cationic gel in the presence of various salts. Nitrate and chloride ions were introduced with either monovalent sodium or bivalent calcium ions. All nitrates and chlorides gave the same swelling ratio a t the same anion concentrations. At pH 7 , for the concentration of sodium sulfate used in our experiments, the major anionic species in solution was SO:-. This bivalent ion had a much stronger effect on the swelling of the gel than monovalent ions-as noted for the anionic gel. This was also demonstrated by the studies with mixtures of Na2S04and NaNO, for which the swelling was intermediate between the monovalent and bivalent data. The theoretical curves in Figure 3 are discussed subsequently.
558
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990
t
't
t
2
1
\
1
1
--*-*-*-
i
4
6
8
q
1
\ \ \
10
PH
Figure 4. Swelling of cationic gel as a function of pH in various salt solutions: (A)NaN03, M; (A)50% NaNO, + 50% NaC1, M; (0) Na2HP04, M; ( 0 )Na2HP04, M; ( 0 ) Na2S04, M; ( e ) Na2S04. M. Ill
In a typical fermentation medium, phosphate is the most concentrated anion. At pH 7 , the major ions are HP042and H,PO,-. The balance between the monovalent and bivalent ions is changed by pH. Figure 4 shows the swelling behavior for pH values between 4 and 8 for sodium phosphate concentrations of lo-, and lo4 M and for three other salts. In Figure 1, the swelling ratio of the cationic gel without salt is constant at a value of 20 between pH 4 and pH 7.5. When NaCl, NaN03, or Na2S04is added, Figure 4 shows that p H has little effect on swelling. For phosphate, the swelling ratio decreases as the pH increases because the phosphate equilibrium shifts from HzP04-to
HPOd2-. Comparison of Theory and Experiment Both gels contained 10 mol 5% ionizable monomer, and we assume that these groups are completely ionized a t pH 7 ; i.e., N = 0.1. The molarity of monomoric units in the polymeric network, n,, is related to the concentration of monomers a t gel formation, nmO,by n m = nmo/X (28) where nmo= 0.712 M. Determination of Parameters. The molarity of salt in the gel phase, n,, is related to the co-ion concentration in the gel, which can be obtained from eqs 26 and 27. For a monomonovalent salt, n, is equal to the co-ion concentration, whereas for bimonovalent salts (anionic gel) or monobivalent salts (cationic gel) n, is one-half of the co-ion concentration. The osmotic coefficients, $J and &, were obtained from experimental results for NaN0, (Hamer and Wu, 19721, CaC1, (Staples and Nuttal, 1977), and Na2S04(Goldberg, 1981). The osmotic coefficient, &,, of the sodium polyacrylate solution changed from 0.55 to 0.65 a t N = 0.1 as n, was varied from 0.01 to 0.25 (Kern, 1939). For 0.5 f: x 5 8, the osmotic coefficient of the gel phase, a, was obtained from the experimental results of Alexandrowicz (1960). In the absence of data, the osmotic coefficient of the calcium polyacrylate solution was assumed initially to be half of that for the sodium polyacrylate solution. However, use of this value underestimated the swelling of the gel, particularly at C 5 M, as discussed below. The osmotic coefficient, &, of a positively charged polyelectrolyte solution (cationic gel) was assumed to be equal to that of a negatively charged polyelectrolyte solution (anionic gel). The concentration of constituent chains, uo, is
Ill1
Ill
I O 1
I l l '
I
Ill
klolarity of Cations
Figure 5 Comparison of theory and experiment for swelling of CaC1, (pH adjusted anionic gel in salt solutions: (A)NaN03, (0) with NaOH), ( 0 )CaCI, (pH adjusted with CaOH,), (solid line) theory.
