8946
J. Phys. Chem. 1996, 100, 8946-8953
Multistate Self-Consistent Field Theory for the Calculation of the Interface of Two Immiscible Electrolyte Solutions A. Vincze,† G. Horvai,*,‡ and F. A. M. Leermakers*,§ Department of Physical Chemistry, Technical UniVersity of Budapest, H-1521 Budapest, P.O. Box 91, Hungary, DiVision of Chemical Information Technology, Technical UniVersity of Budapest, H-1111 Budapest, Gelle´ rt te´ r 4, Hungary, and Department of Physical and Colloid Chemistry, Wageningen Agricultural UniVersity, Dreijenplein 6, 6703 HB Wageningen, The Netherlands ReceiVed: February 7, 1995; In Final Form: NoVember 8, 1995X
A new theory is formulated for the calculation of the interface of two immiscible electrolyte solutions. The theory is based on the self-consistent field theory for (chain) molecules in inhomogeneous systems developed by Scheutjens and Fleer. Complexation equilibrium between ions and carrier molecules is taken into account via the multistate character of the segments. Electrostatic interactions are also included in the theory. The self-consistent equations are derived starting with the optimization of the grand canonical partition function of the system. The set of equations is solved iteratively, giving the equilibrium distributions of the components in the system. Calculations for a model system, consisting of a liquid-membrane phase doped with a carrier molecule in contact with a simple electrolyte solution, are performed to test the applicability of the new theory. These results show that the calculations presented in this paper can be used to interpret or predict important analytical features of ion-selective electrodes.
Introduction Ion-selective electrodes (ISEs) based on liquid-phase membranes play an important role in the current field of electrochemistry. It is well-known that the interface at the membranesample solution boundary has a predominant effect on the performance of the electrode. Investigations of the electrochemistry at the interface of two immiscible electrolyte solutions (ITIES) have produced a number of important results.1-6 While there exist useful formulations describing the potential responses of ISEs on the basis of classical thermodynamic derivations,8,9 less attention has been paid to the physical and chemical structure of the interface. The available analytical solutions describing the potential response of these electrodes contain no information about the structure and conformation of molecules present in the system and offer no molecular interpretation of several nonideal features of these electrodes. It is expected that the statistical thermodynamic modeling which is consistent with the macroscopic thermodynamic analysis can give the molecular insight needed to understand and explain these nonidealities.7 In our earlier paper we have used the self-consistent field (SCF) theory for (chain) molecules in inhomogeneous system developed by Scheutjens and Fleer (SF theory),10 subsequently extended and used to model self-assembling systems by Leermakers et al.,11 to model copolymer systems by Evers et al.,12 and to compute interfacial properties of liquid-liquid interfaces by Barneveld et al.,13 for the modeling of liquidmembrane electrodes based on liquid ion exchangers and liquidmembrane electrodes doped with neutral carriers.14 The SF theory was chosen because it enables the modeling of spontaneous formation of the interface between two immiscible solutions by self-assembling of the components in the system without the presumptions considering the exact place and thickness of the interface. The theory is based on a quasi-crystalline lattice model. The characteristic size of a lattice size is the size of a water molecule or the size of a polymer segment. No information is obtained from the system properties on smaller length X
Abstract published in AdVance ACS Abstracts, April 15, 1996.
S0022-3654(95)00360-1 CCC: $12.00
scales than the size of the lattice units. Typically detailed information on the density distributions of the components in the system on length scales higher than the lattice spacing is readily obtained. Lateral interactions between the different segments are taken into account in a Bragg-Williams approximation with Flory-Huggins interaction parameters.15 The theory accounts for the presence of electric fields generated by the different distributions of the ions across the interface and for the excluded volume effect of molecule segments and ions. We have found that this statistical thermodynamic approach is suitable for describing some of the most important characteristics of ion-exchange type liquid-membrane ISEs. However, using the SF theory, we could not yet take the local equilibrium between the different species into account. Complexation reactions are known to occur between metal ions and so-called carrier molecules, e.g., valynomycin or crown ethers. In this paper we describe an extension of the SF theory, a multistate SCF theory, which enables us to take complexation into account. The new theory accounts for the complexation reaction between the carrier and one or more complexing ions by introducing complex-stability constants as additional parameters. We describe the theory below and present results of calculations for model systems. Theory of Ion-Selective Liquid-Phase Membranes Based on Neutral Carriers Complexation Equilibrium Model. Consider a system with i ) 1, 2, ..., x types of components. In this treatment the components of the system are either molecules or ions occupying a single lattice site (e.g., water and certain ions) or molecules consisting of a chain of so-called segments, each segment fitting into one lattice site. The membrane material (polymer) is modeled by homodisperse chain molecules. The carrier molecules are also considered as chain molecules which possesses a specific segment that represents the active part of the molecule. The components are distributed over a lattice such that each segment occupies exactly one lattice site as shown schematically in Figure 1. © 1996 American Chemical Society
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J. Phys. Chem., Vol. 100, No. 21, 1996 8947
Figure 2. Complexation reaction between the free carrier molecule and a positively charged ion forming a complex carrier molecule. Segment V in the free carrier molecule is in a state called V0, while in the complexed form it is in a state called V1+.
