EUROPHYSICS LETTERS 1 February 1997

Europhys. Lett., 37 (4), pp. 275-280 (1997)

Complexation and precipitation in polyampholyte solutions

R. Everaers, A. Johner and J.-F. Joanny

Institut Charles Sadron (CNRS UPR022)6 rue Boussingault, F-67083 Strasbourg Cedex, France

(received 9 October 1996; accepted in final form 23 December 1996)

PACS. 61.25Hq – Macromolecular and polymer solutions; polymer melts; swelling.PACS. 64.75+g – Solubility, segregation, and mixing; phase separation.PACS. 87.15Da – Physical chemistry of solutions of biomolecules; condensed states.

Abstract. – We discuss multichain effects in polyampholyte solutions of finite concentrationand find that the existing single-chain theories are limited to exponentially small concentra-tions, if the sample contains chains with net charges of both signs. We show that chargedpolyampholytes have a strong tendency to form neutral complexes and to precipitate. Stretchedchains almost play no role in the ideal case of a neutral sample with nearly symmetric chargedistribution. However, for a non-neutral sample, the free counterions accumulate in the super-natant together with charged, elongated chains and the behavior of the solution is dominatedby “impurities.”

Polyampholytes are polymers that carry positively as well as negatively charged groups.Being often water-soluble these molecules offer numerous applications besides providing simplemodel systems for electrostatic interactions in proteins and other biopolymers [1]. A specialclass are quenched polyampholytes, where the charges are predetermined by the chemistry andindependent of the pH. Over the last few years much progress has been made in synthesizingsuch polymers [1] and in understanding their conformations at infinite dilution [2]-[9]. Theapplication of these single-chain theories to experiments on random polyampholytes has,however, led to apparently contradictory results [10], [11].

In the present letter we discuss the solubility of polyampholytes and the composition ofsolutions at finite concentration. For ordinary polyelectrolytes, which carry charges of only onesign, the water-solubility is mostly due to the gain in translational entropy of the counter-ionsin the water phase. The polymers are dissolved in spite of their high electrostatic self-energies,which they minimize by adopting stretched conformations. In contrast, polyampholyte samplescan be self-neutralizing, thus resembling mixtures of oppositely charged polyelectrolytes [12].One can, therefore, expect the formation and precipitation of neutral complexes at finiteconcentrations. For not perfectly neutral samples the free counter-ions have a tendency tostay in the dilute phase, even though the majority of the polyampholytes precipitates. Whatis not clear a priori is the nature of the chains which accumulate in the supernatant to ensurecharge neutrality.

c© Les Editions de Physique

276 EUROPHYSICS LETTERS

Our analysis of the phase equilibrium between a homogeneous bulk and a dilute supernatantis based on the description of polyampholytes in solution as elongated globules [1], [7], [8]. Herewe mostly consider polyampholyte samples with uni- and bimodal net charge distributions, forwhich the main effects can be worked out by simple arguments. We also present some resultsfor randomly co-polymerized samples. A detailed treatment of this more realistic case will begiven elsewhere [13].

Model polyampholyte samples. – For simplicity we only consider net charge polydispersityand treat the chain length N and the fraction f of charged monomers as constants. Samplesare then characterized by a normalized distribution p(δf) for the excess charge per monomerδf . For non-neutral samples with δf0 ≡

∫δf p(δf)dδf 6= 0 the net charge is balanced by a

counter-ion density ci,tot = δf0ctot. Concentrations cδf refer to monomer concentrations inthe dilute phase. If there is phase separation, we use capital letters Cδf to indicate bulkconcentrations. Chain concentrations are given by cδf/N and Cδf/N , respectively.

We consider three different model distributions: (I) Random co-polymerization: p(δf) =√(N/2πf) exp

[−((δf − δf0)2N)/2f

]. Note that the unbiased case with δf0 = 0 is an ex-

ception rather than the rule and that depletion during synthesis leads to broader chargedistributions [14]. (II) Unimodal samples where all the chains have the same net chargedensity δf0. This is a reasonable approximation for (I), if δf0 is larger than the width

√f/N

of the distribution. (III) Bimodal samples which are a mixture of two types of polyampholytes(indicated by subscripts ±) with opposite net charges ±δf of equal magnitude and totalconcentrations ctot

+ = ( 12 + ε)ctot and ctot

− = ( 12 − ε)ctot.

