B. PROPERTIES OF PARTICULAR MEMBRANES TRANSPORT PROCESSES IN ION-SELECTIVE MEMBRANES CONDUCTIVITIES TRANSPORT NUMBERS AND ELECTROMOTIVE FORCES BY J. W. LORIMER, (Miss) E. I. BOTERENBROOD AND J. J. HERMANS Laboratory for Inorganic and Physical Chemistry, The University, Leiden Received 1st February, 1956 The thermodynamics of irreversible processes has been applied to the problem of transport of two ions and solvent through a homogeneous membrane. Measurements of transference numbers of potassium ion and water, conductivities and e.m.f.’s as functions of concentration are reported for a new type of cellulose membrane containing a dissolved polyelectrolyte. It is found that the solvent contribution to the e.m.f. is not negligible, and that transference numbers measured directly are in fair agreement with those derived from e.m.f.data. The results indicate large variations in potassium ion mobility in the membrane. Methods for the evaluation of physico-chemical properties of membranes have depended until recently upon some form of the fixed charge theory of Teorell19 2 and Meyer and Sievers.3 This theory contains several restrictive assumptions 2 which make its quantitative application to membrane phenomena rather uncertain. In particular, a number of workers 4-8 have obtained evidence of large differences between ionic mobilities in membranes and in free solution, the explanation of which falls outside the realm of the fixed charge theory. Staverman,% 109 11 in 195 1, and Kirkwood,lZ in 1954, explored the application of the thermodynamics of irreversible processes to ion transport through membranes.The simplest membrane system to which irreversible thermodynamics can be applied consists of a homogeneous membrane separating two homogeneous solutions each containing the same electrolyte. In this case, discussed in the theoretical part, measurement of the specific conductivity, two transference numbers, two diffusion coefficients and mechanical permeability give six relations from which all six phenomenological coefficients of the thermodynamic theory can be obtained. These coefficients are sufficient to describe other phenomena, such as the e.m.f. of membrane cells and streaming potential. For tests of this theory, homogeneous, reproducible membranes which ex- hibited a marked variation in transport properties at electrolyte concentrations below 0.1 N, were prepared in a wide range of dimensions.The specific conduc- tivity, transference numbers of one ion and solvent, and e.m.f. of a membrane cell have been determined for potassium chloride solutions between 0.001 and 0-1 N, and thus form an important part of an extended programme, the aim of which is to obtain all phenomenological coefficients as functions of concentration. THEORETICAL IRREVERSIBLE PROCESSES IN MEMBRANES Consider a membrane separating two homogeneous solution phases I and 11, each containing rz constituents j . This forms a “ discontinuous ” system as defined 141142 TRANSPORT PROCESSES by de Groot,13 for which the flux of constituent j through the membrane at constant temperature and in units of mole cm-2 sec-1 is Here d& is the difference in total thermodynamic potential of constituent j between phases I and 11, and Ljk is a phenomenological coefficient expressing proportionality between the fluxes Jj and the generalized forces d&.There are n(n + 1)/2 of these coefficients, since Onsager 14 showed that they are symmetrical, Ljk = Lkj. (2) Eqn. (1) was used by Staverman 9,1OY 11 in his treatment of membrane pheno- Since the Ljk’s may be functions of F j , it is important to consider only mena. infinitesimal differences in Fj, given by (3) In this equation, dpj is the purely chemical part (4 ) where aj is the activity of constituent j , R is the gas constant in joule mole-1 deg-1 and T is the absolute temperature. The pressure-volume contribution to dFj is given by the product of the partial molar volume vj in cm3 mole-1 and the pressure difference dp in joule cm-3, while the electrical contribution is given by the product of the faraday (96,500 joule equiv.-I), the valence zi of j (including sign) and the potential difference dE in volts.dpj = dpj + vjdp + FzjdE. dpj = RTd (In aj), The total electric current density in A cm-2 is Solution of (1) and (5) for dE gives i FdE = -I/(FLE) - tJ (dpj + vidp) ; j = 1, 2, . . . 