Figure 1. Schematic two-dimensional representation of the lattice with a possible arrangement of the components. The membrane material is simply modeled as a homopolymer chain A15, the carrier molecule is modeled as A8VA8, and the water and the ions are modeled as single segments.
We define the lattice in such a way that parallel layers each composed of L lattice sites are present. We allow concentration gradient only in one direction that is normal to the interface of interest and ignore inhomogenities within each layer. Thus, a local mean-field approximation is applied in two directions. The layers are numbered by z ) 0, 1, 2, ..., M, M + 1, and reflecting boundary conditions are used both between layers 0 and 1 and between layers M and M + 1. The lattice is positioned in such a way that lattice layer z ) 1 is in the bulk of the membrane phase, whereas layer z ) M is in the bulk of the aqueous phase. The bulk is defined by the condition that all inhomogenities due to the presence of an interface have died out. The lattice is characterized by the coordination number Z and the local organization of the lattice sites with respect to each other (lattice topology), which is specified by the fraction of neighboring sites λ-1, λ0, and λ1, respectively, that a lattice site in layer z has in the layers z′ ) z - 1, z, and z + 1, respectively. (For instance, in a hexagonal lattice Z ) 12, λ0 ) 6/12, and λ(1 ) 3/12). Let us consider that there are κ types of cations present in the system (denoted as Pν11, ..., Pνk k, ..., Pκνκ, respectively, where νk represent the charge of Pνk k) and they can form only 1:1 complexes with the carrier molecules (denoted here as CM) according to the following equation
Pνk k + CM0 T CMk
(1)
where CM0 represents the free carrier molecule and CMk is the complex formed with the kth type of cation. Here we do not yet specify how the complex is formed, but later we shall limit the discussion to carrier molecules where only one segment participates in the complexation reaction. The chemical equilibrium criterion for eq 1 is generally given by the balance of the chemical potentials of the components as
µCM0 + µPkνk ) µCMk
(2)
To model this equilibrium, we have modified the general multistate theory (an extension of the SF theory) given by Bjo¨rling et al.16 In his approach one considers that all the
segments in the system are allowed to have an arbitrary number of internal states. A segment in a given internal state possesses a given amount of internal energy and internal entropy, which both contribute to the total free energy of the system according to
Atot ) ∑∑NA(z)∑RAk(z)(UAk - kBT ln ωAk + z
A
k
kBT ln RAk(z)) + Amix (3) where kB is the Boltzmann constant, ∑z denotes the sum over all layers, ∑A is the sum over all segment types, and ∑k is the sum over all internal states of segment A. Here NA(z) is the number of segments A in layer z, RAk(z) is the probability that segment A is in internal state k, and UAk and kB ln ωAk are the internal energy and entropy of segment A in internal state k, respectively (together they give the internal free energy of segment A, cf. eq 5). The expression kB ln RAk is an entropy term arising from the mixing of internal states, and Amix is the free energy that originates from the mixing of the components in the system. For simplicity, in the model system that we introduce in this paper only one specific type of segment of the carrier molecules (segment V in Figure 1) is allowed to have more than one internal state; all the other types of segments are excluded from the multistate character and are considered to be in internal state 0. However, the theory and the equations are given for the general case. We consider now a specific type of the complexation reaction given by eq 1. The complex CMk is formed by complexing the cation Pνk k to a single segment of the carrier. This segment represents the active part of the carrier molecule CM0 (segment V in Figure 1). By the complexation with Pνk k, it will be transformed from internal state 0 to another internal state k. In this process the complexed ion Pνk k is assumed to be incorporated into segment V. We are going to assume that the affinity that V has for ion Pνk k does not depend on the position of V in the carrier molecule. Therefore the complexation reaction can be written as
Pνk k + VO T Vk
(4)
where VO represents the segment of type V in internal state 0 (in the free carrier molecule), whereas Vk represents the segment of type V in internal state k as shown in Figure 2. We make here the rough assumption that the volumes of V0 and Vk are identical. Moreover, we will assume that V0 and Vk only differ in their electrostatic and internal state properties, but not in their
8948 J. Phys. Chem., Vol. 100, No. 21, 1996
Vincze et al.