Bulk properties. – We concentrate on the generic case of a neutral ensemble of polyam-pholytes in a Θ-solvent with no added salt. The bulk can be viewed as a dense liquidof blobs of ga = b2/l2Bf

2 monomers [2]. Inside each blob the Debye-Huckel polarizationenergy FDH = (−κ3/12π)kBTV is of order −kBT , where κ2 = 4πlBfCbulk, lB = e2/εkBTthe Bjerrum length, and b the monomer size. We usually measure the chain length inunits of these polyampholyte blobs: N = N/ga À 1. The bulk density is of the orderof Cbulk ∼ ga/ξ

3a ≡ ga/b

3g3/2a = lBf/b

4. The bulk chain chemical potential is given byµbulk = Fbulk + kBT log

(Cδfb

3/N)

with a bulk free energy per chain Fbulk of −kBT per blobindependent of the chain net charge [15].

Single-chain properties. – The essential features of the conformations of polyampholytesin solution seem to be captured in the elongated-globule model [7], [8], [1]. Here we follow thenotation in ref. [1]. In a globular state the internal monomer concentration is still approxi-mately given by Cbulk , so that three-body repulsion and polarization contribute Fbulk to thefree energy. The competition between surface tension, Fsurf ∼ N2/3kBT , and electrostatic re-pulsion between the excess charges, Fcoul ∼ (δfN)2(lB/ξaN1/3)kBT , determines the size of theelectrostatic blob, g′e = f/δf2. The molecules resemble a sequence of N/g′e electrostatic blobs,each consisting of g′e/ga polyampholyte blobs. Up to logarithmic corrections the total surface

and Coulomb energies of an elongated globule can be written as Fsurf + Fcoul ∼ N δf2/3kBT

with δf = δfb/lBf3/2. Note that for an ensemble of random polyampholytes N = f/〈δf2〉, so

that a finite fraction of the chains forms elongated globules with a size proportional to N [4], [5].For high net charges, when g′e ≤ ga or δf ≥ 1, the polarization energy can be neglected andthe chains behave as polyelectrolytes. They form a linear sequence of N/ge electrostatic blobs

with ge =(b/lBδf

2)2/3

and Fcoul = (N/ge)kBT = N δf4/3kBT . Summarizing these arguments,

the excess free energy of a single chain in solution relative to the bulk may be approximated

r. everaers et al.: complexation and precipitation in polyampholyte etc. 277

as

Fex(δf , N)kBT

=

N2/3

(1 + δf

2N), δf

2< 1/N (spherical globules),

2δf2/3N , 1/N < δf

2< 1 (elongated globules),

2δf4/3N , δf

2> 1 (polyelectrolytes).

(1)

The chain chemical potential is the sum of its free energy and the ideal gas term for thetranslational entropy µuni(δf , N , cδf ) = Fbulk(N)+Fex(δf , N)+kBT log(cδfb

3/N )− δfN Ψ . Forasymmetric samples, an electrostatic potential difference Ψ = (b/lbf1/2)Ψ develops betweenthe bulk and the dilute phase. In general, Ψ has to be determined from the condition ofcharge neutrality of both phases. Only in simple cases can this tedious procedure be avoidedby considering solubility products.

Dimerization and precipitation for a sample with symmetric charge distribution. – Westart by considering a bimodal sample (III) with ε = 0. In dilute solution the unimers are inchemical equilibrium with neutral dimers and at high enough ctot the sample phase separates.

