3092
J . Phys. Chem. 1987, 91, 3092-3098
relations, we are able to determine recurrence relations for the regular solid spherical harmonic: GKl(F) =
r[(l+ m
k3
+ 1)(1-
m
+ 1)]((21+ 1) cos 0 G;l(r) - rGj!!l(r)l (A 10)
m = 0, 1, .,., I
Appendix B We outline the Runge-Kutta method for determining the numerical solutions for the coupled first-order differential equat i o n ~ . ~We ~ chase , ~ the parameters of the Runge-Kutta algorithm so that we have a Runge-Kutta method of order p + 1 including a Runge-Kutta method of order P. Our choice of method is a 5-order method
+ (5/48)k4 + (27/56)ks
(B4)
k5 = hnf(xn + (1/5)hn,yn k6 = hnf(xn
,
(1/24)k,
+ hn?Yn +
=
l/qkl
k4 = h A x n + hn,y, - k2
G-m ( -r ) = (-)"' Gm I ( -I ) *
- Yn =
(B3)
+ y4k2)
+ 2k3)
(B5)
+ (7/27)k1 + ( 1 0 / 2 7 ) k ~+ (1/27)k4) (B6)
-1 Gf:{(r) = -r sin B e'@G;(?) 21+ 2
Yn+l
k2 = hAXn + f/zhn*Yn+ L/2kl)
+ (1/5)hn,yn + (28/625)k, - (1/5)k, + (546/625)k3 + (54/625)k4 - (378/625)k5
(B7)
4th-order method: Yn+l
- Yn = y6
= (kl + 4k3
+ k4)
(B8)
where k,, k3, and k4 are given by eq B2-B5. We therefore have a fourth-order Runge-Kutta method for determining a numerical solution plus an estimate of the local truncation error T4
(1/8)kl
+ (2/3)k2 + (1/16)k,-(27/56)k5-
+(125/336)k6 (Bl)
(125/336)k6 (B8)
(B2)
When we have calculated T4, we use this information to decide whether the calculated solution has the required accuracy. If not, we reduce the step parameter with a factor 2. Registry No. C1-, 16887-00-6;F, 16984-48-8;benzene radical ion (1-), 34562-85-1; benzene, 71-43-2; fluorobenzene, 462-06-6; nitrobenzene, 98-95-3; aniline, 62-53-3.
kl = hfixn,Yn)
(33) Ralston, A.; Rabinowitz, P. A First Course in Numerical Analysis, 2nd ed.; McGraw-Hill: New York, 1978. (34) Lapidus, L.; Seinfeld, J. H. Numerical Solution of Ordinary Dgferential Equations; Academic: New York, 1971.
Synthetic Polyampholytes. 5. Influence of Nearest-Neighbor Interactions on PotentiometrPc Curves Yves Merle PolymPres, Biopolymeres et Membranes. UA 500, FacultP des Sciences de Rouen, BP 67, F 761 30 Mont-Saint-Aignan, France (Received: August 1 1 , 1986)
Potentiometric curves of polyampholytes have been synthesized by taking into account only nearest-neighbor interactions of charges of opposite sign. Triad distribution of the copolymer has been assumed to facilitate these computations. Long-range interactions have been neglected. These approximations in the computation are approached experimentally in the titration of polyampholytes at high ionic strength (1 M NaCl), where only short-range interactions persist and agreement between comparison and experiment at this salt concentrationlevel is quite good. The polyampholytes examined, copolymers of methacrylic acid and 2-(N,Ndimethylamino)ethyl methacrylate, were obtained in three. different ways: radical initiated copolymerization of the two monomers, and acidic and alkaline hydrolysis of poly[2-(N,N-dimethylamino)ethyl methacrylate].