physicochemical nature. Extensions to the general case are straightforward. The internal free energy of segments Vk and V0 can be defined as int AVk ) UVk - kBT ln ωVk
is determined by the internal free energy value of segment V in internal state k with reference to internal state 0 b ln KCM )k
UVk - UV0 ωVk ) + ln kBT ωV0
(5a) ln
and int ) UV0 - kBT ln ωV0 AV0
(5b)
respectively. The quantities UVk, kB ln ωVk, UV0, and kB ln ωV0 in Bjo¨rling’s approach describe the equilibrium between Vk and V0. In our case, however, this equilibrium also depends on the presence of the cation Pνk k. This can be understood if we realize that the internal free energy of the complex segment int , consists of the free energy due to segment V alone Vk, AVk and the free energy of the cation that is incorporated. It will prove convenient in treating the system to define the part of the internal free energy of segment Vk that is due to segment V alone by subtracting the free energy of the cation. int int )U ˆ Vk - kBT ln ω ˆ Vk ) AVk - µPkνk Aˆ Vk
(6a)
Obviously, for V0 int int ) AV0 Aˆ V0
µˆ CMk ) µCMk - µPkνk
(7a)
Because there is no contribution to the internal free energy of the free carrier molecules due to any other components in the system (e.g., they are not complexes), we may write
µˆ CM0 ) µCM0
(7b)
The consequences of eqs 7a and 7b is that the equilibrium criteria can be simply expressed as
µˆ CMk ) µˆ CM0
(8)
This equilibrium condition is, from a molecular point of view, instructive because we now can find a µˆ CM which does not depend on the internal state of the segment V. If complexes of several types of cations are considered, it is given by κ
µˆ CM ) ∑RCMkµˆ CMk ) µˆ CM0
b fCM φb f b νkφPb kνk 0 CM0 Pk
(10)
In eq 10 fbi and φbi are the activity coefficient and volume fraction of component i in the bulk, respectively. In this way we can use the experimentally available complex stability constant as an additional model parameter to take complexation into account. The advantage of this treatment is that we can handle the complexation equilibrium with a formalism very similar to that of Bjo¨rling et al.16 to find the equilibrium distributions of the segments and the internal states in the system. Equilibrium Distributions. The derivation of the segment and internal state distributions in the system can be realized starting with the grand-canonical partition function, following the line given by Evers et al.12 and Bjo¨rling et al.16 New aspects of the derivation together with some important equations modified by the present approach are given in the Appendix. The internal state distributions {RAk(z)} in the system are the immediate result of the optimization of the grand-canonical partition function and can be given as derived in16
(6b)
since there is no contribution from any other segment. For the same reason we introduce a chemical potential µˆ CMk which defines from µCMk by the chemical potential of the complexing ion
b f CM φb k CMk
RAk(z) )
XAk(z) (11)
∑k XAk(z)
where XAk(z) is given in the Appendix, expressing the statistical weight of finding segment A in layer z in internal state k. Using eqs 10 and 11 with the addition of the mean-field electrostatic potential, we find that b b φV0 φPb kνkf Pb kνkKCM exp{νk(yb - y(z))} k
RVk(z) ) b φV0
(12)
+∑ k
b b φV0 φPb kνkf Pb kνkKCM k
exp{νk(y - y(z))} b
Equation 12 gives the degree of complexation for complex CMk as a function of the local coordinate z. As expected, this quantity depends on the local electrostatic potential, since the charge of the free carrier and that of the complex are different. If we are in the bulk, that is, y(z) ) yb, then we retrieve the expression for the bulk value of the degree of complexation as could be derived in the “classical” way. The calculation of the segment distributions (density profiles) is more complicated due to the conductivity of the segments in the chain molecules. By the optimization of the unconstrained function, the number of molecules i in a given conformation d in the system can be achieved as shown in the Appendix; the result is
(9)
k)0
where RCMk is the probability that a carrier molecule is a complex with ion Pνk k and the sum over all these probabilities equals unity. The equilibrium between CMk and CM0 can be characterized by the complex stability constant of the complex CMk, which
nid L
) Ciλid∏∏GA(z)NA,i(z) d
z
(13)
A
where GA(z) is the free segment distribution function, which represents the statistical weight of finding a segment of type A in layer z if it were not attached to a polymer chain. In the
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J. Phys. Chem., Vol. 100, No. 21, 1996 8949
multistate theory, it is given as the bulk-averaged free segment distribution function of the individual internal states b GA(z) ) ∑RAk GAk(z)