For a globular state the dimer free energy is equal to that of a neutral unimer of twicethe original length. The chemical potential of the dimers is given by µdim(±δf , N , cdim) =Fbulk(2N) + Fex(0, 2N) + kBT log(cdimb

3/2N). The condition µdim ≡ µ+ + µ− of chemicalequilibrium leads to the law of mass action:

cdim / c+c− = (2b3/N) exp[2Fex(δf , N)− Fex(0, 2N)

]≡ KA . (2)

Precipitation sets in, when the concentration is so high that the bulk chemical potential isreached, µdim = µ+ +µ− = 2µbulk. As a consequence, the unimer concentrations in the dilutephase are limited by a solubility product

c+c− ≤ (1/4)C2bulk exp

[−2Fex(δf , N)

]≡ LP (3)

and the dimer concentration by

cdim ≤ (b3/2N)C2bulk exp

[−Fex(0, 2N)

]= KALP . (4)

Note that eqs. (2) to (4) are not restricted to ε = 0.For weakly charged chains the formation of dimers for ctot > 1/KA is preceded by precip-

itation at ctotphase sep = KALP + 2

√LP. The composition of the supernatant is independent of

ctot: cco-ex± =

√LP and cco-ex

dim = KALP. Dimers dominate due to the reduced unimer solubility

for√LP > 1/KA, or, up to logarithms, δf

2> 1/N . Since the criterion is the same as for the

crossover between the spherical and elongated globule regimes, the supernatant contains onlyspherical globules—either of neutral or pairs of charged polyampholytes. To observe elongatedglobules one has to reduce the polymer concentration to ctot < 1/KA ∼ exp[−4δf

2/3N ].

Precipitation sets in for exp[−(2N)2/3] < cco-exdil /Cbulk < exp[−N2/3]. Thus, a symmetric

ensemble of polyampholytes with high chain length N is practically insoluble. Experimentally,such systems always consist of a supernatant in co-existence with a bulk.

Non-neutral samples. – Although these considerations have the benefit of simplicity, theyare not in agreement with most experimental data [10], [11]: i) many random polyampholytesamples are water-soluble at finite concentrations, 2) the supernatant is not always dominatedby neutral chains, and 3) its concentration increases with the total concentration.

278 EUROPHYSICS LETTERS

In the following we argue, that these effects are due to the presence of free counter-ions. Theextreme case is a sample where all polymers have a net charge of equal sign, e.g. the unimodaldistribution (II). In such a case, a polyampholyte molecule can only precipitate together with allits counter-ions. The respective concentrations in the dilute phase are limited by a solubilityproduct, c+(δf0c+)Nδf ≤ Cbulk(δf0Cbulk)Nδf exp[∆Fbulk − Fex(δf , N)], so that

c+ ≤ L1/(Nδf+1)I with LI ≡ CNδf+1

bulk exp[∆Fbulk − Fex] . (5)

The effect is quite dramatic. For δf0N = 1, i.e. a single counter-ion per chain, ∆Fbulk

can be neglected [15]. By effectively reducing Fex by a factor of two the solubility of thesample corresponds to neutral chains which are three times shorter. The counter-ions, on theother hand, experience an electrostatic potential barrier Ψ corresponding to one half of theglobule surface energy. For larger δf0, L1/(Nδf+1)

I becomes independent of the chain length andreaches a minimum of order unity for chains at the crossover between the elongated globuleand the polyelectrolyte regimes. Without discussing the corresponding semi-dilute solutionsany further one may, for all practical purposes, regard such samples as water-soluble.

Impurities and fractionation. – We now turn to the situation where a sample containsa small proportion of free counter-ions. As an example we consider again a bimodal sample(III), but now with a slightly higher proportion of positively charged polyampholytes (ε > 0).

Up to the onset of precipitation the extra chains and counter-ions do not qualitatively changethe behavior of the system. If unimers dominate, there is a slight excess c+ − c− = 2εctot ofpositively charged chains, in the opposite case the concentrations approach cdim = (1−2ε)ctot,c+ = 2εctot and c− = (1/2ε − 1)/KA. However, the composition and concentration of thesupernatant are no longer fixed by establishing a phase equilibrium. At co-existence, theunimer concentrations, not being limited individually but by a solubility product, becomedependent on the total concentration: c+ → 2εctot, while c− = LP/2εctot → 0. Since thedimer concentration is still given by eq. (4) and a constant, the dilute phase may, dependingon ctot, either be dominated by charged or neutral globules (fig. 1).

