J. CHEM. SOC. FARADAY TRANS., 1994, 90(6), 875-878 Forces of Inertia acting on the Aqueous Pore Fluid of Anionic Polyelectrolyte Gels Ngoc-Ty Dang and D. Woermann Institute of Physical Chemistry, University of Cologne, Luxemburger Strape 116, 0-50939Koln, Ger- many Pulses of an inertia electromotive force are generated during deceleration of stiff rods of polyelectrolyte gels with negatively charged ionic groups covalently bound to the matrix of the gel. The gels are loaded with Lit, Cs', Ag+, H' and Ba2+, respectively, and are in swelling equilibrium with water. The values of (m, mass and q, charge of one counterion in the pore field) calculated from the recorded voltage pulse U dt (U, inertia electromotive force and t, time) during deceleration (time constant T z 1 ms) are larger by a factor of at least five than that calculated from the mass and charge of a 'naked' single counterion.This is attributed to an effective mass of the counterions in the pore fluid of the gels which has a larger value than the mass of a naked single counterion. During deceleration the hydrated counterions, together with the mobile water molecules in the pore fluid, are shifted relative to the ionic groups fixed to the stiff matrix of the gels. This generates an electric field causing the observed voltage pulse. The value of of the H' counterion is smaller by a factor of about five than that of the other counterion species (Lit, Cs+, Ag'). It is assumed that this is caused by the chain mechanism of proton migration found in aqueous solutions.Measurements of the electrical conduc- tivity of gels loaded with different counterion species, including H+ ions, reveal that the ratios of the counterion conductivity in the gel phase to that in free solution at infinite dilution have approximately the same value. The inertia electromotive force and the electrical conductivity measurements indicate that the mechanism of H' ion transport in the pore fluid is not modified considerably by the matrix of the gel. In a recent publication' it was demonstrated that pulses of an inertia electromotive force can be generated during the decel- eration of stiff rods of cation-exchange gels loaded with silver ions. The gels were in swelling equilibrium with water. The value of the ratio (rn/q)expcalculated from the experimentally determined voltage pulse U dt [U, inertia-electromotive force, electrical potential difference measured at zero electric current flow with two reversible electrodes (e.g.Ag wires reversible to Ag+ ions) between the ends of a gel rod during deceleration; t time]. The axis of the rod parallel to the vector of deceleration was larger by a factor of four for the SO, gels (larger by a factor of 20 for the -CH,-SO, gels! than the value found by Betsch, Rickert and Wagner' in cor- responding experiments with solid ionic conductors (e.g RbAg,I,). The duration of the voltage pulse is of the order of 1 ms. A detailed description of the physical concept to inter- pret the inertia electromotive force observed in ionic conduc- tors is given in ref.2(a). This concept was adapted to polyelectrolyte gels : During deceleration the force of inertia acts on the matrix of the gel and its aqueous pore fluid in which the counterions (q.Ag+) are dissolved. It is assumed that the matrix of the gels is so stiff that it can follow the force of inertia instantaneously. This is not true for the pore fluid. It is mobile and during deceleration is shifted slightly relative to the matrix in the direction opposite to the vector of deceleration. The Ag+ counterions move with the pore fluid. The slight shift of the pore fluid with the counterions generates an electric field acting along the axis of the gel rod. Its change with time is reflected by the observed voltage pulse. It can be assumed that at each instant of time during deceleration the electric field in the gel phase is just sufficient to decelerate the Ag+ ions at the same rate as the matrix.The time constant, T, for establishment of this stationary state is estimated to be ca. s. [T = rn/(6nqr);rn, r, mass and radius of a single ion, q, viscosity of the pore fluid]. Results of experiments are reported in which forces of inertia act on the aqueous pore fluid between highly cross- linked cation-exchange rods loaded with a mixture of two counterion species ([Li+/Ag+], [Cs+/Ag'], [Ba2 '/Agf] and [H' Ag']). The