For a system which is electroneutral we can set the electric field strengths at the system boundaries equal to zero. We estimate the potential at layer z, yo(z), by applying Gauss’ law
(14)
The free segment distribution function of an internal state is given as the Boltzmann factor of the local potential for an A segment at position z in internal state k
(
)
uAk(z) kBT
GAk(z) ) exp -
kBT
)
u′(z) kBT
+ ∑χAB(〈φB(z)〉 - φBb) + νAk(y(z) - yb) B
kBT
)
β(z) kBT
+∑ i
φib Ni
+
1
φAbχABφBb ∑ ∑ A B
2
C(z-1,z) + C(z,z+1)
(17)
u′(z) is usually called the hard core potential, since it is a function of the Lagrange parameter β(z), which is coupled to the space filling of the system (see the Appendix). The second term of eq 16 is the short range nearest-neighbor interaction contribution, and the last term represents the long range electrostatic interaction contribution to the segment potential. The result given by eq 13 says that the number of molecules i in conformation d can be calculated from the free segment distribution functions which depend on the average segment densities (density profiles). The density profiles are of course fully determined by the summation over all possible conformations. Already for short chains (Ni e 10) this summation contains too many terms to be evaluated, even for modern computers. There is, however, an efficient numerical method, known as the propagator method, to obtain the density profiles from the free segment distribution functions using eq 13, but without the need to generate all the conformations separately.10-13 This is the method we used in our model calculations. Numerical Method. Characteristic for a self-consistent field theory is that the segment potentials are functions of the segment densities and that the segment densities are in turn functions of the segment potentials. The standard way to solve this selfconsistency is the following. Let there be x different segment types (irrespective of the internal degrees of freedom) in the system. For each layer z we introduce x + 1 iteration variables {uAi(z), uiB(z), ..., yi(z)}, where the super index i indicates the fact that these are initial (guessed) values. The first x potentials contain only the u′(z) and the short range interactions, not the electrostatic term; the electrostatic potential is given in the last variable. From these quantities we compute the free segment distribution functions {GAk(z)} for each segment type and for each internal state (eqs 12-14). When the propagation method mentioned above is applied the segment density distributions {φA(z)} can also be found. The potentials {uAo(z), uoB(z), ..., yo(z)}, where the super index o refers to the output character of these quantities, are now recomputed (eqs 16 and 17). For charged systems this is slightly involved.
(18)
where q(z), the overall charge per lattice site in layer z, is
q(z) ) e∑∑νAkφAk(z) A
(19)
k
and the charges in each layer are considered to be located on the midplate (a plate that is perpendicular to the z direction and divides the layer into two halves) of the layer. C(z,z′) is the capacity between the midplates of z and z′
C-1(z,z+1) )
(16)
where
u′(z)
yo(z) )
(15)
where uAk(z) is given by
uAk(z)
q(z) + C(z,z+1)yi(z+1) e/kBT
C(z-1,z)yi(z-1) +
k
(
)
1 1 1 + 2d (z) (z+1)
(20)
where the dielectric permitivity changes half-way between the two midplates from (z) to (z+1). We use a density-weighted average (z) ) o∑AAφA(z), where o is the dielectric permitivity of vacuum and A is the relative dielectric constant of a pure phase of A segments. The short range term uo(z) is enumerated according to
uAo(z) ) ∑χAB(〈φB(z)〉 - φBb)
(21)
B
The difference uAi(z) - uAo(z) should be segment type independent. (Note that we have taken out the electrostatic part as a separate iteration variable). We compute the average difference
A
u′(z) )
uAi(z) - uAo(z) ∑ A (22)
x
We now define a set of x + 1 functions: {gA(z), gB(z), ..., gy(z)}. The first x terms are given by
gA(z) ) 1 - ∑φB(z) + uAi(z) - uAo(z) - u′(z)
A
(23)
B
and the last function is given by
gy(z) ) yi(z) - yo(z)
(24)
Numerically, we solve for the functions given by eqs 23 and b , which we need in eq 24 to be zero. We should note that RVk b b ) (φPb νkkf Pb νkkKCM )/ 14, can be expressed using eq 12 as RVk k b b b (1 + ∑kφPνkkf PνkkKCMk). At the very beginning of the iteration, b is approximated however, f Pb νkk is not yet known; therefore, RVk b by assuming that f Pνkk is unity, and at the end of each step f Pb νkk is recalculated using the following general equation resulting from the theory
f ib )
{(
exp - 1 - ∑ j
φjb Nj
-
1
)}
∑ ∑(φAb - φ*A,i)χAB(φBb - φ*B,i) A B
2
(25) and the necessary corrections are performed.