The increase of the unimer concentration in the supernatant is limited by the solubilityproduct eq. (5) for the excess chains and the counter-ions. Asymptotically, c+ − c− ≈ c+,C+ = (1/2 + ε)Cbulk, and C− = (1/2− ε)Cbulk , so that

c+ ≤ 2εL1/(Nδf+1)I , (6)

where ∆Fbulk = 0 for small ε.

Discussion. – Our results suggest that care has to be taken in the interpretation ofexperiments on samples which contain polyampholyte chains with net charges of both signs:At finite concentrations it is not possible to identify the composition of the dilute phase withthe composition of the sample.

It is worthwhile to illustrate the consequences using the example of randomly chargedchains treated by Kantor and Kardar [4] (case (I) with δf0 = 0). At infinite dilution theaverages for quantities such as the hydrodynamic radius are dominated by the extended chainsin the wings of the sample charge distribution. However, already for total concentrationsas low as Cbulk exp[−N2/3] most of the material is precipitated, so that bulk and samplecomposition coincide. By equating the chemical potentials in the two phases one obtains forthe concentrations in the dilute phase

c(δf) ∼ exp[−δf2N(1 + 2N2/3)/2] . (7)

r. everaers et al.: complexation and precipitation in polyampholyte etc. 279

-2 -1 0 1 20

0.2

0.4

0.6

0.8

1

1.2

1.4

δf N1/2~ ~

spherical globules

elongatedglobules

p(δf

N1/

2 )~

~

elongatedglobules

-17.5 -15 -12.5 -10 -7.5 -5 -2.5 0-20

-17.5

-15

-12.5

-10

-7.5

-5

-2.5

0

log10 (ctot / cbulk)

log 1

0 (c

dil /

cbu

lk)

cdim

c+

c-

Fig. 1. Fig. 2.

Fig. 1. – Composition of the dilute phase vs. total concentration for asymmetric (ε = 0.01) bimodal

distributions of chains with N = 20, ga = 10 and net charge δfN1/2 = 3. The gray shaded areasindicate concentrations for which the dilute phase is dominated by elongated globules.

Fig. 2. – Probability distribution for the excess charge on the unimers: in the supernatant when mostof the chains are precipitated (——); at infinite dilution (- - - - - -).

Compared to the bulk, the distribution of net charges on the unimers in the supernatant isN1/3 times narrower, i.e. there are practically no elongated globules dissolved (fig. 2). Forthe Gedankenexperiment to work, where random polyampholytes with an overall neutralityconstraint are cut into two pieces, the concentrations have to be even lower: the least charged,elongated halves start to form dimers for ctot > Cbulk exp[−2N2/3]. Thus while Kantor andKardar concluded that statistical fluctuations in the net charge density play a dominant role,we believe that for experiments the opposite view is equally relevant: pairs of oppositelycharged chains have a strong tendency to form neutral complexes and to precipitate.

Nevertheless, we find that the extended states play an important role in experiments.The reason is i) the sublinear dependence of the excess free energy on the chain net chargein the elongated globule regime (eq. (1)) and ii) the fact that samples are never perfectlyself-neutralizing and therefore contain free counter-ions. At co-existence, an electrostaticpotential difference Ψ establishes between the phases which determines the composition ofthe supernatant.

For biased random co-polymerization (case (I) with 0 < δf0 ¿ N−1/2) we find two limitingcases [13]. If the bias is exponentially small or if ctot is close to the onset of phase separation,Ψ is so small that the composition of the supernatant is similar to the situation depicted infig 2: the Gaussian peak is shifted, so that spherical, but now on the average slightly chargedunimers dominate. In most cases, however, the potential difference exceeds the critical valueof Ψ = 2 with the consequence that i) the solubility of elongated globules with δf > ( 3

4 Ψ)−3

increases with their net charge, ii) those with the highest charge have the strongest tendencyto accumulate in the supernatant where iii) their concentrations may exceed those in the bulk.To balance the charges on the counter-ions with the limited number of highly charged globules,Ψ adapts its value, so that all chains from the tail of the sample net charge density distributionwith δf > δf

∗are dissolved, where δf

∗is determined by the condition δf0 =

∫∞δf∗ dδf δf p(δf).

