J. Phys. Chem. 1993,97, 51365140
5136
Transport of Cryptates as Model Brownons: Electrical Mobilities and Self-Diffusion Coefficients of Monovalent and Divalent Ions Cryptated by 222 in Aqueous Solutions Sophie Rossy-DeUuc,t Thierry Carhiller,+Pierre Turq,’*t Olivier Bernard,* Nicole Morel-Desrosiers,s Jean-Pierre Morel,$ and Werner Kunzl Laboratoire d’Electrochimie URA 430, Universitt Pierre et Marie Curie, 8 Rue Cuvier, 75252 Paris cedex 05, France, Laboratoire de Physicochimie Thtorique URA 503, Universitt Bordeaux I, 351 cows de la Libtration, 33405 Talence cedex, France, Laboratoire de Chimie Physique des Solutions URA 434, Universitt Blaise Pascal, 631 77 Aubi2re cedex, France, and Laboratoire a o n Brillouin, C.E. Saclay, 91 191 Gif-sur- Yvette cedex, France Received: September 22, 1992; In Final Form: December 29, 1992
Cryptates present a quasi-spherical symmetry and a practically constant size for a charge varying from 0 (uncharged empty cryptand) to 1 or 2 (cryptand containing a monovalent or divalent cation). They are therefore sensible models of heavy and large particles undergoing Brownian motion (brownons). Their transport properties in aqueous solutions, electrophoretic mobilities, and self-diffusion coefficients are in good agreement with this Brownian dynamical model.
K = 5.4 and 9.5, respectively). However, we have chosen to 1. Introduction study the transport properties of Na+222 and Sr2+222because The cryptands, first synthesized by Lehn and co-workers,form radioactive sodium and strontium ions are long-period isotopes very stable 1-1 inclusion complexes, called cryptates, with some and easier to handle, j3-emitters and y-emitters being required alkali-metal and alkaline-earth-metal cations.l*2The cryptation for the electrophoresis and the self-diffusion measurements, process is quite specific, the strongest complexes being formed respectively, according to the custom way of using radiotracers with the ions that best fit the internal cavity of the l i g a ~ ~ d . ~ ? in ~ one of our laboratories. In fact, although Na+ and Sr2+are Another interesting feature of the cryptation process is that it slightly smaller than the internal cavity of 222, they form with transforms a metal cation into some sort of large almost spherical this ligand relatively strong complexes in water (log K = 3.9 and organometallic ion soluble in a variety of solvents. Furthermore, 8.0, respectively). it is possible to vary the charge of the complexwithout significantly The determination of the ionic mobilities and self-diffusion modifying its size, provided the included cation fits the cavity coefficients of cryptates is made in the presence of an excess of well. Because of their large diameter and heavy mass, cryptates supporting electrolyte, the cryptated labeled ion being used as a can thus constitute ideal models of spherical particles undergoing tracer in this medium. From the theoretical laws the data are Brownian motion (model brownons). extrapolated to infinite dilution. The Nernst-Einstein relation The transport properties of charged species in solution give gives control upon the consistency of those extrapolated values, useful information about the ions’ specific properties. For D? and up, which are both related to the friction coefficient {io instance, the apparent charge Z,of an ion can be deduced from by its diffusion coefficient D, and from its electricmobility u1through the Nernst-Einstein relation -D l = -
kBT
u,
Z,e
(1)
where kBT is the Boltzmann factor. At infinite dilution, this equation relates the diffusion coefficient and the electricalmobility to the charge number of the ion. But to interpret the apparent charge of an ion at a finite concentration, it is necessary to take into account the ion-ion interactions that affect both the selfdiffusion coefficient and the ionic mobility. Theoretical models for the treatment of those interactions have been recently developed for simple ele~trolytes.5+~ We present here the first measurements of self-diffusion coefficients and electrical mobilities of two cryptates, Na+222 and Sr2+222, in aqueous solution. The cryptand 222 has been chosen because its complexation with various cations has been largely st~died,33’-~because it is commercially available at a reasonable price, and also because of its almost spherical shape. Among the alkali-metal and alkaline-earth-metal cations, the 222 forms the strongest complexeswith K+(represented in Figure 1, using crystallographic data from Ward et a1.10 and Ba2+ (log
* To whom correspondence should be addressed.