Introduction The potentiometric behavior of polyelectrolytes is usually expressed in terms of the intrinsic pK of the ionizable functional group repeated in the molecule, pKo, and the overall electrostatic potential of their charge at the surface of the polyion, as shown:'
where pKaPpis the apparent pK, a the degree of ionization, e the
elementary charge, k Boltzmann's constant, and T the absolute temperature. The overall potential is strongly dependent on a,on the conformation of the macromolecule (random coil, compact coil, helix, and rodlike molecule), and on the ionic strength. Attempts to anticipate the value of for these different configuration^^.^ have been applied to p~lyampholytes.~In all these approaches, uniform distribution of charges on the surface of the polyion is assumed and nearest-neighbor effects are neglected. Close neighbor interactions between charges were first considered by Marcus to explain the particular behavior of poly(vinylamine) at high ionic ~ t r e n g t h . ~Lifson and Hill independently developed matrix models for the analysis of polyelec-
(1) Overbeek, J. Th. G. BUN.Soc. Chim. Belg. 1948,57, 252. Katchalsky, A,; Shavit, N.; Eisenberg, H. J. Polym. Sci. 1954, 13, 69. Nagasawa, M.; Holtzer, A. J . Am. Chem. SOC.1964,86, 531. Nagasawa, M.; Murase, T.; Kondo, K. J . Phys. Chem. 1965,69,4005. Nagasawa, M. Pure Appl. Chem. 1971, 26, 519.
(2) Katchalsky, A.; Gillis, J. Red. Trau. Chim. Pays-Bas. 1949, 68, 871. (3) Katchalsky, A,; Lifson, S . J. Polym. Sci. 1953, 11, 409. Manning, G. S.; Holtzer, A. J . Phys. Chem. 1971, 77, 2206. (4) Katchalsky, A.; Miller, I. R. J . Polym. Sci. 1954, 13, 57. (5) Marcus, R. A. J . Am. Chem. SOC.1954, 58, 621.
a
pH = pKapp+ log 1-0
0022-365418712091-3092$0 1.50/0
0 1987 American Chemical Society
Synthetic Polyampholytes
The Journal of Physical Chemistry, Vol. 91, No. 11, 1987 3093
trolytes.6 Katchalsky et al. eventually developed a simple expression which anticipated the potentiometric properties of poly(viny1amine) and poly(ethy1eneimine). To achieve this result, the electrostatic interaction between charges of the same sign was presumed to lower the ionization constant.' Bloys Van Treslong showed that such interactions, while small for most polyelectrolytes, are sizeable in polyelectrolytes like poly(vinylamine), poly(ethyleneimine), and copolymers of Faleic acid, where charged groups are close together.* In 1980, Sasaki et al. extended the theory to include all possible interactions (site model); they pointed out that, while the variation of pK is slightly dependent on the second and third neighbor interaction, counterion binding has a major contribution to the apparent pK for polyelectrolytes such as poly(acry1ic acid).9 The nearest-neighbor interaction ("1) treatment of polyampholytes was first developed by Rice and Harris for copolymers of regularly alternating structure.1° Mazur, Silberberg, and Katchalsky then pointed out that nearest-neighbor interactions between charges of opposite sign can be quite important, even in polyampholytes such as copolymers of methacrylic acid and 2vinylpyridine (MMA-2VP) where the ionizable groups are not close together.' In this model, only polyampholytes which contain a large excess of one of the constituent units (acid or base) are considered. It is assumed that one unit is so diluted by the other that they occur as isolated groups. Then, only nearest-neighbor interaction between doublets (-+) are taken into account. All the others are neglected. Because of energy gradients, the arrangement of charges in the polyampholyte is not random. This is taken into consideration by applying the grand partition function to the polyampholyte molecule. With this model, the experimental titration curves of various polyampholytes, such as MAA-2VP or lysozyme, are anticipated. At low ionic strength, the potential 9,,, of eq 2 is the sum of long-range and short-range interactions; a t high ionic strength, long-range interactions are screened by the salt in solution, and only the neighbor group (short-range) interactions remain. The variation of apparent pK, in the neutralization of polyampholytes, in NaCl solutions is mainly due to neighbor interactions. In our attempts to extend the utility of the Mazur, Silberberg, Katchalsky model," we compute the nearest-neighbor interaction for polyampholyte of any composition, from a triad distribution, which can be obtained either from N M R measurement or by calculation. The calculated titration curves are then compared with the experimental potentiometric curves obtained with copolymers of methacrylic acid and 2-(N,N-dimethylamino)ethyl methacrylate, at various compositions and triad distributions.
