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Quantum field theory of polyelectrolyte-counterion condensation
Volume 15 1, number
I,2
QUANTUM
FIELD THEORY
CHEMICAL
PHYSICS LETTERS
OF POLYELECTROLYTE-COUNTERION
7 October
1988
CONDENSATION
T.G. DEWEY Department of Chemistry UniversityofDenver, Denver, CO 80208, USA Received 22 March 1988; in final form 20 July 1988
A simple quantum theory of polyelectrolyte-counterion interactions is presented. A model Hamiltonian is employed which describes both the polyelectrolyte and the counterion as free, spinless fermions. This Hamiltonian is transformed into a form which is isomorphous with traditional Hamiltonians used to describe phase transitions. The difference between this theory and early theories of superconductivity is that the counterion-counterion interaction energies will be quite large and will persist at high temperatures. The counterion condensate is a collective mode resulting from polyelectrolyte-mediated polarizations. Colligative properties for this model are compared with the Poisson-Boltzmann theory and to Manning’s condensation theory
1. Introduction The continued interest in the physical chemistry of nucleic acids has spurred renewed experimental and theoretical efforts in the study of polyelectrolytes. Two main theories dominate the description of polyelectrolytes. They are the Poisson-Boltzmann theory and Manning’s counterion condensation theory (for a comparison of these approaches see ref. [ 1 ] ). The Poisson-Boltzmann theory provides a rigorous theoretical approach in which the levels of approximation can be well defined. Unfortunately, the Poisson-Boltzmann equation for a polyclcctrolyte can only be solved exactly in limiting cases and this approach has not been entirely tractable in describing experimental data. Manning’s theory on the other hand provides a simple mathematical device by which a large body of experimental data may be explained. In Manning’s original theory [ 2 1, he postulated a “condensation” of counterions about the polyelectrolyte which served to relieve an instability in the phase integral. While this theory has been quite successful, it has not provided an intuitive physical picture of the “condensation” phenomena. In the current work polyelectrolyte-counterion interactions are considered in a mode1 quantum system, The goal of this approach is to provide a description of these interactions in a simple system and to provide physical insights into the nature of counterion condensation in such a system. A quantum system is chosen that utilizes the powerful, second-quantization operator algebra of many-particle physics. The mode1 consists of a linear structure containing free particles of spinless fermions (the polyclcctrolyte) which interacts electrostatically with free, spinless fermion particles of opposite charge in the surrounding three dimensions (the counterion). The intent of this mode1 is not so much to present a physically realistic picture of polyelectrolytes, but rather it is to explore counterion condensation in a mathematically tractable mode. Using a Tomonaga transformation [ 3 1, the polyelectrolyte can be represented as a boson field. This transformation converts the Hamiltonian to a form homologous with the Frbhlich Hamiltonian. A second, canonical transformation eliminates the boson (polyelectrolyte)-fermion (counterion) interactions and leads to a form in which an attractive interaction exists between counterions. This results from a counterion’s polarization of the polyelectrolyte’s bosons giving rise to an attractive boson interaction with a second counterion. This boson-mediated attraction is homologous with the effects that lead to the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity. The difference is that in this case the boson energy is the free polyelectrolyte particle energy plus a contribution from the polyelectrolyte electrostatic self-energy rather than a phonon energy. Thus, this is a much higher energy effect and 16
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03.50 0 Elsevier Science Publishers Physics Publishing Division )
B.V.
Volume 15 1, number
1,2
CHEMICAL
PHYSICS LETTERS
7 October
1988
will persist to very high temperatures. Considering this strong attractive force between counterions, a counterion-condensation model is presented which closely resembles a superconducting condensate in the strongcoupling limit. The colligative properties of such a model are examined by assuming realistic functional dependences of the model’s parameters on c, the polyelectrolyte structural parameter. By fitting the proportionality constants for these parameters, quantitative agreement with the Manning model may be obtained.