experiments are carried out as a function of the mole fraction, ZAg+, of the Ag' counterions in the gel in the range 0.05 < tAg+< 1.The presence of Ag+ ion in the gel phase is necessary to be able to use silver wires as reversible electrodes to monitor the electric voltage pulse of the inertia electromotive force. Experimental Polyelectrolyte Gels Two types of cation-exchange gels (type 1, abbreviated SO; and type 2, abbreviated CH,-SO,) in the form of rods, length, 50 mm, radius, 2.5 mm were used. The methods of preparation are given in ref. 3 and 4, respectively. They were loaded with two counterion species ([Ag+/H"], [Ag'/Li'], [Ag+/Cs'], [Ag'/Ba2']) by treatment with aqueous solu- tions containing the nitrates of the corresponding cations and washed free of electrolyte by extended treatment with distilled water. The mole fraction tiof H', Li', Cs' and Ba" in the gel phase was determined analytically C0.05 < ,ti < 1; ii= fii/(n", + fiAg+);fii, amount of substance of ionic species i in the membrane phase].The methods of characterization of the gels are given in ref. 5. The results are given in Table 1. The two gels differ mainly in their water content and fixed ion concentration. The specific electrical conductivity, i;-( =l(Z/A$)c= o, Ap = (i, electric current density passing through the rod; A4, electric potential difference; I, distance between the Ag electrodes in contact with the gel rod) of the gel phase in the absence of co-ions was measured to obtain information about the elec- trical mobility of the counterion species within the gels. rZ-was measured as described in ref.5. The experiments were carried out under the condition cjX < 1 (c,electrolyte concentration; X, fixed ion concentration). The specific electrical conductivi- ty i;-of a gel containing counterions only is related to the molar ion conductivity of the counterions im, by the relation i;-= lm,counter Zcounter-+ F2X2d;,(F, Faraday constant; X, fixed ion concentration ;d, ,hydrodynamic permeability of 8 76 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 1 Characterization of the gels gel counterion fiH2O) Y*(H,O) xll0-j mol (~m-~ pore fluid) cm5 J-' s-' so; Li + H+ 0.573 1 0.5660 0.5768 0.5665 1.56 1.58 2.53 2.65 CH,-SO; cs + Ag + Ba2+ Li + H+ 0.5155 0.4642 0.5068 0.77 16 0.77 13 0.5853 0.5139 0.5780 0.78 15 0.7833 1.74 1.93 1.77 0.427 0.430 2.09 1.32 1.68 9.15 9.07 cs+ 0.7304 0.7643 0.457 6.63 Ag+Ba2+ 0.7106 0.7270 0.7372 0.7619 0.470 0.459 6.05 6.52 yHz0, ygz0, mass fraction of water taken up by the gel; yHzO= m,,dm; m = mge, + mHzo+ mcouncer;ygzo = mHzd(mge,+ mHzo);X,analytically determined fixed ion concentration; J,,, specific mechanical permeability; & = L(jv/AP)A4=o,k=o;L, length of the rod;jv, volume flow density; AP, applied hydrostatic pressure difference.the gel matrix; d;, = 4j,/AP),4=0,dE=o ; j,, volume flow density; AP, hydrostatic press_ure difference; L, length of the rod).6 The term Econ= F2X2d,,describes the convection con- ductivity caused by an electro-osmotic volume flow. The con- tribution of the electric convection conductivity to the total conductivity (kco,,/k) is estimated to be <0.10 for the CH,SOJ gel and to be <0.15 for the SO, gel.For a gel loaded with H+ ions the ratio (kco,,/k) is smaller than 0.04. The results of the conductivity measurements are given in Table 2. Measurement of Inertia Electromotive Force A rod of the swollen gel, free from adherent water, was intro- duced into a Plexiglas tube and was sealed airtight with a screw cap. The position of the rod was fixed by two Plexiglas screws pressing against the top and the bottom of the rod. Two silver wires were introduced through the wall of the Plexiglas tube and connected to silver electrodes. The elec- trodes were tightly pressed against the gel rod by Plexiglas screws at a distance, I = 40 mm, symmetrical to the centre of the rod (L/2).They acted as reversible electrodes for the Ag+counterions present in the gel.The electrodes were con- nected to an amplifier and digital memory-scope to record the voltage pulse, 5 U dt, during deceleration. During decel- eration the axis of the rod is parallel to the vector of deceler- ation. The Plexiglas tube containing the gel rod was placed into a cavity in a cylindrical aluminium block. The alu- minium block was earthed, thereby acting as an electric shield. For the experiment the aluminium block was dropped