’ Universite Pierre et Marie Curie. 1 UniversitC 9 Universite
Bordeaux I. Blaise Pascal. 1 C.E. Saclay.
0022-365419312097-5136$04.00/0
where 7 is the viscosity of the solvent and Ri the hydrodynamic radius of the complex. The experimental data are analyzed in the framework of the new theory of ionic transport,5s6 extendcd here to tracer ions. The results of this analysis are expressed in terms of cryptate effective radii. The influence of the charge and the hydrodynamic conditions (sticking or slipping) can be determined from the comparison of these radii with the structural values.
2. Experimental Section 2.1. Materials. Cryptand 4,7,13,16,21 ,U-hexaoxa- 1,lOdiazabicyclo[8.8.8]hexacosane (Merck; Kryptofii 222), referred to by the abbreviation 222 throughout the text, was kept in a desiccator and used without further purification. NaCl (Merck; suprapur), SrC12.6H20 (Merck; pro analysi), Et4NBr (Fluka; puriss), and Et4NOH (Fluka; purum 40% in water) were used as received. 2*NaC1(/3-and y-emitter), 89SrC12(@-emitter),Na36C1 (&emitter), and [14C]glucose (&emitter), all in aqueous solutions, were purchased from Amersham International. 85SrC12 (y-emitter), in 5.5 mol L-l HCl aqueous solution, was from Dupont de Nemours. Ultrapure water (resistivity of 18 0 1993 American Chemical Society
The Journal of Physical Chemistry, Vol. 97, No. 19, 1993 5137
Transport of Cryptates as Model Brownons
adsorption of the cryptates. Their diameter is about 1 mm and their length is 3 cm. The capillaries containing active solutions are placed in a Teflon basket and immersed in lo0 mL of inactive solution. The basket is slowly rotated (1 rotation/4 min) in order to avoid the accumulation of radioactive substance on the open end of the capillary. The experiments are performed at 298.1 f 0.1 K. The activities are measured at the beginning and at the end of the experiment using a PACKARD Autogamma 800C ycounter with a sodium iodide, doped with thallium, crystal well. The ycounter is optically coupled to a photomultiplier.13
3. Theoretical Section We present here a specialized version of our transport theory, based on the introduction of MSA equilibrium pair distribution functions into the Onsager continuity equation:5.6J4
+
k B T ( q aj)Ahj; + (e,uiAqi - eJujAqi')( Z p j - ZJuj)-Vhj: = 0 (4) Figure 1. Monovalent cryptate structure (K+222).