'
Experimental Section The copolymers of methacrylic acid and 2-(N,N-dimethylamino)ethyl methacrylate were prepared by radical-initiated copolymerization as described by Alfrey and Pinner.'* The polymerization of 2-(N,N-dimethylamino)ethyl methacrylate (DMAEM) was carried according to the procedure in ref 13. The acidic hydrolysis of poly(DMAEM) was performed with concentrated sulfuric acid by a method described e1~ewhere.l~ The alkaline hydrolysis of poly(DMAEM) was carried out with KOH in 2-propanol solution, according to the procedure in ref 13. The polyampholyte titrations were made in the following way: The lyophylized polyampholytes were dissolved in fresh water twice (6) Lifson, S . J. Chem. Phys. 1957, 26, 727. Hill, T. L. J. Polym. Sci. 1957, 23, 549.
(7) Katchalsky, A.; Mazur, J.; Spitnik, P. J . Polym. Sci. 1957, 23, 513. (8) Bloys Van Trcslong, C. J. Recl. Trau. Chim. Pays-Bas. 1978,97, 13. (9) Sasaki, S.; Minikata, A.; Biophys. Chem. 1980, ZZ, 199. (10) Rice, S . A.; Harris, F. E. J. Chem. Phys. 1956, 24, 326. (1 1) Mazur, J.; Silberberg, A.; Katchalsky, A. J . Polym. Sci. 1959,35,43. (12) Alfrey, Jr., T.; Piner, S. H. J . Polym. Sci. 1957, 23, 533. ( 1 3) Merle, Y.; Merle-Aubry, L.; S€l€gny,E. Polymeric Amines and Ammonium Salts, Goethals, E. J., Ed.;Pergamon: New York, 1980.
distilled in quartz, a t or near their isoelectric (IP) depending on whether they are soluble or not at their IP. The polyampholyte molality was chosen to be 0.05. The titrations were carried out a t 25 OC by adding standard HCI or N a O H with a Gilson microburet to the polyampholyte solution contained by a thermostated cell. The 1 M NaCl solution, made by addition of the correct amount of dry NaCl to the titration cell, was kept C 0 2 free by nitrogen flow and the pH was measured with a Corning 130 pH meter 5 min after each addition of reagent. Calculations were made on an Apple 11+ microcomputer using Applespice software for data editing on the printer.
Theory and Principles of Computational Procedures The computations that are to be discussed have been outlined in a previous paper14 where it was pointed out that nearestneighbor interactions can be determined for polyampholytes of any structure, if the triad distribution of their constituent units (acid and base) is known. Let us consider a polyampholyte, which is a copolymer formed by two constituent units, an acidic one, A and a basic one, B. Let p and q be the number of A and B units, a the molar fraction of A (a = p / ( p q ) ) , a the degree of ionization of A, and P the degree of ionization of B. The degree of protonation, g, is then
+
g = [qP
+ P(1 - C Y ) / @
+ d1 = (1 - a)P + 4 1 - ).
(3)
and the position of the isoelectric point is defined by gis = (1 - a)Pis + a(1
-4
(4)
where g,, CY,, and Pi,are the values of g, a,and at the isoelectric point. As the positive and negative charges are in equal number at this point: (1 - a)P,s = by substituting this value of (1 - a)P,, in eq 4, we have gi, = aCYis + a(1
- CY,,) = a
(5)
and the degree of protonation at the isoelectric point is equal to the molar fraction of acidic units. Like Mazur et al." we make the following assumptions in our calculations: The N N I between doublets of opposite sign can be taken in account, while all other interactions are neglected and the charge groupings (-+-) and (+-+) can be regarded as equivalent energetically to two (+ -) doublets. While not absolutely correct, because of steric effects and the repulsion of charges of the same sign, it could provide a good first approximation of the reality. When a basic group is in the vicinity of an isolated acid group, its dissociation constant is multiplied by the factor: exp[e2/ (tk Tr)]
(6)
where t is the dielectric constant of the bulk and r the average distance between the two groups. This factor is the nearestneighbor interaction term x. In the absence of long-range electrostatic interactions, the pK, of the base is changed by this factor:
Similar treatment of the acid function in the immediate vicinity of an ionized basic group only changes the sign of the N N I factor. The acid function becomes more acidic. (8) pK, = p e - x Six different triad arrangements are possible in the polyampholytes, copolymers that are composed of acid (A) and base (B) units: They are the A-centered triads AAA, AAB, BAB and the Bantered triads ABA, BBA, BBB. The bar is drawn above AAB (14) Merle, Y. J . Chim. Phys. 1985, 82, 653.
