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(2) Boens, N. In Luminescence Techniques in Chemical and Biochemical Analysis; Baeyens, W. R. G., De Keukeleire, D., Korkidis, K., Eds.; Marcel Dekker: New York, 1991; pp 21-45. (3) Lakowicz, J. R.; Gryczynski, I.; Laczko, G.; Joshi, N.; Johnson, M. L. In Luminescence Techniques in Chemical and Biochemical Analysis; Baeyens, W. R. G.,De Keukeleire, D., Korkidis, K., Eds.; Marcel Dekker: New York, 1991; pp 141-177. (4) Gratton, E.; Alcala, J. R.; Barbieri, B. In Luminescence Techniques in Chemical and Biochemical Analysis; Baeyens, W. R. G., De Keukeleire, D., Korkidis, K., Eds.; Marcel Dekker: New York, 1991; pp 47-72. ( 5 ) Knutson, J. R.; Beechem, J. M.; Brand, L. Chem. Phys. Lett. 1983, 102, 501-507. (6) LBfroth, J.-E. Eur. Biophys. J . 1985, 13, 45-58. (7) Beechem, J. M.; Brand, L. Photochem. Photobiol. 1986,44,323-329. ( 8 ) Janssens, L. D.; Boens, N.; Ameloot, M.; De Schryver, F. C. J. Phys. Chem. 1990, 94, 3564-3576. (9) Beechem, J. M.; Ameloot, M.; Brand, L. Anal. Instrum. 1985, 14, 379-402. (10) Boens,N.; Janssens, L. D.; De Schryver, F. C. Biophys. Chem. 1989, 33, 77-90. (1 1) Beechem, J. M.; Ameloot, M.; Brand, L. Chem. Phys. Lett. 1985,120, 466-47 2.
(12) Ameloot, M.; Beechem, J. M.; Brand, L. Chem. Phys. Lett. 1 ." 21 1-21 9.
(13) Ameloot, M.; Boens, N.; Andriessen, R.; Van den Bergh, V.;De Schryver, F. C. J . Phys. Chem. 1991, 95, 2041-2047. (14) Andriessen, R.; Boens, N.; Ameloot, M.; De Schryver, F. C. J . Phys. Chem. 1991, 95, 2047-2058. (15) Andriessen, R.; Ameloot, M.; Boens, N.; De Schryver, F. C. J. Phys. Chem. 1992, 96, 314-326. (16) Anderson, D. H. Compartmental Modeling and Tracer Kinetics; Lecture Notes in Biomathematics; Springer-Verlag: Berlin, 1983. (17) Jaquez, J. A. Compartmental Analysis in Biology and Medicine; Elsevier: Amsterdam, 1972. (18) Rabenstein, A. L. Elementary Differential Equations with Linear Algebra, 3rd ed.; Harcourt Brace Jovanovich: San Diego, CA, 1982. (19) Davenport, L.; Knutson, J. R.; Brand, L. Biochemistry 1986, 25, 1186-1 195. (20) Wahl, Ph.; Auchet, J. C. Biochim. Biophys. Acta 1972,285,99-117. (21) Marquardt, D. W. J. SOC.Ind. Appl. Math. 1963, 11, 431-441. (22) Dorn, W. S.;McCracken, D. D. Numerical Methods with Fortran IV Case Studies: Wilev: New York. 1972: DD 221-230. (23) Van den Zegel,*M.; Boens, N:; Daemi,'D.; De Schryver, F. C. Chem. Phys. 1986, 101, 311-335.