2. Model Hamiltonian and transformations The following model Hamiltonian is considered in which at (a) represent the annihilation (creation) operators for a spinless fermion localized in a one-dimensional space of length L and bt (b) represents the operators for a counterion in the surrounding three-dimensional volume, V, H= 1 Iklv,a~a,+(2~)-’
1 V,p(k)p(-k)+
k
k
C &&t&+(N/2V k
C U,(p(q)+p(-q))b~b,-,,
(1)
k,q
p(k) = 1 a&ap+k12 P
and N is the number of counterions, V, is the Fourier transform of the electrostatic interaction potential between polyelectrolyte particles and U, is the transform of the electrostatic potential between counterions and polyelectrolyte particles, Ek and 1k 1uF represent the energy of the counterion and polyelectrolyte particles, respectively. In keeping with the Tomonaga model the polyelectrolyte energy is linear in k. This restriction is not required and may be lifted without formally changing the results. As in most treatments of polyelectrolytes the counterion-counterion interactions are not explicitly considered. The above treatment will correspond to a polyelectrolyte in a salt-free solution. The advantage of choosing the salt-free case is that the colligative properties are well defined in the Poisson-Boltzmann [4] and Manning [ 21 model. The Fourier transform, U,, is identical to the three-dimensional transform for a bare Coulombic interaction when a convergence factor is added which decays exponentially along the axis parallel to the polymer. In the following treatment, two transformations will be performed on the Hamiltonian. First, a Tomonaga transformation (cf. ref. [ 5 ] ) will be performed in order to express the polyelectrolyte particles as a boson field. The resulting Hamiltonian will be homologous to the Frijhlich Hamiltonian with the phonon-electron interaction term representing the polyelectrolyte boson-counterion interaction. A canonical transformation of this Hamiltonian will lead to a form which is quadratic in the counterion terms. The resulting Hamiltonian indicates that the counterions may exhibit a strong attraction to each other. The Tomonaga transformation is made by dividing the density operator, p(k), into two terms and identifying a new set of operators, (Ye,as follows: p(k) =PI (k) +P2(k)
>
p,(k) = & &a~p+w ,
>
p,(k) = &
P1(-k)=aR(k~W~/nTk)“2,
P,(k)=a,(kLu,lrcrk)“*, Pz(k)=ff-k(kLWk/Xrk)“*
4-k12ap+k,2,
>
pz( -k)=at,(kh,/n&)“*,
(2)
with z+ being the Fermi velocity of the polyelectrolyte parwhere wk= Iklv, and &= I/clz+( 1+2VO/~~,)“2 ticles and V. is a constant describing polyelectrolyte charge interactions in an electron-hole excitation model [ 5 1. Using this transformation and the assumption that the density operators obey exact commutator relationships [ 5 1, the transformed Hamiltonian is now given by 17
Volume 151, number
CHEMICAL
I,2
H’= 1 rk(&h+f)+ k
7 October
PHYSICS LETTERS
1k Ekblbk+(R/V)1’2 1 Uqb~bk_,(at,+aq)
1988
(3)
)
k.q
where U” = Uqq(wk/&) ‘12/2x. This expression is homologous with the Frijhlich Hamiltonian and the boson interaction term can be eliminated by a canonical transformation. This transformation is given by H” =e-.YH’ Defining
e”
.
(4)
s as
s=-(7t/v)“2
1 U;blb,( k.k’
gives the transformed H”= 1 Ekb@,+ k
% Or” Ek-Ek, +<_, + Ek-Ep-5,
Hamiltonian,
>
H”,
1 &(a~c~.+~)+(rc/V)“* k
C U;bj$+qbl_gbkbk. k.k’
(E
)z
Et, k-
k
u
rz. -
(5)
v
This transformed Hamiltonian has eliminated first-order terms in the polyelectrolyte boson-counterion interaction. The remaining counterion-counterion term is restricted to the set of terms arising from the commutation of the boson operators. This condition restricts the energy difference in the denominator of eq. (5) to be Ek-Ek_+. A Hamiltonian of this form is the starting point for the BCS theory of superconductivity. A negative Fourier transform for the two-particle interaction potential admits the possibility of an attractive interaction between particles. This occurs when &> Ek- Ek_r In the current case this interaction is likely to be much stronger than in the superconductivity case. In superconductivity <, is the phonon energy and is only comparable to the electron energy at extremely low temperatures. In the current case, $ is the boson energy derived from the polyelectrolyte. It contains contributions from both the polyelectrolyte particle’s kinetic energy and from the electrostatic interactions between polyelcctrolytc particles. Thus this energy term would be expected to be comparable to, if not greater than, the counterion energy. The denominator in eq. (5) is, therefore, expected to yield a negative term over the entire temperature range. Subtle effects in eq. (5 ) lead to the dramatic superconductivity phase transition. It is reasonable that the more pronounced effects expected by this polyelectrolyte mode1 should also lead to a condensation phenomenon. A rigorous description of this polyelectrolyte condensate is possible by an adaptation of the formalism developed by Scalapino et al. [ 61 and this work will be presented in a future publication. For the present we consider a simpler, more intuitive approach comparable to that in the early BCS mode1 of superconductivity [7 ]