MQcm) was obtained by reverseosmosis and filtration (Millipore Milli-Q coupled to Milli-RO). The complexes were prepared in water containing Et4NOH (pH > 12), to avoid protonation of the ligand, and various concentrations of the supporting electrolyte Et4NBr,speciesthat were chosen because the cation Et4N+is too large to penetrate inside the ligand cavity. In order to neglect the adsorption on the capillary, a small amount (10-4 mol L-I) of inactive salt (NaCl or SrC12) was also added. The complexes were prepared by adding 10 pL of active salt mother solution to 3 mL (for the electrophoresis) or 4 mL (for the self-diffusion) of a solution of 222 in the above mixture. In all cases, the cryptand concentration was 3.0 X 10-3 mol L-I and the tracer final concentration was about 10-8 mol L-I. In such conditions, it can be assumed that the tracer ion is totally complexed. The glucose solutions used to evaluate the electroosmotic contribution were prepared in the same way. 2.2. Electropboresis. Electrophoretic mobilities can be followed by substituting an atom of the ligand by its radioisotope or by complexation of a labeled ion, provided the association is total. Since the former method is not easy with a ligand such as 222 we have used the latter one. Our electrophoretic apparatus is a LKB Multiphor thermostated at 5 OC in order to minimize the evaporation. The measurement of the electrophoreticmobility is done using a porous supporting material' 1 ~ 1impregnated 2 with the electrolyticsolution. We use a millipore HAHY 304FO membrane (pore diameter of ca. 0.45 pm); it is checked by chromatography that the ions do not adsorb on the support. The initial deposit must be as thin as possible. The migration is determined after application of an electric field (25 V cm-I) at the extremities of the support during a time of 20-30 min. After drying the strips, in order to stop the diffusion, the position of the deposit is determined using a BERTHOLD &meter TLC linear analyzer LB283 coupled to a microcomputer. The reference ion is Wl-. The electroosmotic contribution is evaluated by measuring the displacement of labeled [ 14C]glucose. If we denoteby dj,dcl,and d,l the electrophoreticdisplacements of the studied ion i, of C1-, and of glucose, respectively, we can express the mobility of i as follows:"
(3) 2.3. Self-Diffusion. We use the open-end capillary method to measure the diffusion coefficients of the sodium and strontium cryptates. The tubes are made of silica in order to minimize the
We only take into account the terms in the continuity equation which only give the limiting law in the relaxation effect.15 In these expressions,relations are needed between the potentials qjland the distribution functions hjj' for nonequilibrium. The equilibrium part of the distribution functions hjp is taken from the MSA. For the nonequilibrium part (linear response to perturbing external force), we use the Poisson-Boltzmann closure relation between the potentials and the distribution functions. This closure relation is satisfying for the Coulombic part of the interactions, and the limitations induced from the hard sphere parts are not severe since in this case the size of the particles appears mainly as a short-range boundary condition for the electrostatic interactions.5~6 3.1. Self-Diffusion of a Tracer in an Electrolyte Solution. In this case, only the relaxation effect 6kl/kl is present for selfdiffusion. We get for the self-diffusion coefficient of the tracer in an electrolyte solution5J6J7
where
In expression 6, oj = DP/keT is the absolute mobility of the particle i and
(7)
hiiO(r) = giiO(r) - 1 where gUo(r) are the equilibrium pair distribution functions, calculated in the mean spherical approximation (MSA) with hard sphere and Coulombic potentials. uii is a distance parameter which is, in our case, the sum of the diameter of ions i (ai) and j (ai):
c is given by
where Q is the relative dielectric constant of the solvent. The integral in eq 6 is evaluated by means of the Laplace transformation G j l ( S ) of rgj,(r):
Gjl(s) = J"gilo(r)e-srdr For small and moderate concentrations, the hard sphere part of
Rossy-Delluc et al.
5138 The Journal of Physical Chemistry, Vol. 97, No. 19, 1993
the Laplace transformation of rgj10can be neglected, and hence only the electrostatic part is taken into account. For this electrostatic part, we used an approximate MSA expression given by Blum:I*
correlation functions. This yields