12, (6) where tj is the mass transport number of constituent j : and with K the specific conductivity in Q-1 cm-1, and a the membrane thickness in cm. If (6) is inserted into (l), where Relations (6) and (10) are the basic equations for this paper. They are formally equivalent to those of Staverman,g. 10911 but are written in a form particularly useful from an experimental point of view.It is to be noted that Ljk and d j k have the dimensions of (concentration x mobility)/(length x F), with concentration in mole cm-3, mobility in cm2 V-1 sec-1 and length in cm. The relationship between this “ discontinuous system ” and the “ continuous system ” used by Kirkwood 12 is too lengthy to be treated here, and will be discussed elsewhere. 15J. W. LORIMER, E . I . BOTERENBROOD AND J . J . HEKMANS 143 DESIGN OF EXPERIMENTS In the case of interest here, consider a univalent cation 1, a univalent anion 2 and a solvent 0. Eqn. (6) and (10) then become Ji = J2 + IIF = tiI/F - Aii(dp + v dp) - dlo(dpo + vodp), Jo = toZ/F - dlo(dp 4- vdp) - Jloo(dp0 + VO~P), (13) (14) and where and it is found that If anion-reversible electrodes in the solution phases are used to measure potentials, FdE = -FIa/K - tl(dp1 + qdp) - t2(d,u2 + v2dp) - fo(dpo + vodp), p = p1 + p2, v = v1 + v2, and tl - t2 = 1, d l 1 = d l 2 = d 2 2 , d l 0 = d 2 0 .must be added to (14) to obtain the total e.m.f. of the cell. The term +(p) contains the effect of pressure on the electrode reactions. With dp = dp = 0, the specific conductivity and two independent transference numbers can be measured. These transport numbers are sufficient to predict the e.m.f. of a membrane cell with dp = I = 0. For a homogeneous membrane with transference numbers which are single-valued functions of p, the value of this e.m.f is E’ = -(2RT/P) (tl - mMto/lOOO)d In a, s:’ where the Gibbs-Duhem equation -dpo = mMdp/1000 has been used, with m the molality of the solution, and M the molecular weight of the solvent.It must be emphasized that this e.m.f. is a measure of a combination of tl and t0,10, 16 and does not involve any unknown single ion activity coefficients. Tangents drawn to a plot of E’ against In a thus yield values of this transference number combination. At dp = dE = 0, again using the Gibbs-Duhem equation, the quantities 4 1 1 - mMdlo/1000, and d l 0 - mMAYoo/1000, may be obtained by measuring the diffusion rates through the membrane of salt and solvent under steady state conditions. Finally, at dp = dE = 0, measure- ment of the mechanical permeability, gives a sixth independent relation which completes the information necessary for computation of the six phenomenological coefficients.Other phenomena such as the streaming potential may then be predicted readily. numbers, and comparison with measured e.m.f.’s are described below. JOPP = - A l O V - ~oovo., dE’jdp = -tlv - tow0 - $(P) Attempts at verification of eqn. (16) by direct measurement of the transference EXPERIMENTAL AND RESULTS MATERIALS AND soLunoNs.-The sodium carboxymethycellulose (NaCMC) and viscose were samples supplied by the Research Laboratories of the A.K.U. Rayon Co., Arnhem, The Netherlands. A 2 % solution of NaCMC in 40 % ethanol + water was prepared1 44 TRANSPORT PROCESSES and centrifuged at 10,000 rev/min for 30 min to remove undissolved material. The con- centration of the resulting clear solution was determined by evaporation of weighed portions at 110" C.Conductimetric titration 17 of three samples of this partially purified NaCMC gave 2-81 0.02 mequiv." of sodium per g of salt (degree of substitution 0-59) and a completely negligible amount of free carboxyl groups. The viscose was stored at 0" C to slow down decomposition. Analyses of this solution by coagulating three weighed samples by the procedure given below, then drying to constant weight in a dry nitrogen atmosphere 18 at 110" C gave 7.22 &- 0.02 % cellulose by weight. Analyses of the regenerated cellulose by the methods of Ludtke 19 and Neale and String- fellow 20 gave 30 & 2 pequiv. COOH/g cellulose. For conductivities and transport numbers, A.R. KCI was recrystallized from conduc- tivity water and dried thoroughly at 700" C.Solutions were prepared by