Theories and Experiments on Nonisothermal Matter Transport in Porous Membranes F. S. Gaeta,* E. Ascolese, U. Bencivenga, J. M. Ortiz de Zlrate,+ N.Pagliuca, G . Pema, S. Rossi, and D. G. Mita International Institute of Genetics and Biophysics of CNR, Via G. Marconi 10, 80125 Naples, Italy (Received: July 8, 1991) The growing body of experimental evidence on nonisothermal matter transport in artificial porous membranes until now has been alternativelyinterpreted within the frame of reference of one of three independent theoretical approaches. According to one, the force driving transport is due to transfer of momentum from thermal excitations to the medium; another assumes this force to be due to transported entropy balance; the third envisages distillation in vapor-filled pores as the particular transport mechanism occurring in hydrophobic membranes. Two of these approaches apply to both hydrophilic and hydrophobic membranes; the other is specific to the case of porous hydrophobic filters and to liquids that cannot permeate them. The two general approaches are complementary, one constituting the thermodynamic representation of a physical process that the other describes in terms of statistical mechanics; the third is incompatible with the other two. The predictions of the alternative models diverge in various ways. Experiments specially designed to investigate the conflicting forecasts have been carried out employing two polar solvents and using a new experimental technique to investigate the behavior of solute fluxes. This article presents a preliminary report of the main experimental results obtained so far and a discussion of their relevance to the theoretical dispute among the different approaches. 1. Introduction The application of temperature gradients to membranes permeated by a fluid phase generally results in transport of matter, and eventually of charge, coupled to the flux of thermal energy.'-" We demonstrated in particular that a system constituted by a nonisothermal grassly porous sheet of hydrophilic or hydrophobic material separating adjoining volumes of aqueous solution exhibits volume flux and selective solute t r a n s p ~ r t ; l subsequently ~-~~ much attention has been paid to this effect by various authors.2b37 Opinions on the physical nature of the causes of nonisothermal membrane transport are divided between the theory, specific to hydrophobic membranes, attributing them to solvent extraction from the heated solution through capillary and the ideas independently put forward by some of us13-23*38-40 and by Tasaka et a1.28,29+34 according to whom the nonisothermal state of a liquid gives rise to forces resulting in matter transport through liquid-filled channels of permeable membranes. In the first of these last two substantially syntonic approaches, a *thermal radiation pressure" arises, owing to momentum exchange among drifting phononic thermal excitations and material particles of the medium.3g40 In the approach of Tasaka et al. the transported entropy of the liquid in the nonisothermal membrane is different from the molar entropy in the external free solutions, and this situation results in the appearance of thermodynamic forces driving matter transport. To whom all correspondence should be addressed. Faculty of Physics, Universidad Complutense, Madrid, Spain.
According to the capillary evaporation hypothesis2e33 solvent transport is on the other hand explained by an evaporation process at the warm side of the membrane channels and condensation at the cold one; matter crosses the membrane as a vapor phase, the liquid beiig kept out of the pores by surface tension. Nonvolatile solutes cannot cross the membrane; their net flux according to this hypothesis should always by equal to zero, and the concentrations in the warm compartment may increase with time only owing to solvent extraction from the liquid on the warm side. However, a more recent development of the capillary distillation t h e ~ r y ~accepts ~ , ~ ' that, by a loss of hydrophobicity, a variable percentage of the pores of the membrane, at any time, might be liquid-filled and part, instead, permeable only to the vapor. Accordingly, when the experiment is performed with closed volumes a transmembrane pressure difference develops and a more complex behavior is observed: solvent, transported as vapor from warm to cold compartment generates a pressure gradient that forces the solution in bulk back through the liquid-filled pores, into the warm compartment. The overall result of the process thus is the dilution of the cold solution, and a net solute flux from the cold to the warm compartment. With the formulation of this hybrid model the phenomenology of nonisothermal solvent and solute transport in hydrophobic membranes can be interpreted within alternative frames of reference that lead to different forecasts on system's behavior. Thermal radiation pressure predicts that the solutes are expected to flow in the pores at individual rates similar to those of ordinary thermal diffusion. while membrane distillation holds that solute
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Nonisothermal Matter Transport in Porous Membranes flux will be determined by the rate of solution back-flow in the liquid-filled channels, Experimental verification on various solutions of different nature may hopefully help to choose between these two forecasts. Other predictions also differ in ways that can be checked by suitable experiments, as for instance in the case of the temperature dependence of the nonisothermal volume flux with pure liquids. Here the radiation pressure theory (RP) holds that the force driving volume fluxes is produced within the liquid filling the pores and that this force linearly depends on average temperature T,, and on transmembrane temperature difference AT. The hybrid membrane distillation theory (MD), on the other hand, predicts an exponential dependence of the fluxes on Tav, owing to the exponential temperature dependence of vapor pressure. The dependence of steady-state pressure from MD point of view is a complex problem, related with the hysteresis effects observed in the measurements of isothermal flow in hydrophobic membranes. The present state of the theory seems unable to do any clear-cut prediction on this point. The expectations of RP on steady-state thermmmotic pressure [(APh)lstare those of a linear dependence on AT and of a dependence on Tavfollowing that of a well-defined characteristic constitutive property of the liquid, as will be shown in the following. The aim of this study is providing quantitative observations on transport phenomena in nonisothermal hydrophobic porous membranes, employing two polar solvents and various solutions of nonvolatile solutes, in conditions where a temperature gradient is the driving force. The results of these experiments will be discussed in terms of the alternative approaches, in an effort to bring out the respective weaknesses and points of strength.