3. Condensate
model
The following model closely parallels the traditional approach of a strong superconductor. The main difference is that in the superconductivity case a spatially homogeneous system is considered. In the polyelectrolyte case the electrostatic interactions will be effectively screened for counterions located far from the polymer. In the following model the attractive interaction between counterions induced by the polyelectrolyte’s boson field is assumed to act only within a volume, V,, surrounding the polymer. The thermodynamic potential difference between the condensate and the normal state, J&-L?“, is given by (cf. ref. [ 81)
where d(x) 18
is the gap function
and g is a perturbation
parameter
representing
the counterion-counterion
in-
Volume 15 I, number 1,2
CHEMICALPHYSICSLETTEKS
7 October 1988
teraction. The quantity g will be treated as an adjustable paramctcr. It is assumed that the gap function is constant within a volume, I’,, surrounding the polyelectrolyte and negligible outside this region. The spatial integration then simply provides the factor V, in the following equations. In superconductivity theory the integration over the variable g’ is converted into one over an energy in which the energy is measured relative to the chemical potential. The limit of this integration is assumed to be small relative to the chemical potential and an energy density of states is defined using this assumption. This assumption is avoided in the following case. Rather the integral is evaluated using the mean value theorem which results in an cncrgy density of states evaluated at an undefined energy level, N(c). This parameter will also be treated as an adjustable one. This approach gives Q,-Q,=K’,N(S)
(hw,)2{1-coth[11gN(5)l}.
(7)
As in the superconductivity case, the limits of the integration were set at an energy fiw,. The result in eq. (7) represents strong coupling in the zero-temperature limit. Extension to higher temperatures follows in an analogous fashion as the superconductivity formalism. Using eq. (7) it is now possible to determine colligative properties in the salt-free case. Although N(r) has not been explicitly determined in this model, we anticipate that it will be proportional to N/V (cf. p. 447 of ref. [ 81) and that the entire dependence for these parameters are contained in N( 5) and in V,. From the above assumption, we have -N-l
%V(~)/aI’=V’-’
The activity coefficient,
&V({)/aN.
(8)
yy, is given by
~~~~=~[(~,-~,>/~~~ll~~=~{1-coth[~lg~t5)l+~lg~(~)-[~lg~(5)lcoth2[~lg~(~)l},
(9)
where K=
[aNg)/alv]
[ Vc(hw,)2/VkT]
.
With the USCof cqs. (8) and (9), the osmotic coefficient,
Qp, is given by
~P=i-(kTN)-Ia(~n,-sz,)/av=1+lnrf-(aV,/aV)(SZ,-SZ,)/I/,.
(10)
These functional forms may be compared with results from the Poisson-Boltzmann ories. These theories utilize the dimensionless structural parameter 5’ defined as 5’ =q2/EkTb,
[ 41 and Manning
[ 21 the-
(111
where q is the electonic charge, e is the bulk dielectric, and b is the average axial charge spacing of the polymer. The colligative properties are given by forr’
forc>l:
lnrY=-&l’
(Manning
theory) ,
@,=l+lnrY
(Manning
and PB theory) ;
Y~=~‘-‘e-‘/2, YE=(2t’)-’
I
Yf/@,=2e-‘/’
(Manningtheory),
YP/#P= 1
(PB theory) .
(12)
At this point one cannot easily compare the quantum field theory with the other two theories because the g, N(r), V, and fro, have not been derived from basic principles. Our approach will be to use appropriate adjustable parameters to establish whether the functional form of eqs. (9) and ( 10) can fit the other theories. To achieve this the functional dependence of the adjustable parameters on I? must be known. These are anticipated using the following physical arguments. First, the axial dimension of V, will be proportional to b, whereas the radial dimension is related to the Debye length. Thus, V, will be proportional to <‘-l. The mean 19
Volume 15 I, number
I,2
CHEMICAL
PHYSICS LETTERS
4
7 October
1988
Fig. I. Plot of the counterion acttvtty coefftcient versus the polyelectrolyte structural parameter. Dashed line is function predicted by Manning’s condensation theory. Solid line is the result oftitting the quantum theory to the dashed line for values oft > 1. Two adjustahle parameters were used as described in the text.