Table 2 counterion Ag + H+ Li + Na+ K+ cs+ Ba2+ H /Ag+ + Electrical conductivity, k, of the cation exchange rods zi so; CH,-SO; 1 1.16 f0.02 0.65 f0.02 1 8.89 f0.25 5.24 f0.05 1 0.99 &-0.02 0.65 f0.02 1 1.58 f0.02 0.95 f0.02 1 2.06 f0.02 1.21 f0.03 1 1.71 & 0.06 1.00 & 0.04 1 0.65 f0.03 0.45 & 0.02 1,+ SO; CH,-SO; 0.10 0.16 7.70 f0.30 4.56 f0.08 0.40 0.37 5.93 f0.14 3.49 f0.04 0.62 0.60 3.59 & 0.10 2.14 &-0.05 0.9 1 0.78 1.99 f0.05 0.78 f0.04 The data refer to gels washed free of electrolyte. T x: 25 "C.2i,molt fraction of counterion species i in the gel phase. onto a plastic support (for further details see ref. 1). The velocity, urnax,of the aluminium block before hitting the bottom plate is given by u,,, = (2~h)"~(9,standard, acceler- ation of free fall; h, drop height; 5 < h/cm < 30; 100 < uma,/cm s -< 250).Results and Discussion The deceleration of a gel rod from a velocity, urnax,to the velocity, u = 0, generates an electric voltage pulse measured by the two silver electrodes in contact with the rod. Accord- ing to ref. 2 this voltage pulse is given by: after strike u dt = (4q)exp lumax (1)before strike U = {&Ag, x = 0) -&(Ag, x = I)) > 0; x, coordinate along the axis of the rod; x = 0, L, coordinate of the lower and upper end of the rod in vertical position; L, length of the rod, 50 mm; I = 36 mm. The value of 5 U dt is obtained by planimetry o----:U(t) curve. The values of (m/q)exp are calculated from the slope of the straight line of the 5 U dt us. u,,, plot. The slope of the 1U dt us.urnaxline [i.e. the value of (n~/q)~~~]depends slightly (i.e. less than +5%) on the pressure applied between the top and the bottom of the gel rod to fix its position in the Plexi- glas tube used for the drop experiments (see Fig. 1). A similar 1 I I 1 15 c Iv) c l a a m 10 0 F1: n v$5 +-+ --+----------i I 1 I I0 0 200 400 600 A4m Fig. 1 Effect of the mechanical force (measured as AL, the change in length of the rod) on the ratio (m/q)expobtained with a gel rod loaded with Ag+ ions. Lo = 50 mm. (a),SO; gel and (m), CH,SO; gel. The Young's modulus of the CH,-SO; gel is smaller than that of the SO, gel. J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 small change in the value of the slope is introduced by reversing the position of the rods in the Plexiglas tube.Care has to be taken to adjust properly the force with which the two silver electrodes are pressed against the surface of the gel rod by Plexiglas screws. Experimentally determined values of (rn/q)expobtained with rods of the SO, and the CH,-SO, gel containing only counterions and no co-ions are shown in Fig. 2. The data are plotted as function of the mole fraction RA,+[ =fiAg+/(iil+ fiAg+)] of the Ag' counterions in the gel phase of the ion pairs [Li+/Ag+], [Cs'/Ag'], [Ba2'/Ag'] and [H+/Ag']. The data show that: The values of the experimentally determined ratio (m/q)expare larger by at least a factor of five than that calculated from the mass and the charge of the naked counterions [(m/q)c,lc/10-3 g A-' s-': H'.1.04 x lo-,; Li', 7.18 x loV2;Cs', 1.38; Ag+, 1.12; Ba2+, 0.71; rn, mass of the naked counterion] ; it is independent of zA,+to a first approximation. The value of (rn/q)expof Ba" ions is smaller, by a factor of two, because each barium ion carries two elementary charges. The values of (rn/q)expobtained with the CH,-SO, gel are larger by a factor of ca. two than that of the SO; gel.? The ratio (m/q)exphas approximately the same value for the counterion species Li', Cs' and Ag' (SO; gel: mean value (rn/q)exp = 5 x lop3 g A-' s-'; CH,-SO, gel: mean value (m/q)expz 11 x lop3 g A-' s-'). For RH+ > 0.5 the value of (m/4)exp for the counterion species Hf is smaller by a factor of ca.five (SO; gel: mean value (rn/q)expx 1 x g A-' s-'; CH,-SO; gel: mean value(m/q),,, x 2 x g A-' s-'). These findings are interpreted by assuming that the hydrated counterions dissolved in the pore fluid of the geis are distributed homogeneously over the cross-section of the pores by the thermal motion of the counterions.'