2. Theoretical and Phenomenological Approaches In this part the MD and RP approaches will be concisely reviewed indicating also some phenomenological expressions that allow the treatment of experimental results. The MD model assumes that nonisothennal volume flux occurs through a process of evaporation at the warm side of the membrane and condensation at the cold one. Thus the driving force will be the difference of vapor pressure between warm and cold side; it will then be possible to derive the trend of nonisothermal fluxes from the temperature dependence of vapor pressure. This can be done, to a first approximation, by using the ClausiusClapeyron equation;36giving a volume flux Jv ( c m 3 d ) : Jv = &5)[e-L/R(Tw+ATJZ) - e-L/R(Tav-AT/Z)] (1) where A is a proportionality constant having the dimensions of length, L is the heat of vaporization of the liquid, and R is the gas constant. 5 ) is the diffusion coefficient of water vapor in the air contained in membrane pores; it has recently been demon~ t r a t e dthat ~ ~ this factor could be important in membrane distillation. From this expression and through a series of algebraic manipulations and approximations,x the following can be obtain&. J,
= &-L/RTwAT
(2)
where 93 is a new constant with dimensions (cm3 s-l "C-l). This expression shows an h e n h i u s dependence on Tav,We can observe that the activation energy is equal to the heat of vaporization of the liquid; by means of another development, using the more precise equation of A n t ~ i n e one , ~ ~obtains a similar nonlinear variation but with a slightly different energy. The MD model applied to experiments performed with solutions of nonvolatile or macromolecular solutes, in the absence of pressure gradients, predicts that no net solute flux is possible through the membrane. The concentration variations observed in both chambers will be due only to solvent extraction from the warm to the cold side of the membrane. In the experiments described below, the situation is different from that for which MD was formulated, thus, a phenomenological model 'ad hoc3@must be developed. To ensure no volume flux, a pressure gradient is applied. Then the vapor flux enters in the cold chamber through the empty pores and is equilibrated by an equal volume flux of solution forced out of this chamber through the filled pores. Therefore a net solute flux will be observed in
sense cold to warm. The equation describing this physical situation is (3) where A* is the hydraulic permeability of the membrane owing to the fraction of liquid-filled pores, AP is the pressure gradient at which the experiment is done, and ([C(r)lcJiis solute concentration in the cold chamber in the course of time. The permeability A* in eq 3 is related to the more commonly used definition of A (see below) by A* = AV, molar volume P being expressed, as usual, in (cm3mol-I). By the conservation of matter law the sum (CwVw+ CcVc) must be constant in time. Considering that no volume change occurs in both chambers and accounting for the relationship between solute flux and variation of concentration, eq 3 can be reduced to a differential equation: (4)
where ci and 0 are positive constants, easily related with chamber volumes, diffusion coefficient, pressure gradient, initial concentration, etc. The suffix C/W means that the equation is similar both for the cold or the warm chamber, the constant ci being equal to zero in the 'cold" case. This equation easily integrated, leads to an exponential behavior in time for solute concentrations in each chamber, with a limit value for t m , where solute flux = 0 and Cw(t-m) = (Cwco)Vw vanishes: Cc(t--) CC(0)VC)/ Vw * Of course, as the concentration gradient grows, diffusion processes appear in the liquid-filled pores simultaneous to the volume flux, and for a more rigorous analysis eq 3 must be modified in this sense. The effect of this will be the disappearance of complete solute transport with some solute remaining in the cold chamber also at t m. It must be noted that in this model only the phenomenological parameter A* appears. Determination of this parameter can be easily done by introducing the measured value of J,(r=O) in eq 3; alternatively this permeability can be measured directly in isothermal flux experiments, as will be shown below. The principal point of this phenomenological model is the existence of liquid-filled pores in hydrophobic membranes. This fact is proven by the existence of isothermal flow at all transmembrane pressures, even at very small ones. The pressure dependence of hydraulic flow is linear at every temperature, which means that the number of liquid-filled channels is constant throughout the explored range of AP and T . Besides, also the observed nonisothermal flux reduction occurring in the first days of experimentation with a new membranezz-23 can be interpreted as an indication of a gradual filling of pores with a liquid phase. This in turn might be attributed to a loss of hydrophobicity in the membrane. The thermal radiation pressure approach:84 a physical theory that has recently been generalized,@can now be applied also to phenomena of nonisothermal volume transport. At present we are able to propose a physical interpretation of nonisothermal matter transport, and we will show that the theory can also be tentatively applied to the new experimental results presented here. A system consisting of an isotropic medium made by a great number of particles interacting among them through short-range forces is now considered. The system can vibrate at many characteristic frequencies, and at any temperature above absolute zero such vibrations spontaneously occur, also in the absence of external excitation. If an amount AQ of thermal energy is fed to the system, a theorem due to Boltzmann and applied by Ehrenfest to the adiabatic case41 states that