weighing, using water of conductivity 1 to 1-6 X 10-6 Sz-1 cm-1. Their pH was between 5.8 and 6.0. Solutions for e.m.f. measurements were made by dilution of stock solutions of A.R. KCl with distilled water, in view of the large volumes required. were mixed in amounts sufficient to give a concentration of approximately 0.01 equiv. sodium/l000 cm3 of membrane in its final condition. The solution was mixed thoroughly with a vibrating stirrer. It could be used for three months if stored at 0" C. It was found desirable to make the cellulose concentration greater than 6.5 % for membranes with good resistance to compression. Membranes were formed by pouring the solution into moulds consisting of two glass plates with Plexiglas separator.The separators were rings machined to within 0.02 mm of a given thickness, with small grooves along their radii to allow diffusion of solution into the interior of the mould. The ring was placed on the lower plate, its interior was filled with excess of solution, and the top plate was pressed on firmly. The mould was placed in 15 % ammonium sulphate (iron-free) in distilled water at room temperature. After about two days, agitation caused the glass plates to slide off the mould. The membrane, with a degree of swelling of 17.2 (the same as the original solution), was removed, and was washed in fresh solution for one day. Its degree of swelling had then reduced to 12, and soluble sulphides and sulphur had been washed out.Finally, it was boiled 5 min in 15 % ammonium sulphate solution to complete the conversion to a water-insoluble, highly-swollen cellulose gel (degree of swelling 7.13). Washing and boiling twice in dis- tilled water removed soluble salts, after which the membrane was placed in saturated KCl for one week to convert it to the potassium form. Before making measurements at a given concentration, the membranes were boiled twice in conductivity water, then kept in conductivity water for 12 h at 80" C, and finally placed in the appropriate KCl solution, which was changed several times over a period of 24 h before the membrane was used. It was found that sodium polystyrenesulphonate, and even the potassium salt of Congo Red (with a micelle-forming anion 21) could be incorporated into membranes in this manner.No Congo Red has been leached out of one membrane in contact with water for a period of over 9 months. SWELLING PROPERTIES Membrane thicknesses were measured by means of a micrometer feeler gauge, so that no pressure was applied during measurement. A membrane could be weighed " in air " with a precision of 0-3 % by blotting, placing in a weighed amount of solution and re- weighing. Its volume and density were determined by the buoyancy method 18 to f 0-3 %. The density of dry cellulose was measured by a similar method, using carbon tetrachloride as the buoyancy fluid. Membranes were prepared in the above manner with a range of thicknesses from 0.1 to 3 mm, and uniform in thickness to 5 0.5 %. Measurement of the ratios of the thickness and diameter of the membranes to the thickness and the diameter of the mould showed that the membrane was swollen isotropically to -I 3 %. Further, successive membranes could be made in the same mould with a reproducibility of f 0.5 % in thickness.The degree of swelling was found to be independent of KCl concentrations between 0.001 and 0.1 N. The density was 1.073 i 0.003 g cm-3 at 25" C. From swelling and analytical data, the concentration of potassium carboxymethylcellulose was calculated to be 0.0126 f 0.0003 equiv./1000 cm3 of membrane. * All limits of error are average deviations from the mean. PREPARATION AND CONDITIONING OF MEMBRANES.