In general, non–self-neutralizing samples behave like mixtures of a soluble and a non-solublecomponent. Concentrating the sample leads to a separation of an almost self-neutralizingbulk and a supernatant containing the counter-ions and the most strongly charged chains.The fractionation process can be repeated by redissolving the bulk until the dilute phase isdominated by weakly charged spherical globules.

280 EUROPHYSICS LETTERS

Clearly, our understanding of the effects discussed in this article can only be as good asour understanding of the properties of polyampholytes in general. While the Debye-Huckelapproximation for the description of dense states and the elongated globule model for chainsin dilute solution capture the essential physics, they can certainly be refined [12], [6], [5]. Forexample, future work on the necklace model [5] in combination with arguments along the linesof those presented here may allow predictions on whether or not particular charge sequenceslead to more soluble molecules than others.

***

The authors thank F. Candau and J. Selb for helpful discussions. RE gratefully acknowl-edges financial support by the French Ministry of Foreign Affairs.

REFERENCES

[1] Candau F. and Joanny J.-F., in Polymeric Materials Encyclopedia, edited by J. C. Salamone

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[2] Higgs P. and Joanny J.-F., J. Chem. Phys., 94 (1991) 1543.

[3] Kantor Y. and Kardar M., Europhys. Lett., 14 (1991) 421.

[4] Kantor Y. And Kardar M., Phys. Rev. Lett., 69 (1992) 61; Kantor Y., Kardar M. and Li

H., Phys. Rev. E, 49 (1994) 1383.

[5] Kantor Y. and Kardar M., Europhys. Lett., 27 (1994) 643; Phys. Rev. E, 51 (1995) 1299.

[6] Wittmer J., Johner A. and Joanny J.-F., Europhys. Lett., 24 (1993) 263.

[7] Gutin A. and Shakhnovich E., Phys. Rev. E, 50 (1994) R3322.

[8] Dobrynin A. V. and Rubinstein M., J. Phys. II, 5 (1995) 677.

[9] Levin Y. and Barbosa M. C., Europhys. Lett., 31 (1995) 513.

[10] Skouri M., Munch J., Candau S., Neyret S. and Candau F., Macromolecules, 27 (1994)69.

[11] Ohlemacher A., Candau F., Munch J. and Candau S., J. Polym. Sci. Phys. Ed., 34 (1996)2747.

[12] Borue V. Y. and Erukhimovich I. Y., Macromolecules, 23 (1990) 3625.

[13] Everaers R., Johner A. and Joanny J.-F., submitted to Macromolecules.

[14] Corpart J.-M., Selb J. and Candau F., Polymers, 34 (1993) 3873.

[15] The bulk properties may change due to the presence of counter-ions. It is important to note thatthe gain in polarization energy of a counter-ion in the bulk is of the order Fbulk/fN and negligiblecompared to the polymer bulk and excess energies. As a consequence, counter-ions accumulate inthe bulk only for want of easily dissolvable polyampholytes with opposite charge. A homogeneousbulk is inflated due to their osmotic pressure with corrections ∆Fbulk/Fbulk ∼ δf0ga which may beneglected as long as the number of counter-ions per blob is small. A higher counter-ion contentmight, as in the case of polyelectrolytes in a poor solvent [16], [17], lead to the formation ofmesophases. In any case we expect |∆Fbulk|/Fbulk < 1.

[16] Borue V. Y. and Erukhimovich I. Y., Macromolecules, 21 (1988) 3240.

[17] Joanny J.-F. and Leibler L., J. Phys. (Paris), 51 (1990) 545.