-
+
-
where (Ek ) is the average kinetic energy associated with vibrations o8"kriod r . Ehrenfest considers the particular case of an adiabatic system; instead we focus our attention on a system at thermal steady state where the condition AQ = 0 follows from the equality of entering and issuing heat fluxes. Such is for instance the case of an isotropic condensed p h a s e s a y a cylinder of a homogeneous liquid thermally insulated on the lateral
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where H = 2IIR. This pressure exists in the liquid phase and inside a liquid-filled membrane pore would push the fluid in the sense of heat flow or against it, depending on the sign of the expression in square brackets. On the other hand, it can be demonstrateda that an expression analogous to eq 9 can also be obtained for radiation pressure on solute particles in a nonisothermal solution. Indeed p(xl,xz) depends only on the difference of [(K/u)(dT/dx)] in x1 and x2and not on the distance between these points. If x1 xz, keeping constant the difference of constitutive properties, one has a physical discontinuity between two adjoining media, as for instance a liquid and a suspended particle. The force on such particle then results as Fhi = 2 4 - k Ti;; d T urt
-
xt
x2
X
Figure 1. Cylindric portion of a homogeneous liquid thermally insulated on its lateral surface. A temperature gradient is applied along axis x , producing a steady heat flux in the medium.
surface-when a steady heat flux flows along its axis (see Figure l), owing to a temperature difference along x. Once thermal effects connected with the initial transient are over, the net exchange of thermal energy of the system with the external world is equal to zero over any time interval, as in the adiabatic case, and from eq 5 for each value of r one then has 6[(Ekyn),]r= 0, i.e., (Eb),r is constant along the x direction. On the other hand, elastic vibrations in the medium satisfy the relation (E,,,), = (Ept), = OSU, where (Ept), is average potential and U total mechanical energy connected with the oscillations. The difference with the adiabatic case of course is that here we have a constant rate of entropy production owing to the flux of thermal energy, a circumstance however that does not affect the form of these relations. Indeed, from these last two relations it follows that -1 -d+r - - 1- dU - 0 r dx
U dx
This means that every local change of vibration frequency will be connected with a variation of mechanical energy AU. More specifically,the percent variation of the period shall be equal and opposite to that of mechanical energy. It is well-known that thermal energy in the condensed phases mostly consists of very high frequency elastic waves. Part of the spectrum of these phononic thermal excitations has been experimentally investigated in liquids by optical methods. Proceeding from a warmer to a cooler region of the liquid, the waves undergo frequency changes owing to thermal expansion of the medium; eq 6 allows us to write dL E F dx = -dU = U d r / r = S(U/Sr) dr (7) where S is cross section of the cylinder. Thus mechanical work is produced by the system when thermal excitations drift down the temperature gradient. The temperature gradient affects the elastic properties, which change along x, and at the same time it also affects the population of thermal phononic excitations, resulting in a net drift of the excitations along this axis, Le., producing a heat flux Jq. The quantity (U/Sr) represents the mechanical work per unit of surface produced in the medium per period, when heat-in the form of high frequency elastic waves-propagates along x. Obviously ( U / S r ) is proportional to heat flux,Le., to the equidimensionalquantity Jq = -K dT/dx, K being thermal conductivity. We may then write
R being an adimensional proportionality constant connecting ( U / S r )and K (dT/dx) within dx; R thus is phonon’s reflection coefficient,due to the nonisothermal condition. We now consider what occurs when thermal excitations drift from x1and xz (Figure 1). Equation 8 can be used to calculate the work dissipated by drifting thermal excitations along any distance down the temperature gradient in the nonisothermal medium. The expression arrived at is40
(t),I);(