8950 J. Phys. Chem., Vol. 100, No. 21, 1996
Vincze et al.
TABLE 1: Segment Representation of the Components Present in the Model (System I) component
segment representation
component
segment representation
PVC chain carrier molecule hydrophobic anion
(A)15 (A)8V(A)8 (A)3C(A)3
water hydrophilic cation hydrophilic anion
W K+ N-
TABLE 2: Segment Parameters in the Model System χ
W
A
C
V
K
N
valence
relative dielectric constant
W A C V K N
0
2 0
10 0 0
-0.1 0 0 0
-5 2 0 0 0
-5 2 0 0 0 0
0 0 -1 0 +1 -1
80 2 20 40 80 80
The use of eqs 24 and 18-20 forces electroneutrality in the systems defined by the layers z ) 1, ..., M. However, we also have to consider the electroneutrality condition for the reference system, that is, the homogeneous system which is in equilibrium with the system of interest. There are two of these systems: one homogeneous phase predominantly composed of water in which a small amount of polymers and ions is dissolved and one homogeneous polymer phase with a little water and ions in it. These two phases are obviously in equilibrium with each other. Usually, we choose the water-rich system to be the reference system, but it can be shown that it is irrelevant which one we choose. When the root of eqs 23 and 24 is available, the sum over the segment densities equals unity in all layers, the average space-filling potential as defined by eq 22 is identical to the individual values of the sum, i.e., the space-filling potential is independent of segment type, and the input and output electrostatic potential profile is identical so that the estimate of the potential in eq 18 becomes the exact value. As in all space coordinates, Gauss’ law applies, so we obey all electrostatic rules. Having the equilibrium segment distribution, the internal distribution, the equilibrium set of the hard core potential, and the equilibrium electrostatic potential profile, we can evaluate the free energy of the system and other thermodynamic characteristic functions (surface tension, etc.) in terms of the segment densities and local potentials. Application of the Model to Ion-Selective Membranes Choice of Parameters. Let us consider a system consisting of an organic liquid phase composed of PVC chains blended with neutral carrier molecules and hydrophobic anions in contact with an aqueous solution containing a hydrophilic salt. This type of system is of great practical importance, since it is the basis of liquid-membrane electrodes doped with neutral carriers.2,3,8,14,17 The segment representation of the components in this model system (system I) is given in Table 1. The segment-segment interaction (χ) parameters in units of kBT, the valence, and the relative dielectric constant of the segments are given in Table 2. We modeled the membrane material with a simple polymer chain consisting of 15 A segments, which can be considered as CH2 units. The carrier is also modeled as a chain comprising 16 segments of type A and one active segment V in the middle of the chain. The hydrophobic anion is also modeled as a chain molecule. The interaction parameter between water (segment W) and CH2 groups (segment A) has been estimated by the
Figure 3. Molar fraction profiles of the components at the interface in system I. The water-bulk concentration of the primary salt (KN) is 1 M.
dependence of the critical micelle concentration (cmc) of surfactant molecules on the number of CH2 groups in their tail to have the value of 2.13 This value assures phase separation between water and the membrane material. The Flory-Huggins interaction parameters for ion-water and ion-membrane material are approximated from the available data for the Gibbs energy of transfer of a given ion from water to an organic solvent.2,3 Negative χ values between water and ions are selected to account for the hydrophilicity of the ions. For the active segment V of the carrier molecule, χ values are chosen to make it slightly hydrophilic. We used a hexagonal lattice type throughout our study. The geometry of the lattice is flat, which implies that the membranewater interface also remains flat. The lattice size is set to 50 layers with a characteristic layer thickness of d ) 0.3 nm. At both sides of the lattice, we have imposed reflecting boundary conditions. The complexation constant for the equilibrium between the hydrophilic cation (K) and the carrier molecule is 105. The amount of the membrane material is θ(A)15 ) 25 equivalent lattice layers. The amount of the carrier molecule was fixed to θ(A)8V(A)8 ) 0.34 and that of the hydrophobic anion to θ(A)3C(A)3 ) 0.07, which was estimated from the composition of liquid membranes usually used in practice. In the process of analyzing the behavior of the model system, the following method was used. For the parameter set presented above, the water bulk concentration of the primary electrolyte (KN) was varied. In each step the equilibrium distribution of the system was calculated resulting in a spontaneous formation of the interface, charge separation, and an interfacial electrostatic potential drop. After calculations the potential was plotted against the logarithm of the primary ion concentration in the bulk of the aqueous phase. In this way “calibration curves” were calculated for a given parameter set. Some of the parameters are then systematically varied to analyze the behavior of the model. Results of the Calculations. The density profiles of the components in system I for 1 M water-bulk concentration of the KN salt, plotted on a logarithmic scale, are shown in Figure 3. It can be seen that the water and the membrane material form an interface by self-assembling, across which all the other components are distributed according to their chemical nature. The transition between the bulk aqueous phase and the bulk organic phase occurs over approximately eight layers (i.e., 2.4 nm). This picture is at variance with the hypothetical model often used in the electrochemistry of ITIES. The latter model assumes the existence of a sharp change in composition at one
Multistate SCF Theory
Figure 4. Volume fraction profiles of the components in system I. The water-bulk concentration of the primary salt (KN) is 0.01 M.