The last term vanishes in the limit of equal dynamic diameters and is always negligible at moderate concentrations. 3.2.2. Relaxation Term. As in the case of the self-diffusion calculation, the relaxation field can be determined starting from the continuity equation. From ref 6 the relaxation field acting on the supporting electrolyte is given by
with
where
and
and
ak H
k' -2 r 1 + ru, Cy2
3.2. Mobility of a Tracerin mElectrolytesolution. By contrast with self-diffusion, all the ions and not only the tracer ions, are affected by the force of an external electric field. This has several consequences: The relaxation field of the tracer ions generally depends also on the relaxation field of the salt. The movements of the ions of the supporting electrolyte decrease the velocity of the tracer ions. This must be taken into account in the calculation of the tracer mobility (electrophoreticcorrection). Thiscorrection is similar to the one made in the calculation of the conductivity of a salt solution. Taking these effects together, the velocity of tracer ion 1 is given by
8, = 8,( 1
+
2)+
68,S'( 1 +
2)
DloZ, DloelE
k,T = k,T
where 3 is the electric field. 3.2.1. ElectrophoreticCorrection. Electrophoretic contributions arise from the hydrodynamic interactionslg*20 of the ions in solution. The first-order term is given by
where
4relE 3 (elwl -ep,) sinh Kd, Ujl bk, = 3e bp,e'kBT(UI 0,) Kd,'jl X
+
+
KqZelwl(P2e2~2+ Pse303)
~u~rhjloe-Kdldr (K:
X
- Kd,*)kBT(Wl + WZ)(WI + 03)
(16)
where 6bIe1is the electrophoretic correction of this velocity, 6kl is the relaxation force acting on the tracer, 6k, is the relaxation force acting on the supporting electrolyte, and 810is the tracer velocity at infinite dilution given by
d,O =
In the same way we get for the relaxation force acting on the tracer
is the Oseen tensor defined by
where q is the solvent viscosity and I is the unity tensor. By substitutingeq 19into eq 18 we get after angular integration
The integral can be analytically evaluated by using the MSA
(25) The intqrals can be calculated as before. Since kl is given by = elE, the expression 8kl/kl can immediately be obtained.
4. Results and Discussion To intrepret our experimental results, we use the theoretical expressions given in the above section. In fact we adjust the diffusion coefficient or the electrophoretic mobility at infinite dilution in order to reproduce the experimental data. From the diffusion coefficient at infinite dilution we calculate a hydrodynamic radius by means of expression 2. Sodium Cryptate. The self-diffusion coefficient of 22Na+, determined in the presence of an excess of cryptand 222 for varying concentrationsof supportingelectrolyte, has been plotted in Figure 2. It varies very slightly with the square root of Et4NBr concentration and shows a linear dependence throughout the concentration range examined. Extrapolation to infinite dilution leads to DO = (0.56 f 0.01) X cm2s-I. Substitution of this value into eq 2 gives a cryptate radius of 4.3 f 0.1 A for perfect sticking and 6.5 f 0.1 A for perfect slipping. The former one is in better agreement with the radius value of ca. 5 A estimated from the molecular model,21 with the Stokes radius of 4.9 A estimated for K+222 in THF from conductivity studits2*and with the radius value of 4.6 A found for K+222and Ba2+222from
The Journal of Physical Chemistry, Vol. 97, No. 19, 1993
Transport of Cryptates as Model Brownons
".' I 0.6
I
!
0
1
@.------e---
$
0.5-
"
I 0.4
1
0.34 0.0
0.3
'
'
0.1
0.3
0.2
0.4
0.5
I
0.6
.qW)
Figure 2. Self-diffusion coefficient D (cm2SKI)of sodium cryptate versus the square root of the supporting electrolyte concentration (mol L-I): 0 , experimental values; broken line, limiting law; full line, our calculations cm2 s-l and r = 4.3 0.1 A). (00 = (0.56 f 0.01) X
*
0.3
I
0.0
0.3
0.2
0.1
0.4
0.5
0.6
.qW) Figure 4. Self-diffusion coefficient D (cm2 s-I) of strontium cryptate versus the square root of the supporting electrolyte concentration (mol L-I): 0 , experimental values; broken line, limiting law; full line, our cm2 s-I and r = 4.4 f 0.1 A). calculations (00 = (0.55 f 0.01) X
1 \
0.1 4 0.0
0.1
0.2
0.3
0.4
I
0.5
0.2
0.0
0.1
0.3
0.2
0.4
0.5
ww)
WW)
sodium cryptate relative to that of Cl-, UCI, versus the square root of the supporting electrolyte concentration (mol L-I): 0 , experimental values; broken line, limiting law; full line, our calculations (00 = (0.56 f 0.01) X 10-5 cm2 s-I and r = 4.3 f 0.1 A).