with suffixes 1 and i standing for “bulk liquid” and “solute particle”. We derived some time ago38an equation identical to (10) on the basis of an analogy with acoustic radiation pressure and applied it to thermal diffusion in liquid solutions4248(Soret effect). On the other hand, each membrane pore filled with solution, in experiments such as those carried out by us (i.e., in the absence of volume flow), is a kind of microscopic Soret cell, into which thermal diffusion should occur with some peculiarities of behavior owing to permeant interactions with the membrane matrix. These differences might turn out to be important in various ways. In the first place convective motions, which plague experiments in Soret cells, will be hydrodynamically quenched within narrow channels. In the second place the interactions of the liquid with the solid membrane matrix alter the structure of the fluid adjacent to the solid phase; within narrow pores a large fraction or all of the liquid in the channel will be affected. Colligative properties are very sensitive to these circumstances; for instance thermal conductivity has been shown to increase very much in water layers adsorbed on mica or glass.49 These modifications become more important when channel radius is smaller. It has been shown16 that solute flux can change sense owing to this cause within membranes of the same nature but having different pore radii. It will also be observed that the solute fluxes obtained in the present experiments, with hydrophobic membranes, have systematically a sense opposite to the corresponding fluxes in ordinary thermal diffusion in the liquid in bulk and to solute fluxes in hydrophilic membranes. Tasaka et al.” ascribe similar inversions of the sense of volume flow in hydrophobic with respect to hydrophilic membranes to the change of the value of transported entropy of water through channels having different properties. We attributed such i n v e r ~ i o n s to ~ ~changes + ~ * ~ of the K / u of the liquid, and in a certain sense this amounts to the same kind of assumption as that of Tasaka. Assuming that our separation process is thermodiffusive, then in a multicomponent solution, referring with the suffix i to any of the molecular or ionic species present in solution, the time dependence of individual fluxes Ji should conform to the phenomenological equation
where a, bM, and r are membrane void volume, thickness, and pore tortuosity, respectively, AT is the applied temperature difference, and ([C(t)lwJiis the solute concentration in the warm chamber at each time; Di and D: are ordinary diffusion and thermal diffusion coefficients of the ith component. Equation 11 states that at each instant the net flux of a solution component is equal to the difference of the opposite fluxes due to ordinary and thermal diffusion. The substitution of [([C(~)]W)~ + ([C(t)]c)i]/2in place of the initial concentration Coin the last term of eq 11 accounts for the changes of composition with time in the semicells and of average concentration in the pores during
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solute transfer across the membrane. Equation 11 can be easily converted to a differential equation for the evolution in time of the concentrations in each chamber, its analytical form turning out to be similar to eq 4. Then the two approaches hybrid membrane distillation and thermodiffusion lead to similar formal predictions for the time evolution of the results, both in accordance with the observed experimental behavior. From the experiments the [Js(t)],can be calculated, in particular it is possible to derive initial fluxes, when concentration gradients do not yet exist. In these conditions [C(0)lw = [C(0)lc = C, and the first term of the second member of eq 11 disappears. Accordingly, the coefficient of thermal diffusion D'can be calculated independently from D. At steady state, Le., at 1 = a,solute flux goes to zero, and from eq 11 we can get the ratio D'/Di = si(K1), known as Soret coefficient:
If one measures the concentrations obtained in conditions near steady state, it is possible to calculate si and also D (as D' has been determined in the previous step). Now, having determined by extrapolation from experimental data the initial solute fluxes [J,(O)li, in the fixed-volume experiments, we can immediately derive the initial thermodiffusivedrift velocity of solute particles [U(0)li in the standing liquid that fills membrane pores:
d
a
9 t 86
aM being the thickness of the membrane. In their steady-state motion solute particles work against viscous forces equal and opposite to the thermal driving force, pi = 6rqlr,[U(0)li= -Pi,where q1is the viscosity of the solution and ri particle radius. Substituting in this last relation [U(0)li with the expression in the third term of eq 13 and expressing Fithby the relation obtained from the theory (eq lo), assuming also AT/aM dT/dx, we have
k
Figure 2. Schematic representation of an experimental unit: C, C' = semicells; Man = manometer; P = flow pipe; R regulable-height water rcservoir; MM = membrane; th = thermocouple housings; n = supporting nets; TI, T2= cryothermostats; Si = three-way stopcocks. When the apparatus is assembled, the two semicells are juxtaposed, gripping tightly the interposed membrane.