related to the concentration of constituent chains per unit volume in the dry state, Vd, by UO = VdUO (29) where uo is the polymer volume fraction at gel formation; mol/cm3 was used here uo = 0.037. A value of Vd = 7 X based on previous work on ionic copolymer gels of acrylamide (Ilavsky and Hrouz, 1982). The number of statistical segments per chain, n, which is a function of fixed charge density, quality of the solvent, added salt, temperature, etc., was used as a free parameter to fit the experimental data. The best values of n are approximated by n = 4OX-O3 (anionic gel)
(30)
and
n = 2 4 X q 3 (cationic gel) (31) These expressions were obtained using swelling data for monovalent counterions. They were then used for bivalent counterions. Theoretical Predictions. The theoretical predictions for the swelling of the anionic gel in monomonovalent (NaNO,) and bimonovalent (CaCl,) salt solutions are compared to experimental data in Figure 5. This figure also includes the theoretical predictions for the monomonovalent salt in the ideal case (a = 4 = 1) and for the case where the nowGaussian distribution of chain extension is neglected, Le., n = 03. For a high swelling ratio, the contribution to swelling pressure due to mixing of the polymer with solvent, rl,is negligible. By use of n given by eq 30, the theoretical predictions agree well with experiment for monomonovalent salts. If the non-Gaussian term is neglected, the predictions are very poor a t low M). If the osmotic coefficients concentrations (C, < are assumed to be unity, the swelling ratio is also badly overpredicted. In Figure 5 two sets of experimental data are presented for CaC1,. For the open circles, the pH was adjusted with NaOH; hence, this system contained small concentrations of monovalent as well as bivalent cations. The filled circles
Ind. Eng. Chem. Res., Vol. 29, No. 4, 1990 559 Table I. Equivalent Fraction of Na+ in Gel Phase, f, and Osmotic Coefficient, @, of the Anionic Gel CaCl,, M X f @ 0.55 10-7 16.8 0.10 0.44 10” 13.0 0.03 0.01 0.44 10-5 12.5
represent experiments in which a single bivalent ion, Ca2+, was present. T o obtain these data, dialyzed gel particles (which were formed from sodium acrylate) were first equilibrated with M CaC1, solution t o replace the residual sodium. The particles were then swollen in 10-4-10-7 M CaC1, solutions while the p H was adjusted with Ca(OH),. When the p H was adjusted with NaOH and air was excluded from t h e vessels, about 3 X IO* M NaOH was required t o bring the p H to 7 . Figure 5 shows that a t concentrations of CaC1, below lo4 M the swelling ratio increased above the value for the system containing only Ca2+. In this case, the swelling of the gel tends t o its behavior in the presence of monovalent counterions, and it can be predicted by the theory as shown in Figure 5. Similar behavior in mixed mono- and bivalent systems was exhibited by poly(methacry1ic acid) gels (Katchalsky and Zwick, 19551, by copolymer gels of acrylic acid (Ricka and Tanaka, 1984), and by our cationic gel (see Figure 3). If air were not carefully excluded, additional NaOH was required to maintain the pH. At concentrations below M CaCl,, any swelling ratio between the pure Ca2+and Na+ values could then be obtained depending upon the experimental procedure. The data for the system containing only Ca2+were well represented by the theory with n from eq 30 and cpp = 0.44-see Figure 5. The latter value was somewhat larger than t h e expected value of 0.33, which is one-half of the value for sodium polyacrylate solutions. In the mixed cation system, the gel polyion was neutralized by a mixture of monovalent and bivalent cations. The swelling behavior was investigated by noting that the concentration of NaOH in the external solution was approximately 3 X lo4 M. The equivalent fraction of Na+ for the gel phase, obtained through eq 24 for various CaC12 solutions, is shown in Table I. For CaC1, concentrations of M and above, there is too little sodium present inside the gel to affect swelling. At lower concentrations, where the equivalent fraction increases, we calculated the value of the osmotic coefficient, 9,required to match the data by using eq 30 and the theory described earlier. The results are shown in Table I. The osmotic coefficient, 9, for the mixture of monovalent and bivalent counterions is higher than that for bivalent ones. These results are consistent with theoretical expectations as discussed earlier. Figure 3 also shows the comparison of the theoretical predictions with experimental data for swelling of the cationic gel in salt solutions. There is good agreement between theory and experiment for the effect of mono- and bivalent anions on the swelling of the cationic gel. The osmotic coefficient of bivalent counterions, cpp, equal to 0.44 was used t o fit the experimental results.
