2559
IRREVERSIBLE POTENTIOMETRIC BEHAVIOR OF ISOTACTIC PMA 3.
N, N,,, MI,,,,
= fractional flux of species j at the surface of segment
Variables
number of electroactive species number of segments of working electrode number of segments of length required to represent semiinfinite linear diffusion Nllm = number of segments of length required to represent cell thickness Mt = number of time increments of the calculation DID = diffusion parameter used in finite difference equations FJ'(Z,k) = fractional concentration of j in the center of the segments of coordinates ( I & ) after applying finite difference equations F , ( l , k ) = same as F,'(Z,k) but before application of finite difference equation F,O(I) = fractional concentration of j a t the surface of segment Z of the working electrode = = =
EO' RlU
R" V Vmax,
Z of the working electrode current due to background reactions = current due to background reactions at segment I of the working electrode = total current = potential measured between segment Z of the working electrode and the reference electrode = standard formal potential = resistance (dimensionless) between solution segments at the surface of the electrode = uncompensated resistance = trial potential = positive and negative limits of potential
Vmin
Irreversible Potentiometric Behavior of Isotactic Poly(methacry1ic Acid) by J. C. Leyte,* H. M. R. Arbouw-van der Veen, and L. H. Zuiderweg Gorlaeus Laboratories, Department of Physical Chemistrg, University of Leyden, Leyden, T h e Netherlands (Received February 7, 1572)
Experimental evidence is presented for the irreversibility of the potentiometric titration curve of isotactic poly(methacry1ic acid). A description in terms of thermodynamically irreversible conformation changes is given and it is shown that the dissipation due to the transition of a monomeric unit is independent of its state of dissociation.
I n recent literature,2 some attention has been given to irreversible behavior of macromolecular systems because of the implicit importance of thesc phenomena in the quest for molecular mechanisms for information storage. It has been pointed out3 that hysteresis loops in physical-chemical properties of macromolecules offer, at least in principle, the possibility of storing, on a molecular level, information about the history of the system. The investigated macromolecular systems are, however, rather complicated from a physical-chemical point of view: RNA from several sources, mixtures of poly A and poly U, etc. I n all these systems complicated chemical equilibria as well as polymer conformational equilibria shift simultaneously as a function of the driving physical variable (usually pH). We wish to report the occurrence of hysteresis in a relatively simple polyelectrolyte molecule, poly(methacry1ic acid) (PMA).
Experimental Section Isotactic poly(methy1 methacrylate) (PITMA) w-as synthesized according to the standard p r ~ c e d u r e . ~ Molecular weights were estimated from intrinsic viscos-
ities of the CHC13 solutions using the relation [7] = 4.8 X (&f,]O.sO given by G ~ o d e . For ~ the polymer = 3.9 X lo5. used in this investigation we found The tacticity of the PNIMA was determined from nmr spectra (100 MHz) of CHCL solutions, run a t 60". The signals from the a-methyl group showed an isotactic triade content of 95-98%. Hydrolysis of the ester w'tts achieved by dissolving 6 g of the dry ester in 300 ml of oxygen-free 96% H2SO4 solution. After maintaining the system for 10 hr at room temperature the ester was dissolved. The temperature was raised to 60" ; after 2 lir the solution was cooled to 0" and 1.2 1. of distilled water was added t o the yellow solution. After filtering and washing, the
a,
(1) Dedicated to Professor Dr. H. Veldstra on the occasion of his retirement from the chair of Biochemistry of the University of Leyden. (2) A. Katchalsky, "International Symposium on Macromolecules' Leiden, 1970," Butterworths, London, 1971, p 368. (3) A. Katchalsky, A. Oplatka, and A . Litan in "Molecular Architecture in Cell Physiology," T. Hayashi and A. G. Szent. Gyorgyi, Ed., Prentice-Hall, New York, N. Y., 1966, p 3. (4) G. C. Overberger, Macromol. Syn., 1 , 29 (1963). (5) W. E. Goode, F. H. Owens, R. P. Fellemann, W. H. Snijder, and J. E. Moore, J. Polym. Sci., 46, 321 (1960).
T h e Journal of Physical Chemistry, Val. 76, N o . 18,1572
2560 polymer was dissolved in dimethylform- mide and dialyzed for 3 weeks against the daily refreshed solvent in an effort to remove the yellow color of the polymer. The solution remained slightly yellow. From the ir spectra of the sodium salt it was concluded that hydrolysis was complete. A portion of the obtained isotactic PMA was reesterified;6 the nmr spectrum of the resulting product was exactly the same as the spectrum of the PMMA used for the hydrolysis. The nmr spectra of DzO solutions of the Ka salt of isotactic PMA yielded an estimated spin-coupling constant J = 15 Hz and an estimated shift of 0.1 Hz for the methylene protons in agreement with the results of Muroga, et aL7 Potentiometric titrations were performed using a Radiometer titrator (TTTI) and titrigraph (SBR 2) with a TTA 3 titration set allowing titration under IT2. The temperature was maintained at 20". A calomel electrode K401 and a Type A glass electrode were used, both from Radiometer. I n view of the length of the titration time (7 hr for a single titration curve) the calomel electrode was checked for KCl leakage which was found to be negligible for the PMA solutions used (these contained at least 0.02 equiv 1.-' of KC1).