The use of three distinct apparatuses allowed us to compare results obtained under different conditions. The general features of an experimental unit, as represented in the figure, comprise the two semicells C and C' connected either to a flowmeter or to a manometer and a manostat R. These alternative connections were effected by means of stopcocks SI, Having measured q1and D', and knowing membrane porosity S2and S3,S4, S5, with the purpose of measuring volume flow at t and the solvated particle radius ri, we can now calculate the value the flow pipe P or determining steady-state pressure on the maof the first term of eq 14. Unfortunately the present state of the nometer (Man). The use of the adjustable-height container R, theory does not allow calculation of H from first principles, since fffled with solution, allowed isothermal permeability measurements not enough is known about phonon-particle interactions in the to be made precisely under the same conditions of nonisothermal liquid state, to consent to an evaluation of the scattering of phonons flow. Another purpose of the manostat system was to contrast by particles having dimensions comparable with phonon wavenonisothermal volume flow, so that we could determine solute length. Anyway, assuming H to be approximately constant with transport in the absence of bulk transport of the solution, Le., under temperature in a given solution, one may also derive the trend the condition Jv= 0. Alternatively this condition was ensured of [(klu),- ( k / ~ )and, ~ ] where the (K/u), of the dispersing liquid phase is known, one can also determine the trend of the ( K / u ) ~ by closing both stopcocks, giving access to the cold semicell, after feeding into it solution under an overpressure equal to that which of solvated particles. was going to be produced by the temperature gradient at steady 3. Experimental Section state. Volumes initially fed into each semicell were compared with A p t u s , Materials, and Metbods. The apparatus employed those extracted at the end of the runs, to ensure that no net flow in this study is essentially the same as that already described by had occurred. Density and viscosity of the liquid phases were us in the investigations of thermoosomotic transport of ~ a t e r . ~ z ~measured ~ by a Mohr-Westphal balance and by Ostwald-UbThe experimentalsetup however had to be modified to some extent, belhode viscometers, in order to calculate the passive forces op to meet the exigency of making accurate analysis of solute posing thermoosmotic volume flux through membrane channels transport. A schematic representation of the apparatus is given and resistance to thermodiffusive solute drift relative to the in Figure 2. Three slightly different units were used in the present surrounding liquid. study, each one best suited for use with one kind of solution. Cell Solute concentrationsin the original and in the treated solutions design was optimized in view of solution confiement, by inserting were determined by means of an optical double-beam spectrotwo rigid metallic nets to support the membrane during experiphotometer, and when this was not feasible, as in the case of ments; membrane bulging however could not be avoided, as the solutions of NaCl, KCl, and CaC14, cation concentrations were forces acting on the membrane amount in some experiments to a s d by a Varian atomic absorption spectrophotometer,while more than 500 N. Other experimental facilities which had to be chlorine concentrations were independently determined by adjusted to meet experimental conditions consist of devices, exchemical quantitative analysis. With poly(ethy1ene glycol) the ternal to the cells, designed for the measurement of steady-state measurement of concentrations was effected by determining the pressures and for the assessment of final component concentration. refraction index of solutions using a Baush and Lomb refrac-
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TABLE I: Porow Partitions Employed in This Study and Their Principal Characteristics, As Specified by the Manufacturesa membrane pore size empty vol bubble point in membrane type membrane structure thickness, pm radius, pm fraction, % water, kg/cm2 Gelman TF-1000 PTFE supported on polypropylene net 150 0.50 80 0.492 Gelman TF-450 PTFE supported on polypropylene net 175 0.225 80 1.406 Gelman TF-200 PTFE supported on polypropylene net 175 0.10 80 2.8 12 Millipore FGLP PTFE supported on polyethylene 130 0.10 70 2.900 Celgard 2400 polypropylene film 25 0.02 38 >3.0 Celgard 2402 polypropylene film 50 0.02 38 >3.0
'Pore sizes (fourth column) are operatively defined by the smallest particle size which the partition is able to retain. Empty volumes (fifth column) are defined as ratio of inner empty volume to total membrane volume.