Figure 5. Mobility usr of strontium cryptate relative to that of CI-,ucl,
the standard molal volumes measured in water and combined to the volumes of the cavity calculated by using the scaled-particle theory.23 This indicates that sodium cryptate seems to diffuse chiefly under sticking conditions. The electrophoretic mobility of sodium cryptate, with respect to C1-, has been plotted in Figure 3. It shows a quasi-linear variation with the square root of Et4NBr concentration. Extrapolation to infinite dilution leads, in diffusion units, to DO = (0.56 f 0.01) X 1t5cm2s-' and, from eq 2, to a cryptate radius of 4.3 f 0.1 A for perfect sticking. The DO value deduced from the electrophoretic mobilities is in excellent agreement with that found by extrapolating the self-diffusion coefficients. This gives strong support to the use of the Nernst-Einstein relation for the understanding of the transport properties of these electrolytes. Strontium Cryptate. The self-diffusion coefficient measured in the presence of an excess of cryptand 222, for varying concentrations of supporting electrolyte, is less accurate for 85Sr2+ than for 22Na+. It has been plotted versus cII2in Figure 4. It appears that it does not vary linearly through the whole range of concentration examined. The extrapolation to infinite dilution requires the full theoretical expressions given in the previous section. It leads to DO = (0.55 f 0.01) X le5 cm2 s-l which corresponds to a cryptate radius of 4.4 f 0.1 A, a value comparable to that found for Na+222. The electrophoretic mobility of strontium cryptate, with respect to C1-, also shows a slight curvature with the square root of the concentration of the supporting electrolyte (Figure 5 ) . Extrapolation of these values by using the theoretical expressions given above leads, in diffusion units, to DO = (0.54 & 0.01) X l t 5 cm2
s-I. This corresponds to a cryptate radius of 4.5 f 0.1 A for perfect sticking. Again, both theelectrophoreticand self-diffusion methods lead to very close results. It must be underlined that the self-diffusion coefficients determined here by two different methods and the radius values calculated for Na+222 and Sr2+222are very similar for both cryptates.
Figure 3. Mobility
U N of ~
versus the square root of the supporting electrolyte concentration (mol L-I): 0 , experimental values; broken line, limiting law; full line, our calculations (00= (0.54 0.01) X 1 t 5 cm2 s-I and r = 4.5 f 0.1 A).
*
5. Conclusion Cryptates, because of their size, quasi-sphericity, and solvation properties, are good candidates for model brownons. The charge effect between monovalent and divalent cations affects only the strength of the ionic interactions, giving a sharper variation of the transport coefficients with the concentration of the supporting electrolyte for the divalent cryptate. However, the intrinsic part of the friction coefficient of a cryptate is strongly related to its size. The hydrodynamic radius calculated here is not only, as required, independent of the transport coefficient used for its determination, but appears to be also independent of the charge of the cryptated cation. This shows that the dynamic characteristics of the hydration of these monovalent and divalent species are not fundamentally different. Obviously the radii determined from transport coefficients do not compete with crystallographic data for precise structural information. They are only hydrodynamic quantities which are related to the bulk size of the cryptated ion and do not give information on the internal structure of the cryptated ion. The new theory, some of us have developed recently for the variation of the transport coefficients with the concentration of
5140 The Journal of Physical Chemistry, Vol. 97, No. 19, 1993 a supporting electrolyte, has been adapted here to the case of tracer ions. It appears that this extension is required to interpret the variation of the transport coefficients of the divalent cryptate, whereas the limiting laws seem to contain the essential of the phenomenon observed with the monovalent cryptate, even at relatively high concentration of the supporting electrolyte. An interestingfeature of cryptates is their solubilityin a variety of solvents. Model brownons of this type can thus be studied in various polar and nonpolar media. For instance, small-angle neutron scattering measurements have just been reported for solutions of 222, K+222, and Ba2+222in acetonitrile, measurements from which solute-solute correlation functions have been inferred.24
Acknowledgment. The authors would like to express their gratitude to Nathalie Prulibre for her technical assistance.
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