3094
Merle
The Journal of Physical Chemistry, Vol. 91, No. 11, 1987
and BBA to call attention to the fact that reverse configurations, BAA and AAB, equally possible, are energetically indistinguishable. Triad distributions can be deduced from N M R spectra,15 but more accurate estimates can be obtained from computations based on chemical considerations of polyampholyte preparation: for example, (a) from the reactivity ratios, on addition of the two monomers onto the chain, during their copolymerization,I6 and (b) from the relative rate constants of the reaction, when polyampholytes are prepared by partial reaction of one unit of a homopolymer, e.g., ester hydrolysis of poly[2-(N,N-dimethylamino)ethyl methacrylate] (PDMAEM).I7 The six different triad configurations of A and B sites that are possible dictate their respective characterization by six different ionization constants. If we now consider the degrees of ionization corresponding to each basic or acidic group with the same electrostatic surrounding, we find that, for the B a n t e r e d triads (basic groups), (1) where the two nearest-neighbor groups are ionized (triad A-BA-): pH = p@
1-01 + 2~ + log -
TABLE I: Flow Sheet for the Calculation of g, as a Function of pH, for a Given Value of aN,the Intermediate Value of (Y at Each Step of the Iteration"
Calculation of q l 8 qz, qs by (12), (13), (14)
a
-
Calculation of p by (15) an
Calculation of a s , a z , a1 by (19), ( Z O ) , (21) Calculation of ps, p2, p1 Calculation of a,, by (16)
(9)
+,
PI
(2) for the case where one nearest-neighbor group is ionized (triads ABA- and BBA-):
I no
~
pH = p@
1-P2 + x + log P2
and (3) for thecase where the neighbor groups are not ionized (triads ABA, BBA, and BBB) 1 -P3 pH = p a log -
+
P3
Similar relationsips except that the NNI term has the sign (-) are obtained with the A-centered triads. In order to obtain a reasonably accurate simulation of the potentiometric properties of such systems, Le., the prediction of pH as a function of the degree of protonation, the proportion of ionized neighbor groups must be known. Such information is available by application of the grand partition function to the polyampholyte molecule. However, when the intrinsic pKs of the base and of the acid are sufficiently different ( p e - p e > 3), neutralization of the base is nearly complete before the protonation of the acid begins and a,ai,cy2, cy3 = 1. Conversely, P, PI, P2, P3 N 1 over the neutralization range of the acid. A good approximation of the ionization of acidic groups in the protonation range of the base is obtained by multiplying the quantities of acidic sites by a,the average of al,a2,and a3. For resolution of the best value of g at a given P3, a reiterative program, starting with an a value of 1, has been employed. Assessment of the ionization of basic groups in the acid protonation range is accomplished in the same way, by multiplying the quantities of basic sites by 0. The reiterative program for such estimate of a,0, and g has the following form: We let F be the molar fraction of a particular triad in the polyampholyte; q l , q2, q3,p3, p2, and pl, the amounts (also in molar fraction units) of ionizable groups involved respectively in relations 9-1 1 and in the corresponding relations on the A-centered triads, are obtained by multiplying their molar fractions by product of the degree of ionization of opposite neighbor groups, as shown for the B-centered triads. 41 =
F A B A ~ ~
(12)
(15) Merle, Y . ;Merle-Aubry, L. Macromolecules 1%2, IS, 360. (16) Ito, K.; Yamashita, J. J . Polym. Sci. Part A 1965, 3, 2165. (17) Klesper, E.; Gronski, W.; Barth, V. Makromol. Chem. 1971, 150, 223. Klesper, E.; Strasilla, D.; Barth, V. Reaction on Polymer, Moore, J. A., Ed.; Reidel: Boston, 1913. Merle, Y . J . Polym. Sci., Polym. Phys. Ed. 1984, 22, 525.