value energy density, N( <), and fiw, for the counterion will be a function of the kF for the counterion particle. While k, of the polyelectrolyte particle will be proportional to r’, this parameter for the counterion is to first order merely related to the volume and number of particles. Thus, N( <) and iron arc independent of 5’. Because the coupling of counterions is mediated by the polyelectrolyte electrostatics, it is anticipated that the coupling factor g will be proportional to r. With these considerations, three adjustable parameters are used to fit the colligative properties of the Manning model. They are glv( 5) (proportional to 5’ ), K (proportional to r’ ~ ’ ) and N(c) ( fiwD)* i3VJa V (proportional to r’-’ ). Non-linear least-squares fits were performed to determine the proportionality constants for these parameters. In all of these fits only the region r’ > I was used. If the following assignments are made: glv(r) = d? and K- 1.19/f, then eq. (9) gives the curve of ~7 versus c shown in fig. 1. The expected curve for the Manning theory is also shown for comparison. As can be seen, quite accurate fits are possible. The region c < 1 would not be expected to be fit well by the quantum theory if, as predicted by Manning theory, “condensation” does not occur. The osmotic coefficient was calculated using the above fitted parameters and independently adjusting the remaining term in eq. ( lo), N( 5) (hw,,)’ X dV,/dV. The fit to Manning’s theory for tip versus r’ is shown in fig. 2 in which the adjusted term equalled 1.66/r’. Fig. 3 shows the fit to the Poisson-Boltzmann result (qb=yP ) when the adjusted term equalled I .57/
Fig. 2. Plot of the osmotic coeffkient versus the polyelectrolyte structural parameter. Dashed line is the function predicted by Manning’s condensation theory. Full line is quantum theory fit using the adjusted values from fig. 1 and one additional adjustable parameter. See text for details.
20
Fig. 3. Plot of the osmotic coefftcient versus the polyelectrolyte structural parameter. Dashed line is the function predicted from the Poisson-Bohzmann theory. Full line is quantum theory tit using the adjusted values from fig. 1 and one additional adjustable parameter. See text for details.
Volume 15 1, number I,2
CHEMICAL PHYSICS LETTEKS
7 October 1988
r. As can be seen from these plots, the fit to Poisson-Boltzmann theory is better than to Manning’s theory for the osmotic coefficient parameter. The purpose of the above exercise is to demonstrate that the functional form of the colligative properties derived from this quantum theory is capable of representing the very different functions derived from the Poisson-Boltzmann and Manning theory. The purpose was not to propose specific values for the parameters of the model.
4. Discussion A simple quantum model of polyelectrolyte-counterion interaction has been examined. The use of the second quantization notation served to link this approach to other more traditional models of phase transitions in quantum systems. This allows one to visualize counterion condensation as a collective mode of the system resulting from the mediation of counterion interactions by the polarization of the polyelectrolyte. The transformation of the polyelectrolyte fermion number operators to a boson field was the key step in establishing these relationships. This relatively crude model provides an encouraging fit to the colligative processes of a salt-free polyelectrolyte and provides a starting point for more rigorous and more realistic models. This fit required assumptions concerning the functional dependence of the model’s parameters on C. While these assumptions were based on physical arguments, a more complete theory is required to establish them from first principles. This may be achieved by employing a more general Hamiltonian in eq. ( I ) in which the counterioncounterion electrostatic interaction is explicitly introduced. Future work will accomplish this using the model of Scalapino et al. [ 61.
References [ I ] C.F. Anderson and M.T. Record, Ann. Rev. Phys. Chem. 33 ( 1982) 191. [2] G.S. Manning, J. Chem. Phys. 51 (1969) 924. [ 31 S. Tomonaga, Progr. Theoret. Phys. (Kyoto) 5 (1950) 544. [4] S. Lifson and A. Katchalsky, J. Polymer Sci. 13 (1954) 43. [ 5 ] G. Mahan, Many particle physics (Plenum Press, New York, 198 I ) pp. 311-3 17. [6] D.J. Scalapino, J.R. Schrieffer and J.W. Wilkins, Phys. Rev. 148 (1966) 263. [7] N.N. Bogoliubov, Soviet Phys. JETP 7 ( 1958) 41. [ 81 A.L. Fetter and J.D. Walecka, Quantum theory of many-particle systems (McGraw-Hill, New York, 1971) pp. 439-454.