Figure 5. Charge density profiles for different electrolyte bulk concentrations of the primary salt in system I.
defined layer. Our model would give such a sharp interface only if the interaction parameter between the water (W) and the A segments of the polymer were extremely positive. The distribution of various ionic species across the interfacial layers is best appreciated if comparison is made between systems of type I with different concentrations of the hydrophilic salt KN. Figures 3 and 4 show species distributions for two different concentrations of KN in the bulk aqueous phase: 1 M in Figure 3 and 0.01 M in Figure 4. The notable difference is that while the concentration of the hydrophilic anion N in the organic phase is negligible at low KN concentration (Figure 4), it becomes comparable with the concentration of the carrier and hydrophobic anion when the KN concentration is high (Figure 3). What really happens at high KN concentration is that a considerable quantity of KN is extracted into the organic phase. Most of the extracted K is complexed by the carrier, so we do not find a high concentration of free K in the membrane, but the extraction of N is very obvious on the logarithmic scale. This extraction of KN into the organic phase explains the nonideal potentiometric behavior of liquid membrane ISEs at high analyte concentrations simply documented by experimental work. The second notable observation from the comparison of Figures 3 and 4 is that in Figure 3 there is a accumulation of complexed carrier V1 segments in the interfacial layers, while in Figure 4 the hydrophobic anion A3CA3 is accumulated instead. Since these two ions have opposite charge, we can expect conspicuous differences in the charge distribution curves. Figure 5 shows that there is a tremendous change indeed in the shape of these curves as the bulk aqueous KN concentration is changed from 1 to 0.001 M. At 1 M the organic side bears the excess positive charge of the double layer, while at 0.01 and
J. Phys. Chem., Vol. 100, No. 21, 1996 8951
Figure 6. Potential response “calibration” curve for system I.
0.001 M this side is negatively charged. At 0.1 M there is little charge accumulated at all near the interface. This picture is rather instructive and goes against intuitive considerations about ISEs. It is argued that the membrane with the cation-complexing carrier in it is always positively charged against water because of the preferential uptake of complexable hydrophilic cations compared to that of hydrophilic anions. Our model shows that the extent of hydrophobicity of the A3CA3 ions also play an important role in the charge separation. Electroanalytical chemists are first of all interested in the calibration properties at ISEs, i.e., in the relationship between the interfacial potential difference and the concentration of the ion to be measured, ion K in our notation. Ideal electrodes show so-called Nernstian behavior; i.e., the potential difference against the log of the concentration of K in the aqueous phase is a straight line with the slope RT (ln 10)/F (about 59 mV at room temperature). Deviations from this behavior have been experimentally observed at high and low analyte concentrations and in the presence of certain cations or anions.3 Our model allows the calculation of potentiometric calibration lines, and it can be used for the interpretation of deviations from ideality. Figure 6 shows a calibration line for the ion K. Nernstian behavior is observed between 0.001 and 0.1 M, but a negative deviation from the straight line is noted at 1 M. As noted above in relation to Figure 3, this deviation can be explained by considerable extraction of KN into the organic phase, which does not occur at lower KN concentrations. The effect of interference from various cations is shown in Figure 7. Here calibration lines are shown for systems II and III which differ from the previously discussed system I by the presence of another hydrophilic salt in the system: PS. Cation P forms much weaker complex with the carrier than does K, the respective stability constants being 10-5 and 105. In system II, the concentration of PS in the bulk of the aqueous phase is constant at 0.01 M and the concentration of KS is varied. In system III, KS is at 0.01 M and PS is varied. In qualitative agreement with the experimental facts, system II gives virtually the same calibration line as that of system I, i.e., PS does not interfere with the measurement of K. In system III, the calibration line is merely horizontal; in other words, the concentration of P cannot be measured in the presence of 0.01 M K. These results show that the calculations presented in this paper can be used to interpret or predict important analytical features of ISEs. Further calculations relating to the interference by hydrophobic anions have also been successful.18 It is also of great importance that there are new experimental approaches to measure the distribution profiles of different
8952 J. Phys. Chem., Vol. 100, No. 21, 1996
Vincze et al. that the sum of the probabilities that a segment is in a given internal state adds up to unity for each segment with more than one internal state. We can define an unconstrained function f by introducing Lagrange mulitpliers as
f ) ln
Ω
-
U - U*
Ω*
kBT
+∑
ni(µˆ i - µˆ i*)
+
kBT
i
γA(z)(NA(z) - ∑NAk(z)) ∑z β(z)(L - ∑i ∑c nicNic(z)) + ∑z ∑ A k (a.3) In eq 12 Ω*, U*, and µˆ i* are the degeneracy, the energy, and the chemical potential in a suitable reference system, Nci (z) is the number of segments of molecule i in conformation c in layer z, and NAk(z) is the number of segments of type A in state k in layer z. Now the equilibrium condition can be expressed as
Figure 7. Calibration curves for system II and system III.