Calculation of g by ( 3 )
#Similar calculations are made to obtain g starting from a3 in the acid neutralization range of g. Similar relationships express p3, p2, and p 1 as a function of the A-centered triads and p. The values of P and a can be calculated by the following relationships: (16) + p2az + P3a3)/a The following relationships between PI, P2, P3, a3,a2, and a1then a =
@,.I
apply:
P2
1 - P3 - lox
(18)
The reiteration procedure used to predict the pH as a function of g, the degree of protonation, during neutralization of the base sites has been used in the following way: Starting with an arbitrarily selected value of P3, we calculate P2 and PI with relations 18 and 17; eq 11 yields then the pH. With a assigned a value of 1, relations 12, 13, and 14 yield q,, q2,and q3 and the corresponding value of 6 is obtained from eq 15. In eq 19, 20, and 21, we then obtain cy3, a2,and a1by using P3. With the preceding value of P, we can then calculate p3, p z , and p 1 from their relationship to P. A revised value of a (aN)is obtained from eq 16. The calculations are repeated with the new value of (aN)replacing the first value used. The computations are repeated until the difference laN- a1 is smaller than the wanted approximation. The value of g is then calculated, using eq 3. The same kinds of calculations are made to obtain the neutralization curve in the acid range of g, in this instance, an a3value is arbitrarily selected to initiate the cycle of computations with
Synthetic Polyampholytes the INTERPROCHE program written in Applesoft Basic. This program is available on request.'* A flow sheet of the program is presented in Table I.
Choice of Parameters There is some uncertainty in the assignment of values to p g and p e . The acidic ionization constants of the corresponding unit models, 2-(N,Ndimethylamino)ethyl acetate and pivalic acid, respectively, are, in pure water, 8.50 and 5.03.19 The apparent pKs determined with eq 1 in neutralization studies that were carried out in 1 M NaCl solution varied for poly(DMAEM) from 7.70 to 8.10 and for poly(methacry1ic acid) (PMA) from 4.80 to 5.40.20 These results agreed well with those reported earlier for PMA8 and for poly(DMAEM)," and the values selected, p e of 5.35 (carboxylic group) and p e of 8.00 (dimethylaminoethyl group), were based on the best rationalization of these results that could be affected: In the first series (series C) of polyampholytes prepared by radical-initiated copolymerization of the two monomers, the distribution of the A and B sites predominantly random is Markovian (first order).15 In the two other series (A and B), obtained by partial acidic hydrolysis (series A) and alkaline hydrolysis (series B) of the methacrylate] homopolymer,13 poly[2-(N,N-dimethylamino)ethyl the distributions are not Markovian, presenting predominantly short sequences of the same unit (blocklike copolymers). This characteristic is more important in the A series.15 For copolymers of series C, triad distribution is identified with molar fraction assigned as a function of the reactivity ratios rA and rB,of the copolymerization of the two monomers into the ultimate polyampholyte16(program DISTRICO'~). The values of rAand rBthat follow are taken from ref 21: rA = 0.45, rB = 0.98 The triad distributions of series A and B were sought by using the relative rate constant given in ref 22 in Bauer's SEQDIST program which is adapted into Applesoft Basic.23 In this approach, the triad distribution is obtained directly for a given value of the acid molar fraction (program DISTRIREA'~). Small differences are observed between the calculated triad distribution of certain copolymers and their direct determination by 13CN M R spectroscopy. This is attributed to the relatively poor accuracy of the N M R method and complications introduced by hydrolysis.I5 The calculated distributions are more reproducible in each series. The relative rate constants chosen for these calculations are23 for series A: K1 = 6 , K2 = 6 for series B: K, = 8, K2 0.2 where K ,and K2 are the ratios of the respective hydrolysis rate of triads BBA and ABA relative to that of the BBB triad.
Numerical Evaluations ( A ) Model-Based Results. Triads compositions in hypothetical polyampholytes with equal quantities of acid and base units ( a = 0.5) and of various distributions are examined with the model in order to predict the influence of nearest-neighbor interaction and of triad distribution. First, copolymers of two extreme distributions are considered: (a) In the one extreme, the diblock copolymer (DB), there is no mixing of the molecular units of one polymer with those of the copolymer, and only two kinds -of triad, AAA and BBB, are present. The triads AAB and ABB, which can occur only at the junction of the two blocks, are present in negligible quantity, in this hypothetical high-molecular-weight polyampholyte and can be disregarded. (18) Listings of Programs DISTRICO, DISTRIREA, and INTERPROCHE will be sent on request. (19) Fashman, G . D. Handbook of Biochemistry ad Molecular Biology. Physical and Chemical Data, Vol. I; CRC Press: Cleveland, 1976. (20) Merle, Y . , to be published. (21) PrBdny, M.; Sevcik, S . Makromol. Chem. 1985, 186, 11 1. (22) Merle, Y . ;Merle-Aubry, L. Macromolecules 1983, 16, 1009. (23) Bauer, B. J. Macromolecules 1979, 12, 704.