Conclusion The degree of swelling of an ionic gel in a salt solution is determined largely by the concentration of the mobile counterions. At a fixed composition, counterions of higher charge cause a larger shrinkage of the gel. Complex formation can increase this effect. At low concentrations of salts, stray counterions have a strong effect on swelling. Small concentrations of counterions, present from acid or
base addition to adjust pH, may determine the swelling behavior. Our cationic gel swells about 400 times its dry weight at pH values between 3 and 7. Its swelling is still appreciable (about 200 times dry weight) in M phosphate solution over the same range of pH. This gel is a good candidate for biological applications. A thermodynamic model based on the additivity rule for the osmotic pressure of polyelectrolyte-salt solutions described the effect of salt on the swelling of ionic gels. Theoretical predictions agreed with the experimental results for swelling of ionic gels in monomonovalent salt solutions. Assuming the osmotic coefficient is equal to unity and neglecting the non-Gaussian distribution of chain extension gave predictions well above the data at low salt concentrations. The effect of bivalent counterions on the swelling of ionic gels was also well represented by the theory if no complex formation occurred. The effect of a salt solution containing both mono- and bivalent counterions on swelling behavior was predicted a t low salt concentration using ideal Donnan equilibrium.
Acknowledgment This work was supported by the Natural Sciences and Engineering Research Council of Canada.
Nomenclature a = CMz+/CNa+2 C = salt concentration, M (2, = molarity of anion in the external solution CA = molarity of anion in the gel phase Cc = molarity of cation in the external solution Cc = molarity of cation in the gel phase C, = molarity of mobile ions in the external solution Ct = molarity of mobile ions in the gel phase f = fraction of ionic sites neutralized by Na+ AG = Gibbs free-energy change l G 1 = free-energy change due to mixing of polymer with solvent AGz = elastic free-energy change AG3 = free-energy change due to mixing of ions with solvent M = swollen gel mass M, = original mass of gel at preparation n = number of statistical segments per chain n, = molarity of component i n, = molarity of monomer np = molarity of polymer n, = molarity of salt in polyelectrolyte-salt mixture n, = molarity of counterions for negatively charged polymer/ bimonovalent salt system n_ = molarity of co-ion for negatively charged polymer/bimonovalent salt system R = gas constant T = absolute temperature V = gel volume V, = molar volume of solvent u = polymer volume fraction uo = polymer volume fraction at gel formation X = swelling ratio, M/M, x = concentration of dissociated monomers/concentration of salt in polyelectrolyte-salt system 2, = charge of anion Zc = charge of cation Greek Letters = degree of ionization
(Y
vd
= concentration of constituent chains per unit volume in
dry state LJ,,
= concentration of constituent chains per unit volume at gel formation
I n d . Eng. Chem. Res. 1990, 29, 560-564
560
= osmotic pressure = ideal osmotic pressure 7rp = osmotic pressure of salt-free polyelectrolyte solution rS = osmotic pressure of polyelectrolyte-free s a l t solution x1 = osmotic pressure of polymer solution T:, = elastic c o m p o n e n t of osmotic p r e s s u r e T , = osmotic p r e s s u r e of mobile ions = osmotic coefficient of gel phase 4 = osmotic coefficient of e x t e r n a l solution 4p = osmotic coefficient of salt-free polyelectrolyte solution & = osmotic coefficient of polyelectrolyte-free s a l t solution x = polymer-solvent interaction parameter K
Tideal
Registry No. N a N 0 3 , 7631-99-4; AgNO,, 7761-88-8; KMnO,, 7722-64-7; KzS04, 7778-80-5; Ca(NO,),, 10124-37-5; CaCI,, 10043-52-4; CoCl,, 7646-79-9; CU(NO,)~,3251-23-8; La(NO&, 10099-59-9; NaCl, 7647-14-5; Na2S04, 7757-82-6; N a 2 H P 0 4 , 7558-79-4; (acrylamide)(N,N'-methylenebis(acry1amide))(sodium acrylate (copolymer), 33882-67-6; (acrylamide (N,N'-methylenebis(acrylamide))([ 3-(methacrylamido)propyl]trimethylammonium chloride) (copolymer), 98587-56-5.