-viSCOSe and 2 % NaCMC SolutionJ . w. L O R I M E R , E . I . BOTERENBROOD AND J .J . HERMANS 145 RETENTION OF CMC Measurement of the membrane resistance at 0.005 N KC1 before and after passing direct current through it, or before and after pressing solution through it, indicated no measurable losses of CMC. Qualitative tests of the membrane equilibrium solutions with 2 : 7-dihydroxynaphthalene-H2SO417 similarly gave negative results. One membrane was used repeatedly for e.m.f. measurements over a period of two months without any evidence of loss of CMC. It appears safe to conclude, therefore, that this method of incorporating polyelectrolytes into cellulose membranes is quite satisfactory. SPECIFIC CONDUCTIVITY The specific conductivity of the membranes was measured in a cell similar to that described by Manecke and Bonhoeffer,22 and by means of a Philoscope a.c.bridge. A piece of sheet plati- num was soldered to the end of each of two brass plugs A, A' (fig. 1). The plugs were then turned and threaded to fit holes in the two Plexi- glas discs B, B'. These electrodes were screwed into place, the platinum-Plexiglas joint was sealed by cementing with acetone, and the electrodes were coated with platinum black. The two halves of this cell could be lined up reproducibly by means of pins C , C' and corresponding sockets, and brass screws through holes D held the cell together. The cell was supported by a handle E in such a way that the membrane was held horizontally between the two cell halves. In this way, very slight pressure on the membrane was sufficient to prevent leakage. :: F FIG. 1 .-Conductivity cell.The cell was filled through tubes F, F' by means of a capillary pipette. All measure- ments were carried out with the cell in a thin rubber bag in a grounded water thermostat at 25.00 i 0.02" C. Steady resistance readings were obtained within about 1/2 h. FIG. 2.-Resistance as a function of thickness ; 0.01 KCl alone 0 ; with membrane 0. A number of Plexiglas rings of various thicknesses could be used to separate the two halves of the cell by known amounts. This permitted determination of the resistance of the cell as a function of thickness of a layer of KCl solution. The cell resistance with membrane was also measured as a function of membrane thickness. A typical deter- mination is shown in fig. 2.146 TRANSPORT PROCESSES Least-square lines were computed from the data.If the slope of the line with KCI alone is ( R / a ) ~ c l , and with the membrane is (Ria),, the specific conductivity of the mem- brane is where KKCl is the specific conductivity of the equilibrium KCl solution, obtained from data tabulated by Gunning and Gordon.23 The lines had the same intercept at a = 0 within the experimental error of f 2 %. Thus, the effective area for conduction was the same for membrane and solution. Results at both 1000 and 50 c/s were identical at all concentrations to f 0.5 %. Since evaluation of the specific conductivity involves a ratio of resistances, correction for the solvent conductivity was found to be important only at 0.001 N. The good proportionality between resistance and length indicates that any refraction of the current lines 24 in the area between the two Plexiglas plates is negligible, and that the membranes do not show any large inhomogeneous regions.Table 1 gives the specific conductivity of the membranes and for comparison, that of KCl, as a function of the nor- mality of KC1 (C). Each value of K is the average, computed as shown above, for eleven membranes. TABLE l.-SPECIFIC CONDUCTIVITIES (IN f2-l Cm-l) OF MEMBRANES AND OF KCl SOLUTIONS AT VARIOUS CONCENTRATIONS (C) OF KCl C( equiv./l.) 1 0 4 ~ 1 0 4 ~ ~ c 1 0*1000 75.9 129.0 0.04989 39.5 66.8 0.009993 9-02 14.1 0.004968 5.32 7.12 0.00 1068 2.36 1 -57 E.M.F. MEASUREMENTS The e.m.f. of the cell : Ag, AgCl/KCl (Ci)/M KC1 (C2)/AgCl, Ag was measured by means of a Leeds and Northrup type K-2 potentiometer and a Pye portable galvanometer of variable sensitivity. The silver chloride electrodes were prepared by the method of Brown.25 The cell consisted of two identical halves between which the membrane was clamped. Each half (fig.3) had a Plexiglas chamber I into which a glass electrode chamber I1 with a narrow orifice A was cemented with black wax. The area of the membrane through which diffusion could take place was found to affect the depen- dence of the e.m.f. on flow-rate markedly. Consequently, the membrane B was pressed against interchangeable disc C with holes exactly opposite each other. All measurements reported here were made with holes 1 mm in diameter with edges tapered to permit free access of flowing solution to the membrane surfaces. Solutions and cell were kept in separate water thermostats at 25.0 f 0.1" C.A Plexi- glas ring D, slightly thinner than the membrane and greased, prevented electrical leakage between cell interior and