tometer. Finally, with bovine serum albumine a quantitative analysis consisting of a colorimetric titration was adopted. This is a standard procedure with thymine-containing proteins, based on the interaction of phenol-folin-ciocaulten reactant with thyrosine residues. Temperatures across the membrane were registered by thermocouples adjacent to membrane faces; temperatures at the thermostats being kept constant to better than 0.1 OC. Stirring of the fluid contained in the semicells is necessary to avoid the formation of immobilized layers adjacent to membrane faces. Such layers reduce the temperature drop across the membrane and, when working with solutions, create profiles of concentrations adversely affecting solute transport. These effects of membrane polarization were greatly reduced by tilting the apparatus some 6 - 1 2 O arc, warm side up. In this way convective countercurrents develop, ensuring a better efficiency of membrane transport. The best operating conditions must be empirically determined by a series of preliminary experiments. All metallic parts of the apparatus in contact with the working fluid are made of stainless steel, including manometer, stopcocks, and metallic pipes; the flexible tubing is made of Tygon; the graduated flow pipe is made of glass. The gaskets which ensure a tight grip of the flanges on the membrane are of polyethylene, polypropylene, or nylon. The porous partitions were the same as already employed by us in recent researches. A list of the membrane types used, together with some information about their principal structural characteristics relevant to the present research, is given in Table I. As already discussed by us22*23 membrane properties generally change with time, particularly during the first days of functioning. After this initial stage, they exhibit a good stability, being unaffected by the substanm employed in this study. This allowed the protracted use of the same porous sheet in a large number of experiments, consenting an excellent reproducibility. In most cases an apparatus was not disassembled proceeding to the study of a new solute, a circumstance which obviously helped to make the comparison among the behavior of different permeants more stringent and reliable. The apparatus was washed carefully after each experiment. Besides normal cleaning and rinsing of cells and of hydraulic circuits, it was found that a short run with pure water under a high-temperature gradient was necessary. For the same reason, before starting a new run,the initial filtrate through the membrane had to be spilled out, and the apparatus refilled, in this way dilution of solutions with the retained water can be avoided. Similar procedures were also applied with solutions in ethylene glycol. All chemicals used in this investigation were pure-pro-analysis grade, and the water employed for solution preparation and washing procedures was doubly distilled, prepared in our laboratory. The Closed-CellSystem. The experiments done with the apparatus of Figure 2, equipped with hydrophobic membranes were carried out in isothermal and nonisothermal conditions, with water or ethylene glycol and solutions of nonvolatile solutes in both solvents. In each case we investigated various nonequilibrium thermodynamic properties, namely, hydraulic permeability, pressure at steady state, and nonisothermal solute fluxes. In these different measurements the same membrane was employed until it remained operative and, in general, we tried to keep the experimental conditions similar as much as possible throughout this
study, to facilitate comparison of results. Since the experimentswith solutions were performed employing a new experimental technique where the absence of net volume flow is imposed, we will describe how these particular conditions were obtained. The simple expedient of closing all four stopcocks S1,S2, S3, and S4, after filling the cells with equal volumes of solution to abolish volume flux, is unaffective since membrane bulging on the low-pressure side cannot be avoided. A pressure gradient, due to liquid transfer into the cold semicell,induces forces of sufficient intensity to inflect the membrane and the supporting net toward its warm side. A sizeable fraction of the total volume thus may be transferred into the cold semicell during the initial stages of a run. In such a case the final composition of the liquids extracted from the two compartments will depend on the combination of an initial stage of transfer in presence of volume flux and a subsequent phase in which solute and solvent cross the membrane in countercurrent, subject to the condition Jv = 0. The interpretation of results would then become difficult since only measurement of initial and final volumes and concentrations can be done and the determination of initial rates of solute transport would not be possible. This would greatly impair the elaboration of experimental data, reducing the chances of gaining insight on the physics of nonisothermal membrane matter transport. To overcome this difficulty an alternative procedure was devised, which proved to be quite satisfactory. A preliminary run is carried out before