21
I,2
QUANTUM
FIELD THEORY
CHEMICAL
PHYSICS LETTERS
OF POLYELECTROLYTE-COUNTERION
7 October
1988
CONDENSATION
T.G. DEWEY Department of Chemistry UniversityofDenver, Denver, CO 80208, USA Received 22 March 1988; in final form 20 July 1988
A simple quantum theory of polyelectrolyte-counterion interactions is presented. A model Hamiltonian is employed which describes both the polyelectrolyte and the counterion as free, spinless fermions. This Hamiltonian is transformed into a form which is isomorphous with traditional Hamiltonians used to describe phase transitions. The difference between this theory and early theories of superconductivity is that the counterion-counterion interaction energies will be quite large and will persist at high temperatures. The counterion condensate is a collective mode resulting from polyelectrolyte-mediated polarizations. Colligative properties for this model are compared with the Poisson-Boltzmann theory and to Manning’s condensation theory
1. Introduction The continued interest in the physical chemistry of nucleic acids has spurred renewed experimental and theoretical efforts in the study of polyelectrolytes. Two main theories dominate the description of polyelectrolytes. They are the Poisson-Boltzmann theory and Manning’s counterion condensation theory (for a comparison of these approaches see ref. [ 1 ] ). The Poisson-Boltzmann theory provides a rigorous theoretical approach in which the levels of approximation can be well defined. Unfortunately, the Poisson-Boltzmann equation for a polyclcctrolyte can only be solved exactly in limiting cases and this approach has not been entirely tractable in describing experimental data. Manning’s theory on the other hand provides a simple mathematical device by which a large body of experimental data may be explained. In Manning’s original theory [ 2 1, he postulated a “condensation” of counterions about the polyelectrolyte which served to relieve an instability in the phase integral. While this theory has been quite successful, it has not provided an intuitive physical picture of the “condensation” phenomena. In the current work polyelectrolyte-counterion interactions are considered in a mode1 quantum system, The goal of this approach is to provide a description of these interactions in a simple system and to provide physical insights into the nature of counterion condensation in such a system. A quantum system is chosen that utilizes the powerful, second-quantization operator algebra of many-particle physics. The mode1 consists of a linear structure containing free particles of spinless fermions (the polyclcctrolyte) which interacts electrostatically with free, spinless fermion particles of opposite charge in the surrounding three dimensions (the counterion). The intent of this mode1 is not so much to present a physically realistic picture of polyelectrolytes, but rather it is to explore counterion condensation in a mathematically tractable mode. Using a Tomonaga transformation [ 3 1, the polyelectrolyte can be represented as a boson field. This transformation converts the Hamiltonian to a form homologous with the Frbhlich Hamiltonian. A second, canonical transformation eliminates the boson (polyelectrolyte)-fermion (counterion) interactions and leads to a form in which an attractive interaction exists between counterions. This results from a counterion’s polarization of the polyelectrolyte’s bosons giving rise to an attractive boson interaction with a second counterion. This boson-mediated attraction is homologous with the effects that lead to the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity. The difference is that in this case the boson energy is the free polyelectrolyte particle energy plus a contribution from the polyelectrolyte electrostatic self-energy rather than a phonon energy. Thus, this is a much higher energy effect and 16
0 009-2614/88/$ ( North-Holland
03.50 0 Elsevier Science Publishers Physics Publishing Division )
B.V.
Volume 15 1, number
1,2
CHEMICAL
PHYSICS LETTERS
7 October
1988
will persist to very high temperatures. Considering this strong attractive force between counterions, a counterion-condensation model is presented which closely resembles a superconducting condensate in the strongcoupling limit. The colligative properties of such a model are examined by assuming realistic functional dependences of the model’s parameters on c, the polyelectrolyte structural parameter. By fitting the proportionality constants for these parameters, quantitative agreement with the Manning model may be obtained.