Results and Discussion As shown in Figure 1 the potentiometric titration curve of isotactic PRiIA is irreversible. The vertical bandwidths of the acid and alkaline titration curves demonstrate a relatively serious irreproducibility for both pathways. However, the hysteresis loop is clearly present, there being B range of degrees of neutralization in which acid and alkaline titration curves never overlap even on repeated cycling. From Figure 2 it may be seen that after an initial time dependence the pH of solutions with equal degrees of neutralization prepared in the A and the B regions become stable with an appreciable pH difference. It is therefore safe to conclude that either or both of the titration paths are along a series of metastable states. Titration curves of isotactic PMA have of course been published before8 in connection with model calculations but only the alkaline titration curve (B) was given and the irreversibility of the potentiometric titration was not reported. I t has been demonstrated previouslyg-" that the potentiometric titration curve of atactic PMA can be consistently interpreted by assuming the existence of a compact conformation at low degrees of neutralization (the a form) and an extended conformation (b form) at high charge densities. I n the region from 10 to 30% neutralization a conformational transition takes place. I n the same region both the acid and the alkaline titration curves of isotactic PMA show the peculiar weak dependence of pH on the degree of neutralization that is characteristic for the conformational transition The Jozirnat of Physical Chemistry, Vol. 76, N o . 18, 1978
LEYTE,ARBOUW-VAN DER VEEN,AND ZUIDERWEG
"O
t
0.4
0.2 + H
Figure 1. Boundaries for titration cycles of isotactic PMA: A, titration by addition of HCl; B, titration by addition of NaOH. The figure summarizes the results of six titration cycles on three different PMA solutions, each containing 0.002 equiv 1.-1 of PMA and 0.02 A4 KCl. The degree of neutralization of the polyacid is indicated by C Y .
0
60
120
180
M
t (min.)
240
Figure 2. Time dependence of pH of solutions of isotactic PMA. The degree of neutralization ( C Y ) of both solutions is 0.22. B was prepared by addition of NaOH to a solution in which CY % 0 while A was obtained by addition of HCl to a PMA solution in which CY = 0.44. The solutions contained 4.4 X 10+ equiv L-1 of PMA and 5 x IO-* M KCl.
in atactic PRIIA. A similar transition therefore appears to occur in the isotactic polyacid. I n this case, however, one or several irreversible processes take place during the titration cycle. Now, reversion of the titration direction after titrating along A to a 'v 0.2 did not produce a significant hysteresis loop. The irreversible processes responsible for the hysteresis are therefore confined to the alkaline titration curve B and possibly a portion of the acid titration curve at very lo^ degrees of neutralization. It has been showna that the area enclosed by a hysteresis loop in a titration curve is, apart from a numerical factor, equal to the sum of the irreversible changes in the Gibbs free energy during the cycle which equals (6) W.L. Miller, W. S. Brey, Jr., and G. B. Butler, ibid., 54, 329 (1961). (7) Y.Muroga, I. Noda, and M. Nagasawa, J . Phys. Chem., 73, 667 (1969). (8) M . Nagasawa, T. Murase, and K. Kondo, ibid., 69, 4005 (1965). (9) J. C. Leyte and M . Mandel, J. Polym. Sci., Part A-8, 1879 (1964). (10) J. C. Leyte, Polym. Lett., 4, 245 (1966). (11) M. ,Mandel, J. C. Leyte, and M. G. Stadhouder, J . Phys. Chem., 71, 603 (1967).
2561
IRREVERSIBLE POTENTIOMETRIC BEHAVIOR OF ISOTACTIC PMA the energy dissipation AQ. The relation with thermodynamically irreversible conformation changes in PMA (assuming all other processes to be fast on the time scale relevant for pH measurements) may be derived as follows. The thermodynamic potential associated with the monomeric unitsg will be indicated with p(A) and p (HA) for the units bearing dissociated and undissociated carboxylic acid groups, respectively. The different polymer conformations (random coil, ordered conformations, partially ordered conformations with a variable ordering range) in which the monomeric units may occur will be denoted with subscripts:
If conformational equilibrium is realized along the titration curve p(HA)t is independent of the index and, using eq 4, the right-hand side of eq 6 is zero. Any cyclic integral of pH corresponding to a titration cycle disappears of course in this case. I n the absence of conformational equilibrium every irreversible i + ,jtransition contributes and an integral of pH along a closed loop will not equal zero. Designating the change in nzAdue to a transformation 1 + i with dnizAand noting that the (HA)a F1 H+ (A), equilibrium does not contribute to d(ntA n?IA) we have
+
dntA
dNi, P(A)J,. . . . For an isothermal and isobaric change in the Gibbs free energy G per mole of polyelectrolyte molecules we may then write dQ
=
C(p(A)a dntA z
+ p(HA)i dniHA)
(1)
Acid dissociation equilibrium will be assumed to be established within every class i of monomeric units. p(H+) = p(HA)i - p(A)t = = . ..
(2)
b(HA)ji = b(A)ji
i
+ dniHA)= 0
(4)
Elimination of all P ( A ) ~from (1) yields (5) which is essentially the titratfion equation -p(H+)xdniA
=
i
dG
- Cp(HA)$(dniA+ dnrHA) (5) %
For a titration cycle returning the polymer molecules to their original states the integral of G along the cycle is zero "
pH d a
=
-
1
2.303ZR T
4
X
+
Tp(HA)r(dn$A dnHA) (6)
+
(7)
The dntl are, of course, antisymmetric in the indices and using (7) and (3) we may therefore rewrite the summation in (6) C ~ ( H A ) i d n i z= a,Z
l/zC,I [p(HA)i %
p(HA)zl dniz = Cb(HA)iz dn,i = Z