thermostat. Solution flowed into the cell from Mariotte bottles at E. The flow was regulated by stopcocks connected to outlet F. The cell was assembled, filled with the appropriate solutions and allowed to stand over- night in order to permit establishment of an approximate steady state in the membrane. Measurement of the e.m.f. as a function of flow-rate then gave constant values for rates greater than about 100 cm3 min-1 (fig. 4) after about 10 min. These values for various ratios CI/C2 are given in table 2, along with average deviations from the mean of several measurements. TABLE 2 C1 (equiv./l.) C2 (equiv./l.) 0.1 0.05 0.05 0.02 0.02 0.0 1 0.0 1 0.005 0-005 0.002 0002 0*001 e.m.f.(mV) 17.6 & 0.1 25.8 h 0.1 22.0 * 0-2 23.6 0.2 37.9 i 0.2 29.9 f 0.3J . w. LORIMER, E . I . BOTERENBROOD AND J . J. HERMANS 147 TRANSFERENCE NUMBERS Experiments were carried out in a cell based upon a design of Remy.26 The membrane was clamped between two Plexiglas flanges cemented to calibrated, closed glass chambers of about 50 cm3 capacity. A silver sheet anode 27 and a fused silver chloride cathode 28 served as electrodes. Calibrated capillary tubes projected from each chamber, and terminated in horizontal sections which carried glass scales. The cell was operated in a water thermostat at 25" C. A rubber ring prevented electrical leakage at the membrane.Current was passed through the cell from a 24 V accumulator supply, and measured by noting the potential drop across a stan- dard resistor in series with the cell. Analyses of the solutions in each chamber were made con- ductimetrically before and after electrolysis, sufficient charge being passed to cause a 10 % change in concentration. Measurement of the displacement of the solution menisci in the capillaries permitted :'i E FIG. 3.-E.m.f. cell. calculation of the solveni transport and the change in volume of each compartment. Passage of one faraday through the cell corresponds to the transfer of ti equivalents of potassium ion and to equivalents of water from anode to cathode compartment. Although 30 - 28 v W 26 24 FIG. 4.-Effect of flow- rate on e.m.f. ; 0*002-0.001 KCl : 0 ; 0.05-0.02 KC1 : 0.100 200 300 FLOW RATE (m'/min) the data obtained should be sufficient to calculate t l with a precision of 0.3 %, the re- producibility of the concentration change at the cathode side was only about 8 %. The anode side gave lower and more irregular results. Possible irreversible reactions at the anode are being investigated, and more complete data will be published later. The cathode results and the solvent transport numbers are given in table 3, together with the mean deviations. TABLE 3 C (equiv./l.) I 1 t0 0.01 0.68 f 0.05 130 & 10 0.005 0.70 & 0.05 170 & 30 0.00 1 0.82 & 0.05 300 & 70148 TRANSPORT PROCESSES DISCUSSION The cellulose-polyelectrolyte membranes described here are similar in principle to the collodion-polyelectrolyte preparations of Neihof.29 The excellent re- producibility in thickness and the possibility of continuous variations in fixed charge concentration make them especially useful models for the investigation of general membrane properties.Comparison of the magnitude of the solvent transference numbers in table 3 with those for potassium ion by means of eqn. (16) shows that the contribution of solvent transfer to the e.m.f. can be as high as 4 %. One other evaluation of this effect has been made by Graydon and Stewart,35 who attributed deviations from ideal fixed charge theory to solvent transfer. Some older work exists, but it is concerned either with ill-defined membranes,26 or solely with electro-osmotic pressure (see summary by Schmid).34 150 50 0.5 1.0 1.5 log a lo /ao.ooc FIG.5.4.m.f. as a function of activity of KC1 By suitable addition of the data of table 2, the e.m.f. for any given concentration difference may be obtained. This has been done in table 4, where for all e.m.f.’s, C2 = 0.001 N. The difference in activity between C1 and 0.001 N is also given,33 and the e.m.f. E’is plotted as a function of loglo(a/ao.ool) in fig. 5. The two straight lines in this figure give the e.m.f. for two ideal cases in which the transference numbers are constant and equal to 1 or 3, and solvent transport is negligible. Differentiation of an empirical function fitted to