2. Model Hamiltonian and transformations The following model Hamiltonian is considered in which at (a) represent the annihilation (creation) operators for a spinless fermion localized in a one-dimensional space of length L and bt (b) represents the operators for a counterion in the surrounding three-dimensional volume, V, H= 1 Iklv,a~a,+(2~)-’
1 V,p(k)p(-k)+
k
k
C &&t&+(N/2V k
C U,(p(q)+p(-q))b~b,-,,
(1)
k,q
p(k) = 1 a&ap+k12 P
and N is the number of counterions, V, is the Fourier transform of the electrostatic interaction potential between polyelectrolyte particles and U, is the transform of the electrostatic potential between counterions and polyelectrolyte particles, Ek and 1k 1uF represent the energy of the counterion and polyelectrolyte particles, respectively. In keeping with the Tomonaga model the polyelectrolyte energy is linear in k. This restriction is not required and may be lifted without formally changing the results. As in most treatments of polyelectrolytes the counterion-counterion interactions are not explicitly considered. The above treatment will correspond to a polyelectrolyte in a salt-free solution. The advantage of choosing the salt-free case is that the colligative properties are well defined in the Poisson-Boltzmann [4] and Manning [ 21 model. The Fourier transform, U,, is identical to the three-dimensional transform for a bare Coulombic interaction when a convergence factor is added which decays exponentially along the axis parallel to the polymer. In the following treatment, two transformations will be performed on the Hamiltonian. First, a Tomonaga transformation (cf. ref. [ 5 ] ) will be performed in order to express the polyelectrolyte particles as a boson field. The resulting Hamiltonian will be homologous to the Frijhlich Hamiltonian with the phonon-electron interaction term representing the polyelectrolyte boson-counterion interaction. A canonical transformation of this Hamiltonian will lead to a form which is quadratic in the counterion terms. The resulting Hamiltonian indicates that the counterions may exhibit a strong attraction to each other. The Tomonaga transformation is made by dividing the density operator, p(k), into two terms and identifying a new set of operators, (Ye,as follows: p(k) =PI (k) +P2(k)
>
p,(k) = & &a~p+w ,
>
p,(k) = &
P1(-k)=aR(k~W~/nTk)“2,
P,(k)=a,(kLu,lrcrk)“*, Pz(k)=ff-k(kLWk/Xrk)“*
4-k12ap+k,2,
>
pz( -k)=at,(kh,/n&)“*,
(2)
with z+ being the Fermi velocity of the polyelectrolyte parwhere wk= Iklv, and &= I/clz+( 1+2VO/~~,)“2 ticles and V. is a constant describing polyelectrolyte charge interactions in an electron-hole excitation model [ 5 1. Using this transformation and the assumption that the density operators obey exact commutator relationships [ 5 1, the transformed Hamiltonian is now given by 17
Volume 151, number
CHEMICAL
I,2
H’= 1 rk(&h+f)+ k
7 October
PHYSICS LETTERS
1k Ekblbk+(R/V)1’2 1 Uqb~bk_,(at,+aq)
1988
(3)
)
k.q
where U” = Uqq(wk/&) ‘12/2x. This expression is homologous with the Frijhlich Hamiltonian and the boson interaction term can be eliminated by a canonical transformation. This transformation is given by H” =e-.YH’ Defining
e”
.
(4)
s as
s=-(7t/v)“2
1 U;blb,( k.k’
gives the transformed H”= 1 Ekb@,+ k
% Or” Ek-Ek, +<_, + Ek-Ep-5,
Hamiltonian,
>
H”,
1 &(a~c~.+~)+(rc/V)“* k
C U;bj$+qbl_gbkbk. k.k’
(E
)z
Et, k-
k
u
rz. -
(5)
v
This transformed Hamiltonian has eliminated first-order terms in the polyelectrolyte boson-counterion interaction. The remaining counterion-counterion term is restricted to the set of terms arising from the commutation of the boson operators. This condition restricts the energy difference in the denominator of eq. (5) to be Ek-Ek_+. A Hamiltonian of this form is the starting point for the BCS theory of superconductivity. A negative Fourier transform for the two-particle interaction potential admits the possibility of an attractive interaction between particles. This occurs when &> Ek- Ek_r In the current case this interaction is likely to be much stronger than in the superconductivity case. In superconductivity <, is the phonon energy and is only comparable to the electron energy at extremely low temperatures. In the current case, $ is the boson energy derived from the polyelectrolyte. It contains contributions from both the polyelectrolyte particle’s kinetic energy and from the electrostatic interactions between polyelcctrolytc particles. Thus this energy term would be expected to be comparable to, if not greater than, the counterion energy. The denominator in eq. (5) is, therefore, expected to yield a negative term over the entire temperature range. Subtle effects in eq. (5 ) lead to the dramatic superconductivity phase transition. It is reasonable that the more pronounced effects expected by this polyelectrolyte mode1 should also lead to a condensation phenomenon. A rigorous description of this polyelectrolyte condensate is possible by an adaptation of the formalism developed by Scalapino et al. [ 61 and this work will be presented in a future publication. For the present we consider a simpler, more intuitive approach comparable to that in the early BCS mode1 of superconductivity [7 ]