*6 An electri- cal space charge density ?, is generated having a sign opposite to that of the fixed ionic groups (?, = -oFX; o,sign of the fixed ionic groups; w = -1 for gel with cation exchange properties). During deceleration the mobile hydrated counter- ions and the water molecules in the pore fluid are moved relative to the fixed charges bound covalently to the matrix. This process generates an electric field which causes the hydrated counterions to be decelerated at the same rate as the matrix.Therefore, the mass which determines the value of (m/q)expis larger than that of the naked counterion. rn is con- sidered as an effective mass to which the counterions and a certain number of water molecules contribute. With this assumption the data obtained with the systems [Li'jAg'] and [Cs+/Ag+] can be understood. Values of the effective mass meff calculated from the ratio (rn/q)exp obtained from the experiments with the systems [Ag'], [Li+/Ag+] and [Cs+/Ag+] are given in Table 3 using a mean value of (m/4)exp = 5 x lop3g A-' s-l for the SO, gel and a mean value of (rn/q)exp= 11 x 10-g A-' s -' for the CH,-SO; gel. The contribution of the mass of the naked counterions to the effective mass meff can be neglected to a first approximation.This reflects the finding that the value of (m/q)expis independent of the nature of the counter- ion species. Values of the ratio of the number of water mol- ecules NHzOto the number of mobile counterions NcOunterin the pore fluid [i.e. (NHzO/Ncounter)exp]calculated from meff are also given in Table 3. The values of (NHzO/Ncoun~er)expcan be 6- I 45-c I -I-* I I--_I I EJ, m *I 04- F\ 2 F-2-:3 h - A A--A - ,/ ' /,i'/ _- I 1 1 I 14 l2 I I . m10 4 // 8)-/ 01 / / I /'.4' ' 0' I I 1 1 1 1 0 0.2 0.4 0.6 0.8 1 RA, f Fig. 2 Experimentally determined values of (m/q)expobtained with rods of (a) the SO; gel and (b) the CH,-SO; gel loaded with a pair of counterions: (m), [Li+/Ag']; (+), [Cs+/Ag+], (A),[Ba2+/Ag'] and (@), [H+/Ag+]).The gels contain no co-ions. TAg+is the mole fraction of Ag+ counterions [TAg-= fiA8-,'(fii + fiAgt)]. tration and the water content of the gels [see Table 3; y$,o = mH20/(mHz0+ rnge,)]. It turns out that CQ. 70% of the water molecules taken up by the SO, gel contribute to the rneff of the mobile counterions. The corresponding value for the CH,-SO; gel is ca. 50%. It can be expected that the water molecules present in the gel do not contribute to meff. A fraction of them will adhere to the hydrophilic pore wall (boundary layer). They can be considered immobile on the timescale of deceleration.The data obtained with the counterion pair [H'/Ag+] for RH+ > 0.5 indicate that the H+ counterions behave differ- ently, during deceleration, to the Li+, Cs' and Ag+ counter- ions. For both kinds of gels the ratio (rn/q)exp has values smaller by a factor of ca. five (see Fig. 2). The number of water molecules contributing to the effective mass of a counterion is smaller by a factor of ca. 10. It is assumed that compared with the corresponding values (NH20/Ncounter)analthis effect is caused by the chain mechanism of proton migra- calculated from the analytically determined fixed ion concen- t The (rr~/q)~~~value of the CH,-SO; gel reported for Ag-counterions' is larger. It turned out that this is an artefact caused by an inadequate fixation of the softer gel rod in the Plexiglas tube in the drop experiments.tion in water. Measurements of the direct current electrical conductivity, E, of the gels reveal that the ratio of the counterion conductivities in the gel phase (I,.,,, H+/jm,i) neglecting the small contribution of the convection conduc- tivity [(ilcon/k)< 0.151 and that in free solutions at infinite dilution (i;, i}H+/lLg,(i = Li', Na', K', Cs+, Agf) have J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 3 Results of the evoluation of the inertia electromotive force so; Ag + Li+/Ag+ Cs+/Ag+ H+/Ag+,gAg+< 0.5 CH,-SO; Ag + Li+/Ag+ Cs+/Ag+ Hi/Ag+, ZAg+ -= 0.5 (0.80) (0.80) (0.80) (0.16) (1.76) (1.76) (1.76) (0.32) (m/q)exp,ratio of the effective mass of a counterion and its charge; NH20,number of water molecules; NcOunter,number of counterions; gAg+, mole fraction of Ag+ counterions in the gel phase; ( ), mean values, see Fig.2. mH20= 0.3 x g. approximately the same value: (Irn,H+/&,,, i), (A:, H+/A:, i), [H+/Li+]: (9.0) (9.0); [H+/Na+]: (5.6) (6.9); [H+/K+]: (4.3) (4.7); [H+/Cs+]: (5.2) (4.5); [H+/Ag+]: (7.7) (5.65). This indicates that the mechanism of the H+ ion transport in the pore fluid is not modified considerably by the matrix of the gels. The inertia electromotive force measurements give the same information as the measurements of the electrical conductivity of the gels. The observed effect of the action of forces of inertia on the pore