(U - U*)
species at liquid-liquid interfaces, to which the model presented above can serve as a unique theoretical reference.19
∂ ∂f
Summary A new theory has been developed which enables the modeling of the interface of two immiscible electrolyte solutions. The unique feature of the model is that complexation equilibrium is also taken into account by introducing the multistate character of the molecule segments. Semimolecular information (density profiles, local composition, conformation of molecules, etc.) about the interface can be obtained together with the electrostatic potential and charge density profiles in the system, without any presumption about the interface. The advantage of this coursegrained SCF model is that relatively complex systems with many components can be modeled. This allows the study of ionselective liquid membranes based on carriers, systems widely used in the analytical practice. Acknowledgment. The authors are indebted for the PECO 1079 E6 project grant.
)
∂ln Ω/Ω*
∂nid
k BT
-
∂nid
+
∂nid
µˆ i - µˆ i* kBT
∑z β(z) Nid(z) ) 0
∀ nid ∈ {nic} (a.4)
and
U - U* kBT
∂
∂f ∂ln Ω/Ω* ) ∂NAk(z) ∂NAk(z) ∂NAk(z) γA(z) ) 0
∀ A, k, z (a.5)
The derivation of the terms ln Ω/Ω*, (U - U*)/kBT, and (µˆ i µˆ i*)/kBT and the differentiation of eqs a4 and a5 were performed following the guide lines given by Evers et al.12 including the multistate character of the system as done by Bjo˜rling et al.16 From the condition given by eq a5 we arrive at
Appendix Evers et al.12 have given the grand-canonical partition function for inhomogeneous systems in which all molecules have segments that could only have a single internal state. The expression of this partition function to allow for segments with multiple states is given by
Ξ({µˆ i},M,L,T) )
∑
∑
( )
Q({nic},{RAk(z)},M,L,T) exp
all{nci} all{RAk(z)}
ω ˆ Ak ω ˆ A0
-
1
∑χAB〈φB(z)〉 2B
Ω({nic},{RAk(z)},M,L,U{nic})
(a.1) RAk(z) )
B
( )
exp -
U{nic} kBT
-
kBT
νAky(z) - γA(z) ) 0 (a.6)
where
Q({nic},{RAk(z)},M,L,T) )
U ˆ Ak - U ˆ A0
We can eliminate the Lagrange multiplier with the help of the constraint ∑kRAk(z) ) 1, getting the following result
niµˆ i
∑i k T
-ln RAk(z) + ln
XAk(z)
∑k XAk(z)
(a.7)
where XAk(z) is given by
(a.2)
{RAk(z)} is a set of internal state distributions in the system, µˆ i is the chemical potential of molecule i as defined by eqs 7a and b, Q is the canonical partition function, and Ω is the degeneracy of the system. The optimization of the grand-canonical ensemble should take place under the constraint that the layers are exactly filled and
XAk(z) )
{
exp ln
ω ˆ Ak ω ˆ A0
-
1 2
∑χAB〈φB(z)〉 B
U ˆ Ak - U ˆ A0 kBT
}
- νAky(z)
(a.8) The differentiation given by eq a.5 is performed by realizing
Multistate SCF Theory
J. Phys. Chem., Vol. 100, No. 21, 1996 8953
( )
u′(z)/kBT, we can write eq a.12 in a simple form as
that