The Journal of Physical Chemistry, Vol. 91, No. 1 1 , 1987 3095 TABLE 11: Triad Distribution of Various Hypothetical Copolymers with Equal Amounts of Acid and Base Functionality (a = 0.5)O
polymers AAA AAB DB 0.5 0 RA
0
AE BE CE
0.249 0.205 0.080
0 0.218 0.257 0.240
0
BAB
ABA 0
0.5 0.033 0.038 0.181
0.5 0.048 0.084 0.181
BBA
BBB
(B)
~
0 0 0.188 0.165 0.240
0.5 0
0.264 0.251 0.080
1 3.52 3.00 1.66
a ( B ) is the number-average length of B units. DS, diblock, RA, regularly alternating; AE, acid hydrolysis; BE, alkaline hydrolysis; CE, copolymerization according to the terminal model.
TABLE III: p H at Half-Neutralization of the Base and the Acid Sites of Polyampholyte CE Given for Different Values of the Nearest-Neighbor Interaction Term x 0 0.2 0.5 0.9 1.5
8.00 8.24 8.373 9.096 9.807
5.35 5.109 4.745 4.254 3.543
2; 0.2 OA a6 0.8 g Figure 1. Influence of the nearest-neighbor interaction term x on the titration curves of polyampholyte (CE): 1, x = 0; 2, x = 0.2; 3, x = 0.5; 4 x = 0.9; 5 , x = 1.5.
(b) In the other extreme, the equimolar polyampholyte with a regularly alternating structure (RA) has only two kinds of triads, BAB and ABA. This copolymer can be prepared with monomers which do not attach to themselves, The triad compositions of the three copolymers determined from the data obtained in their preparation for study are then examined with (c) the acid hydrolysis model (sample AE), the alkaline hydrolysis model (sample BE), using the relative rate constants Kl and K2 obtained earlier, and the copolymerization models, using the reactivity ratios rA and rB (sample CE) obtained earlier. The triad compositions of each polyampholytes are given in Table 11. Influence of Nearest-Neighbor Interaction on the Titration Curues. The effect of the use of different x (the NNI term) values from 0 to 1.5 on the shape of the neutralization curves is examined for the polyampholyte (CE). This range of x value is the most common encountered in this polyampholyte (CE) example of the highest random distribution case. As expected, for x = 0, the titration curve is similar to that of a mixture of micromolecular units. But even for an x value as low as 0.2, the influence of N N I is visible. In addition, the symmetrical redistribution of A- and B-centered triads in the projected product (Table 11) produces symmetry at the isoelectric point. The titration curves are shown in Figure 1, and the pH's
Merle
3096 The Journal of Physical Chemistry, Vol. 91, No. 11, 1987 TABLE I V pH at Half-Neutralization of the Base and Acid URits of Mfferent PolvamPeolvtea with Vnrioua Triad Distribution’ polyampholytes DB AE BE CE RA
PHU-0.5) 8.00 8.470 8.544 9.096 9.800
P%-OS)
5.35 4.863 4.759 4.254 3.550
‘See Table 111; an identical nearest-neighbor interaction term ( x = 0.9) is employed.
half-neutralization are given in Table 111. The contribution of N N I to pH increases with x ; when x = 1.5, the pH of half-neutralization is decreased or increased by 1.807 pH units, depending on the acid or base range of neutralization. Though this is an extreme case, it can be encountered in polyelectrolytes (poly(viny1amine))’ or in small molecules (8alanine). I9 The value of the N N I factor, x , which fits the experimental titration best, is 0.9. Though this value seems high for the relatively long distance between the dimethylaminoethyl and the carboxylic groups, the amine group is on a lateral flexible chain, and the two ionizable groups are probably brought close together by electrostatic attraction. 7 CH2
2\
0
0.2
0.4
0.6
0.8
g
Figure 2. Influence of the triad distribution on the titration curves; nearest-neighbor interaction term x = 0.9, DB, diblock; AE, acid hydrolysis; BE, alkaline hydrolysis; CE, copolymerization; RA, regularly alternating structure.