fluid of a polyelectrolyte gel could be used in the construction of a sensor for acceleration and deceleration.This device would be more sensitive than a device already developed2 for this purpose using an ionic superconductor as sensor. It is assumed that an H30+ ion in the pore fluid of the gels is strongly connected to three H20 molecules by hydrogen bonds forming a H90: complex. This assumption is based on the accepted structure of the primary hydration shell of a H30+ ion in free The mechanism of charge motion consists of proton transfers within the hydrogen bonds of the H,Ol complex which is accompanied by associ- ations and dissociations of hydrogen bonds at the periphery of the complex. The proton oscillates continuously and very quickly (time constant of the order of s) within the complex. Motion over larger distances is determined by the migration of the whole hydrogen bond complex (rotation of water molecules into orientations in which they can accept or donate protons).The time constant of this 'structural diffusion' process over the distance of a hydrogen bond is estimated to be s.' This special mechanism of charge motion of the H'ions in aqueous solutions leads to the assumption that during decel- eration the H+ ions escape the HgOf complexes moved by the forces of inertia together with the mobile water molecules of the pore fluid at time intervals that are short compared with the time constant of deceleration (1 ms). They migrate by structural diffusion in the direction opposite to that of the electric field generated by the force of inertia.Therefore, the electric field generated by the forces of inertia in a gel loaded with H+ ions is smaller than that generated in a gel loaded with Ag+, Li+ and Cs+ ions, respectively, under the same conditions. The measured pulse of the inertia electromotive force is smaller. Structural inhomogeneities in the two types of gels, formed by regions with locally higher and lower values of the fixed ion concentration (cross-linking), could explain the depen- dence of (rn/~&~in the [H+/Ag+] system for ZAg+ > 0.5 (see Fig. 2). Such inhomogeneities show up in other types of experiments with the same gels.'0." On the basis of these experiments it is expected that the Ag+ and H+ are not dis- tributed uniformly in the gel phase. The Ag+ counterions will be located preferentially in the highly cross-linked regions of the gels.12 Conclusion The experiments demonstrate that forces of inertia acting on the aqueous pore fluid of anionic polyelectrolyte gels loaded with different counterion species (Li+, Cs', Ag', H+ and Ba2+) during deceleration produce a voltage pulse.It is larger than that calculated from the mass and charge of a single counterion. The measured voltage pulses allow us to determine an effective number of water molecules per mobile counterion in the pore fluid causing this effect. The experi- ments reveal that the mechanism of H+ ion transport in the pore fluid is not modified considerably by the matrix of the gel, confirming measurements of the electrical conductivity of the gels.We thank H. Rottger for his help in the early stages of the experiments. References 1 H. Rottger and D. Woermann, Ber. Bunsenges. Phys. Chem., 1992, %, 623. 2 (a) M. Betsch, H. Rickert and R. Wagner, Ber. Bunsenges. Phys. Chem., 1985, 89, 113; (b) M. Betsch, H.Rickert and R. Wagner, Solid State Zonics, 1986, 18,19, 1193; (c) W. Koch and H. Rickert, Solid State Zonics, 1988,28-30, 1664; (d) W. Koch and H. Ricker in High Conductivity Solid Ionic Conductors, Recent Trends and Applications, ed. T. Takahashi, World Scientific, New Jersey, 1989, p. 64. 3 G. Manecke, 2. Phys. Chem., 1952,201,193. 4 E. Phillipsen and D. Woermann, J. Membrane Sci., 1984, 17, 139. 5 G. Wiedner and D. Woermann, Ber. Bunsenges. Phys. Chem., 1975,19,868. 6 R. Schlogl, Stofltransport durch Membranen, Steinkopff Darm- stadt, 1964, p. 75. 7 M. Eigen and L. DeMaeyer, in The Structure of Electrolyte Solu- tions, ed. W. J. Hamer, Wiley, New York, 1959, p. 64. 8 T. G. Fillingim, N. Luo, J. Lee and G. W. Robinson, J. Phys. Chem., 1990,94,6368. 9 G.W. Robinson, J. Phys. Chem., 1991,9!5, 10386. 10 H. Rottger and D. Woermann, Langmuir, 1993,9, 1370. 11 C. Richter and D. Woermann, to be published. 12 F. Helfferich, Zonenaustauscher, Verlag Chemie, Weinheim, Bergstrak, (1959), p. 173. Paper 3/06094B; Received 12th October, 1993