these e.m.f. data gave the quanti- ties ( t l - rnMto/1000) in table 4. Values of tl were then obtained from the solvent data of table 3.These may be compared with the directly-measured values in table 3. At 0.01 the agreement is good, but the direct values appear to be too low at lower concentrations. The general trend of tl as a function of concentra- tion is similar to that found in oxidized Cellophane by direct measurement and TABLE 4 c.l (equiv./l.) 0.1 0.05 0-02 0-0 1 0.005 0.002 0.001 e.m.f. tmV) 156.8 139-2 113.4 91.4 67.8 29.9 0.0 log10 ta/aoood 1 -902 1-625 1.253 0.969 0.680 0.299 o*ooo I1 t l (direct) 0 1 - 0.52 0.55 0.60 0.66 0-77 0.85 0.87 0.018 mto) (e.rn.f.1 0-68 0.78 0.87 0.68 0.70 0.82J . w. LORIMER, E. J. BOTERENBROOD AND J . J . HERMANS 149 by e.in.f.31 In that case, however, comparisons by Wright 34 showed that the e.m. f. transference numbers were lower than the directly measured values. Further investigation of the equivalence of transference numbers derived from these two methods is being carried out.The specific conductivities resemble those obtained for a more highly selective membrane at higher concentrations by Clarke and co-workers,32 who also found an intersection of the specific conductivity against concentration curves for membrane and solution. On the basis of the fixed charge theory, the membrane conductivity should be greater than that of the solution if the mobilities in membrane and solution are the same.24 Although the internal salt concentration should be known in order to obtain mobilities in the membrane,sy 69 7 estimates based on simple Donnan equilibrium and the data of tables 1 and 4 indicate that the potassium ion mobility for the membranes considered here varies from about 37 at 0.1 N to 18 at 0.001 N, compared with about 73 in free solution.Further interpretation of these variations in transport properties will be reserved until more complete knowledge of the thermodynamic phenomenological coefficients is obtained. The authors wish to thank Miss A. Wijnand and Mr. J. T. Semeyns de Vries van Doesburgh for invaluable assistance with the e.m.f. measurements, and Prof. A. M. Liquori (Rome) for helpful preliminary work during his stay at Leiden. 1 Teorell, Proc. SOC, Expt. Biol., 1935, 33, 282. 2 Teorell, Progress in Biophysics, 1953, 3, 305. 3 Meyer and Sievers, Helv. chim. Acta., 1936, 19, 649. 4 Wright, Trans. Faraday SOC., 1953, 49, 95. 5 Wright, Trans.Faraday Soc., 1954, 50, 89. 6 Manecke and Otto-Laupenmuhlen, 2. physik. Chem., 1954, 2, 336. 7 Hills, Kitchener and Ovenden, Trans. Faraday Soc., 1955, 51, 719. 8 DespiC and Hills, Trans. Faraday, SOC., 1955, 51, 1260. 9 Staverman, Chem. Weekblad, 1951, 47, 1. 10 Staverman, Trans. Faraday SOC., 1952, 48, 176. 1 1 Staverman, Acta Physiol. Pharmacol. Neerl., 1954, 3, 522. 12 Kirkwood, in Ion Transport Across Membranes (Academic Press, Inc., New York, 13 de Groot, Thermodynamics of Irreversible Processes (North-Holland, Amsterdam, 14 Onsager, Physic. Rev., 1931, 37, 405. 15 Hermans and Lorimer, in preparation. 16 Scatchard, J. Amer. Chem. SOC., 1953, 75, 2883. 17 Eyler, Klug and Diephuis, Ind. Etrg. Chem. (Anal.), 1947, 19, 24. 18 Hermans, Contribution to the Physics of Cellulose Fibres, Elsevier (Amsterdam, 1946). 19 Ludtke, 2. angew. Chem., 1935, 48, 650. 20 Neale and Stringfellow, Trans. Faraday Soc., 1937, 33, 881. 21 Robinson and Garret, Trans. Faraday SOC., 1939, 35, 771. 22 Manecke and Bonhoeffer, 2. Elektrochem., 195 1, 55,475. 23 Gunning and Gordon, J. Chem. Physics., 1942, 10, 126. 24 Schmid and Schwarz, Z. Elektrochem., 1951, 55, 295. 25 Brown, J. Amer. Chem. SOC., 1934, 56, 646. 26 Remy, 2. physik. Chem., 1925, 118, 161. 27 Jones and Dole, J. Amer. Chem. SOC., 1929, 51, 1073. 28 LeRoy and Gordon, J. Chem. Physics, 1938, 6, 398. 29 Neihof, J. Physic. Chem., 1954, 58, 916. 30 Neale and Standring, Proc. Roy. SOC. A., 1952, 213, 530. 31 Clarke, Marinsky, Juda, Rosenberg and Alexander, J. Physic. Chem., 1952, 56, 100. 32 Harned and Owen, The Physical Chemistry of Electrolytic Solutions (Reinhold, 33 Wright, J . Physic. Chem., 1954, 58, 50. 34 Schmid, 2. Elektrochem., 1951, 55, 229. 35 Graydon and Stewart, J . Physic. Chem., 1955, 59, 86. 1954), p. 119. 1952), p. 54. pp. 200ff. New York, 2nd ed., 1950), p. 369.