3. Condensate
model
The following model closely parallels the traditional approach of a strong superconductor. The main difference is that in the superconductivity case a spatially homogeneous system is considered. In the polyelectrolyte case the electrostatic interactions will be effectively screened for counterions located far from the polymer. In the following model the attractive interaction between counterions induced by the polyelectrolyte’s boson field is assumed to act only within a volume, V,, surrounding the polymer. The thermodynamic potential difference between the condensate and the normal state, J&-L?“, is given by (cf. ref. [ 81)
where d(x) 18
is the gap function
and g is a perturbation
parameter
representing
the counterion-counterion
in-
Volume 15 I, number 1,2
CHEMICALPHYSICSLETTEKS
7 October 1988
teraction. The quantity g will be treated as an adjustable paramctcr. It is assumed that the gap function is constant within a volume, I’,, surrounding the polyelectrolyte and negligible outside this region. The spatial integration then simply provides the factor V, in the following equations. In superconductivity theory the integration over the variable g’ is converted into one over an energy in which the energy is measured relative to the chemical potential. The limit of this integration is assumed to be small relative to the chemical potential and an energy density of states is defined using this assumption. This assumption is avoided in the following case. Rather the integral is evaluated using the mean value theorem which results in an cncrgy density of states evaluated at an undefined energy level, N(c). This parameter will also be treated as an adjustable one. This approach gives Q,-Q,=K’,N(S)
(hw,)2{1-coth[11gN(5)l}.
(7)
As in the superconductivity case, the limits of the integration were set at an energy fiw,. The result in eq. (7) represents strong coupling in the zero-temperature limit. Extension to higher temperatures follows in an analogous fashion as the superconductivity formalism. Using eq. (7) it is now possible to determine colligative properties in the salt-free case. Although N(r) has not been explicitly determined in this model, we anticipate that it will be proportional to N/V (cf. p. 447 of ref. [ 81) and that the entire dependence for these parameters are contained in N( 5) and in V,. From the above assumption, we have -N-l
%V(~)/aI’=V’-’
The activity coefficient,
&V({)/aN.
(8)
yy, is given by
~~~~=~[(~,-~,>/~~~ll~~=~{1-coth[~lg~t5)l+~lg~(~)-[~lg~(5)lcoth2[~lg~(~)l},
(9)
where K=
[aNg)/alv]
[ Vc(hw,)2/VkT]
.
With the USCof cqs. (8) and (9), the osmotic coefficient,
Qp, is given by
~P=i-(kTN)-Ia(~n,-sz,)/av=1+lnrf-(aV,/aV)(SZ,-SZ,)/I/,.
(10)
These functional forms may be compared with results from the Poisson-Boltzmann ories. These theories utilize the dimensionless structural parameter 5’ defined as 5’ =q2/EkTb,
[ 41 and Manning
[ 21 the-
(111
where q is the electonic charge, e is the bulk dielectric, and b is the average axial charge spacing of the polymer. The colligative properties are given by forr’
forc>l:
lnrY=-&l’
(Manning
theory) ,
@,=l+lnrY
(Manning
and PB theory) ;
Y~=~‘-‘e-‘/2, YE=(2t’)-’
I
Yf/@,=2e-‘/’
(Manningtheory),
YP/#P= 1
(PB theory) .
(12)
At this point one cannot easily compare the quantum field theory with the other two theories because the g, N(r), V, and fro, have not been derived from basic principles. Our approach will be to use appropriate adjustable parameters to establish whether the functional form of eqs. (9) and ( 10) can fit the other theories. To achieve this the functional dependence of the adjustable parameters on I? must be known. These are anticipated using the following physical arguments. First, the axial dimension of V, will be proportional to b, whereas the radial dimension is related to the Debye length. Thus, V, will be proportional to <‘-l. The mean 19
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Fig. I. Plot of the counterion acttvtty coefftcient versus the polyelectrolyte structural parameter. Dashed line is function predicted by Manning’s condensation theory. Solid line is the result oftitting the quantum theory to the dashed line for values oft > 1. Two adjustahle parameters were used as described in the text.