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I. Theory. J . Polym. Sci., Polym. Phys. Ed. 1975, 13, 253-262. Helfferich, F. Equilibria: Swelling. Ion-Exchange; McGraw-Hill: New York, 1962. Hirokawa, Y.; Tanaka, T. Volume Phase Transition in a Non-Ionic Gel. J . Chem. Phys. 1984, 81, 6379-6380. Ilavsky, M.; Hrouz, J. Phase Transition in Swollen Gels: 4. Effect of Concentration of Crosslinking Agent a t Network Formation on the Collapse and Mechanical Behaviour of Polyacrylamide Gels. Polym. Bull. 1982, 8 , 387-394. Katchalsky, A. Polyelectrolytes. Pure Appl. Chem. 1971, 26, 325-373. Katchalsky, A.; Michaeli, I. Polyelectrolyte Gels in Salt Solutions. J . Polym. Sci. 1955, 15, 69-86. Katchalsky, A,; Zwick, M. Mechanochemistry and Ion Exchange. J. Polym. Sci. 1955, 16, 221-234. Katchalsky, A.; Cooper, R. E.; Upadhyay, J.; Wasserman, A. Counter-ion Fixation in Alignates. Chem. Soc. J . 1961,5198-5204. Kern, W.Der Osmotische Druck Wasseriger Losungen Polyvalenter Sauren und Ihrer Salze. Z . Phys. Chem. 1939, A184, 197-210. Kitchener, J. A. Recent Developments: Ion-Exchange Resin Membranes. Ion-Exchange Resins; Methuen: London, 1957. Konak, C.; Bansil, M. Swelling Equilibria of Ionized Poly(methacrylic acid) Gels in the Absence of Salt. Polymer 1989, 30, 677-680. Kou, J. H.; Amidon, G. L.; Lee, P. I. pH-Dependent Swelling and Solute Diffusion Characteristics of Poly(Hydroxyethy1 Methacrylate-Co-Methacrylic Acid) Hydrogels. Pharm. Res. 1988,5, 592-597. Panayitou, C. G.; Vera, J. H. The Quasi-Chemical Approach for Non-Randomness in Liquid Mixtures. Expressions for Local Surfaces and Local Compositions with an Application to Polymer Solutions. Fluid Phase Equilib. 1980, 5,55-80. Peppas, N. A. Dynamically Swelling Hydrogels in Controlled Release Applications. In Hydrogels in Medicine and Pharmacy; Peppas, N. A., Ed.; CRC: Boca Raton, FL, 1987; Vol. 3. Prange, M.; Hooper, H. H.; Prausnitz, J. M. Thermodynamics of Aqueous Systems Containing Hydrophilic Polymers or Gels. AIChE J . 1989, 35,803-813. Otake, K.; Inomata, H.; Konno, M.; Saito, S. A New Model for the Thermally Induced Volume Phase Transition of Gels. J . Chem. Phys. 1989, 91, 1345-1350. Ricka, J.; Tanaka, T. Swelling of Ionic Gels: Quantitative Performance of the Donnan Theory. Macromolecules 1984,17,2916-2921. Ricka, J.; Tanaka, T. Phase Transition in Ionic Gels Induced by Copper Complexation. Macromolecules 1985, 18, 83-85. Siegel, R. A,; Firestone, B. A. pH-Dependent Equilibrium Swelling Properties of Hydrophobic Polyelectrolyte Copolymer gels. Macromolecules 1988, 22, 3254-3259. Staples, B. R.; Nuttal, R. L. The Activity and Osmotic Coefficients of Aqueous Calcium Chloride at 298.15 K. J . Phys. Chem. Ref. Data 1977, 6, 385-407. Sun, C.-L. Thermodynamics of Polyelectrolyte Gels. Ph.D. Thesis, University of Southern California, Los Angeles, 1986. Tanaka, T. Collapse of Gels and the Critical Endpoint. Phys. Reu. Lett. 1978, 40, 820-823. Treloar, L. R. G. Non-Gaussian Chain Statistics and Network Theory. The Physics of Rubber Elasticity; Clarendon: Oxford, 1958. Gill, S. J. Interaction of Cupric Ions with Polyacrylic Wall, F. T.; .4cid. J . Phys. Chem. 1954, 58, 1128-1130. Received for reuiew J u l y 11, 1989 Accepted October 20, 1989
Prediction of Equilibrium Data of Adsorptions from Liquid Mixtures Wei-Rong Ji* and Y. C. Hou Chemical Engineering Thermodynamics Laboratory, Zhejiang University, H a n g t h o u , Zhejiang, PRC
On the basis of solution theory and surface thermodynamics, an activity coefficient model for the adsorbed phase with no adjustable parameters is presented. In combination with the phase-exchange adsorption model, it can predict the equilibrium data of nonideal adsorption from binary liquid mixtures. Five systems with nonideal adsorbed phases are tested, The results are satisfactory. Many efforts have been made on the nonideality of adsorbed solutions. A few models d e a l i n g with n o n i d e a l 0888-5885/90/2629-0560$02.50/0
adsorbed s o l u t i o n s of a d s o r p t i o n f r o m gas m i x t u r e s (AFGM) have been published (Suwanayuan and Danner, G 1990 American Chemical Society