∂f ∂NAk(z) ∂NA(z) ) ∑∑ ∑ ) z A k ∂NAk(z) ∂NA(z) ∂nid ∂nid ∂f
uA(z) ) u′(z) - kBT ln ∂f
∑z ∑ ∑ A k ∂N
Ak(z)
d (z) RAk(z) NA,i
µˆ i - µˆ i* -ln + ln λid - ln Ni - 1 - ∑β(z)Nid(z) L kBT z U ˆ Ak - U ˆ A0 d NA,i (z)∑RAk(z) + ln RAk(z) + νAky(z) + ∑z ∑ kBT A k ω ˆ A0 φ* B,i ∑B χAB 〈φB(z)〉 - 2 + ln ωˆ ) 0 (a.9) Ak nid
(
[
]}
)
which can be rearranged in the form
nid
) Ciλic ∏ ∏{GA(z)} z
A
(
)
and GA(z) ) exp(-uA(z)/kBT) and
kBT
)
kBT
+
1
∑ ∑ A B
2
{ [
∑k
RAk(z)
φAbχABφBb
k
1
∑B χAB(〈φB(z)〉 - φBb) -
2
}
νAk(y(z) - yb) )∑
b RAk GAk(z)
(a.14)
k
where GAk(z) is the distribution function of the free segment A in state k at position z. When eqs 15, 16, a.7, a.8, a.13, and a.14 are combined, eq 11 can be rewritten as
GAk(z) GA(z)
(a.15)
References and Notes
µˆ i - µˆ i* φjb ln Ci ) -ln Ni + - 1 + Ni ∑ + kBT j Nj Ni ∑∑(φAb - φ*A,i)χAB(φBb - φ*B,i) 2 A B U ˆ Ak - U ω ˆ Ak ˆ A0 b b N R ln R + (a.11) + ln ∑ A,i∑ Ak Ak k BT ω ˆ A0 A k
β(z)
{
b GA(z) ) ∑RAk exp -u′(z) -
(a.10)
where Ci is defined as
uA(z)
(a.13)
k
b RAk(z) ) RAk
d NA,i(z)
L
∑XAkb
Using eqs a.6 and a.13, GA(z) can be expressed as
The result is
{
∑k XAk(z)
+∑ i
φib
+
Ni
U ˆ Ak - U ˆ A0
+ ln RAk(z) + νAky(z) + kBT ω ˆ A0 U ˆ Ak - U ˆ A0 ∑B χAB〈φB(z)〉 + ln ωˆ - RAkb k T + ln RAkb + Ak B ω ˆ A0 b χ φ + ln (a.12) ∑B AB B ωˆ Ak
] [
]}
Using eqs a.6 and a.7, and combining the first three terms into
(1) Amstrong, R. D.; Horvai, G. Electrochim. Acta 1990, 35, (1), 1-7. (2) Koryta, J. Sel. Electrode ReV. 1991, 13, 133-158. (3) Rakhmanko, E. M.; Yegorov, V. V.; Gubevich, A. L.; Lushchik, Ya. F. Sel. Electrode ReV. 1991, 13, 5-111. (4) Kakiuchi, T.; Senda, M. Bull. Chem. Soc. Jpn. 1983, 56, 13221326. (5) Benjamin, J. Science 1993, 261, 1558-1560. (6) Amstrong, R. D.; Proud, W. G.; Todd, M. Electrochim. Acta 1989, 34 (7), 977-979. (7) Stafiej, J. J. Electroanal. Chem. 1993, 351, 1-27. (8) Morf, W. E.; Ammann, D.; Pretsch, E.; Simon, W. Pure Appl. Chem. 1973, 36, 241. (9) Eugster, R.; Spichiger, U. E.; Simon, W. Anal. Chem. 1993, 65, 689-695. (10) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619. (11) Leermakers, F. A. M.; Scheutjens, J. M. H. M.; Lyklema, J. Biophys. Chem. 1983, 18, 353-360. (12) Evers, O. A.; Scheutjens, J. M. H. M.; Fleer, G. J. Macromolecules 1990, 23, 5221-5233. (13) Barneveld, P. Ph.D. Thesis, Wageningen, 1991. (14) Vincze, A.; Horvai, G.; Leermakers, F. A. M.; Scheutjens, J. M. H. M. Sen. Actuators B 1994, 18-19, 42-46. (15) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1971. (16) Linse, P.; Bjo¨rling, M. Macromolecules 1991, 24, 6700-6711. (17) Nieman, A. T.; Horvai, G. Anal. Chim. Acta 1985, 170, 359-363. (18) Vincze, A.; Horvai, G.; Leermakers, F. A. M. Electroanalysis 1995, 7, 877. (19) Thoda, K.; Umezawa, Y.; Yoshiyagawa, S.; Hashimoto, S.; Kawasaki, M. Anal. Chem. 1995, 67, 570-577.
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