value energy density, N( <), and fiw, for the counterion will be a function of the kF for the counterion particle. While k, of the polyelectrolyte particle will be proportional to r’, this parameter for the counterion is to first order merely related to the volume and number of particles. Thus, N( <) and iron arc independent of 5’. Because the coupling of counterions is mediated by the polyelectrolyte electrostatics, it is anticipated that the coupling factor g will be proportional to r. With these considerations, three adjustable parameters are used to fit the colligative properties of the Manning model. They are glv( 5) (proportional to 5’ ), K (proportional to r’ ~ ’ ) and N(c) ( fiwD)* i3VJa V (proportional to r’-’ ). Non-linear least-squares fits were performed to determine the proportionality constants for these parameters. In all of these fits only the region r’ > I was used. If the following assignments are made: glv(r) = d? and K- 1.19/f, then eq. (9) gives the curve of ~7 versus c shown in fig. 1. The expected curve for the Manning theory is also shown for comparison. As can be seen, quite accurate fits are possible. The region c < 1 would not be expected to be fit well by the quantum theory if, as predicted by Manning theory, “condensation” does not occur. The osmotic coefficient was calculated using the above fitted parameters and independently adjusting the remaining term in eq. ( lo), N( 5) (hw,,)’ X dV,/dV. The fit to Manning’s theory for tip versus r’ is shown in fig. 2 in which the adjusted term equalled 1.66/r’. Fig. 3 shows the fit to the Poisson-Boltzmann result (qb=yP ) when the adjusted term equalled I .57/
Fig. 2. Plot of the osmotic coeffkient versus the polyelectrolyte structural parameter. Dashed line is the function predicted by Manning’s condensation theory. Full line is quantum theory fit using the adjusted values from fig. 1 and one additional adjustable parameter. See text for details.
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Fig. 3. Plot of the osmotic coefftcient versus the polyelectrolyte structural parameter. Dashed line is the function predicted from the Poisson-Bohzmann theory. Full line is quantum theory tit using the adjusted values from fig. 1 and one additional adjustable parameter. See text for details.
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r. As can be seen from these plots, the fit to Poisson-Boltzmann theory is better than to Manning’s theory for the osmotic coefficient parameter. The purpose of the above exercise is to demonstrate that the functional form of the colligative properties derived from this quantum theory is capable of representing the very different functions derived from the Poisson-Boltzmann and Manning theory. The purpose was not to propose specific values for the parameters of the model.
4. Discussion A simple quantum model of polyelectrolyte-counterion interaction has been examined. The use of the second quantization notation served to link this approach to other more traditional models of phase transitions in quantum systems. This allows one to visualize counterion condensation as a collective mode of the system resulting from the mediation of counterion interactions by the polarization of the polyelectrolyte. The transformation of the polyelectrolyte fermion number operators to a boson field was the key step in establishing these relationships. This relatively crude model provides an encouraging fit to the colligative processes of a salt-free polyelectrolyte and provides a starting point for more rigorous and more realistic models. This fit required assumptions concerning the functional dependence of the model’s parameters on C. While these assumptions were based on physical arguments, a more complete theory is required to establish them from first principles. This may be achieved by employing a more general Hamiltonian in eq. ( I ) in which the counterioncounterion electrostatic interaction is explicitly introduced. Future work will accomplish this using the model of Scalapino et al. [ 61.
References [ I ] C.F. Anderson and M.T. Record, Ann. Rev. Phys. Chem. 33 ( 1982) 191. [2] G.S. Manning, J. Chem. Phys. 51 (1969) 924. [ 31 S. Tomonaga, Progr. Theoret. Phys. (Kyoto) 5 (1950) 544. [4] S. Lifson and A. Katchalsky, J. Polymer Sci. 13 (1954) 43. [ 5 ] G. Mahan, Many particle physics (Plenum Press, New York, 198 I ) pp. 311-3 17. [6] D.J. Scalapino, J.R. Schrieffer and J.W. Wilkins, Phys. Rev. 148 (1966) 263. [7] N.N. Bogoliubov, Soviet Phys. JETP 7 ( 1958) 41. [ 81 A.L. Fetter and J.D. Walecka, Quantum theory of many-particle systems (McGraw-Hill, New York, 1971) pp. 439-454.
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