N*(CH3)2
The acid function becomes more acidic and p e is lowered by 2x as well. This result is due to the symmetry of the effect.
CH3
The neutralization range of the polyampholytes AE, BE, and CE falls within the limits defined by the two hypothetical extremes (DB and RA). The polyampholytes AE and BE are more ordered than C E and their titration curves are more like those developed for the DB case. Table IV lists the pH of half-neutralization for the five different triad distributions examined in the hypothetical polyampholytes. From Figure 2 which presents these results graphically, we see that the triad distribution has a strong influence on the titration curves of polyampholytes. (B) Comparison with Experimental Results. Triad distributions in 12 polyampholytes have been estimated by the procedure described earlier (5 in series A, 3 in series B, 3 in series C). These estimates are listed in Table V. For each sample the numberaverage length of amine units ( B ) was calculated by using the formula developed by Itoi6 as shown:
CH3
Influence of Triad Distribution on the Titration Curves. TO examine the influence of triad distribution on the shape of titration curves, triad distributions of a hypothetical polyampholyte were varied while maintaining composition ( a = 0.5) and N N I ( x = 0.9) fixed. In the diblock copolymer (DB), there is no nearestneighbor interaction -since heterogeneous triads such as AAB, BAB, ABA, and BBA are absent. The result is that the computed titration curve mimics that of a mixture of the acid and base monomer units that constitute the polyampholyte copolymer. The pH of half-neutralization of such a functional unit is equal to its intrinsic p K PH(po.5) = pf$ = 8.00
-
~ F -B 2(1 - a ) FB = (B) = FAB FE FA= + ~ F B A B
The regularly alternating polyampholyte has the strongest effect on the apparent pK. It contains only the heterogeneous triads, BAB and ABA, where the N N I is greatest. When the intrinsic pKs of base and acid functions are sufficiently different, the neutralization of the base occurs at a more basic pH and the p e is increased by 2x. pH(B-0.5) = pf$ + 2~ = 9.80
The blocklike character of series A and B is indicated by the greater magnitude of (B) in copolymer of equal acid content. Potentiometric curves in pure water and in 1 M NaCl solutions are compared to the synthesized curves in Figure 3 for one sample of each series, chosen for their nearly identical acid content (66% for A4, 60% for B3, and 61% for C3).
TABLE V Triad Distribution of Different Copolymers of 2-(N,N-Dimethyl.mino)ethyl Methacrylate and Methacrylic Acid samoles AI A2 A4 A5 A7 B2 B3 84
co
c1 c3
a 0.28 0.48 0.66 0.80 0.95 0.52 0.60 0.66 0.19 0.40 0.61
AAA 0.08 1 0.230 0.423 0.615 0.882 0.218 0.275 0.321 0.002 0.033 0.176
-
AAB 0.151 0.215 0.217 0.175 0.067 0.265 0.291 0.308 0.036 0.163 0.303
BAB 0.048 0.035 0.020 0.010 0.002 0.037 0.034 0.031 0.152 0.204 0.131
ABA 0.025 0.046 0.058 0.055 0.027 0.089 0.113 0.133 0.036 0.136 0.205
-
BBA 0.197 0.192 0.142 0.085 0.017 0.161 0.133 0.104 0.269 0.299 0.156
BBB 0.498 0.282 0.140 0.060 0.007 0.230 0.154 0.103 0.506 0.165 0.030
(B) 5.83 3.65 2.65 2.05 1.408 2.83 2.17 1.83 4.76 2.10 1.38
‘ 0 is the molar fraction of acid content; ( B ) is the number-average length of units B. Series A: acid hydrolysis. Series B: alkaline hydrolysis. Series C: radical-initiated copolymerization.
The Journal of Physical Chemistry, Vol. 91, No. 11, 1987 3097
Synthetic Polyampholytes
TABLE VI: pH at Half-Neutralization(Calculated and Measured) of the Base and Acid Sites of Experimental Polyampholytes'
samdes ~
~~
A1 A2 A4 A5
A7 B2
B3 B4
co c1
c3 I 21 0
.
AI
'\
-.
0.2
a4
0.6
0.8
'\
2
2
,