each series of experiments, at the same average temperature and temperature difference chosen for the series. Equal volumes of solution are introduced into each semicell, all four stopcocks are closed, and the pressure increase with time in the cold chamber is followed on the manometer. Once the steady-state pressure difference is obtained, the solution is extracted from both semicells, and volumes are accurately measured. The apparatus is then refilled with fresh solution, introducing into each chamber a volume equal to that found in it under steady-state conditions; the pressure in the cold semicell at the end of this filling operation reaches its steady-state value. In this way net volume flow is eliminated, as can be checked by comparing final volumes with initial fillings. Component's fluxes in experiments carried out according to this method satisfy the condition jVtotal = jVsolvcnt + J solute = V 0 (15) (15 ) is analogous to the equation of thermodiffusive transport within a Soret cell. This fact justifies a phenomenological model based on the assumption that solute transport occurs by way of thermal diffusion.16J8J9 Let us now describe the results of experiments of solute transport in absence of volume flux, and then we shall proceed with the measurements of isothermal and nonisothermal volume flow in the same system. 4. Reaults
Solute Transport in the Absence of Volume Flux. The experiments in which the volumes of the two semicells are held fixed makes it possible to measure only final concentrations of the solutions extracted from the two compartments. Thus the solute fluxes that may be deduced from these data are time-averaged over the duration of the experiment. To evaluate initial fluxes, one must determine the time dependence of solute transport, by performing a series of experiments of different duration, under
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Nonisothermal Matter Transport in Porous Membranes Mcmbranc TF-200: T $
E''=30%; T
17.S.C : ATM,CI =
30.c
: ATIS*
~
2S.C
6
aver-
---
_______ ____-----
2
3
4
w)
-------
.-c w.r_n t .n t_i o.n_( ~. _C.) _I . _ . _
aver-
. _ . _ . _ . - . - . _ . _ . - . - . - . - . - . - e - - - . - . -
1
camantration (
5
6
7
t h x 10- 4( m )
Figure 3. Time-dependent solute concentrations in the two chambers, in the course of fixed-volume experiments. Membrane TF-200; NaCl aqueous solution (A);bovine serum albumin in universal buffer solution (0).
time x
IO-^( SOC )
Figure 4. As in Figure 3, but for aqueous solutions of poly(ethy1ene glycol) (2000 daltons) (A);phenylalanine in aqueous physiological solution (0) and human hemoglobin in phosphate buffer solution (0);the H b initial concentration is 4 X lo-* (mol an"), 3 orders of magnitude smaller than that of Phe and PEG 2000.
the same conditions. From a series of such measurements the initial solute fluxes can be deduced by extrapolation to t = 0 of the fluxes measured in the course of time. The best procedure to determine initial solute flux consists in plotting the fluxes, calculated by the method described here below, from experiments of various duration, as a function of time, on d o g paper. Such fluxes at t = 0 m r under the particularly simple conditions Ac = 0, J, = 0,AP # 0, and AT # 0. Reported in Figures 3-5 are some typical results of these experiments; the group of measurements reported in the fgures have
been obtained with a Gellman TF-200 membrane, employing solutions of electrolytes, amino acids, poly(ethy1ene glycol), hemoglobin, and bovine serum albumine, in the solvents water and ethylene glycol. Inspection of the fgurcs show some aspects which deserve being mentioned. The average concmtration increaseswith time in every case. This instrumental effect derives from the difference of the solution volumes filling the warm and cold compartments V, and V,, the former being smaller than the latter, as we said, owing to membrane bulging. Since solute flux proceeds from the larger
Gaeta et al.
6348 The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 Y . m M t F - l 0 0 : la, = 26%; AT=mZ
351 ~
CaCI,/W,O
a
301
n
I
I 25
I-
20
1
I
I
i 151
I I
time x
sec)
Figure 5. As in Figures 3 and 4, but for CaCl, in the solvents water (A) and ethylene glycol (0). respectively.
volume of the cold solution to the smaller warm compartment, the concentration increase in the latter overcompensates for the decrease in the former; this also accounts for the intriguing skewness of the curves. It should also be observed that solute transport continues until great concentration differences are produced but never proceeds until complete disappearanceof the solute in the cold chamber. In Figure 3 the much slower rate of transport of the macromolecular compound bovine serum albumin relative to NaCl is evident; accordingly the last concentrationratio measured at the end of an experiment lasting a full day is still far from steady-state conditions. In Figure 4 three solutes of different molecular weight in aqueous solutions evidence a difference of behavior, that unfortunately cannot be univocally ascribed to a single well-defined cause, because many circumstances, such as solution composition, viscosity, solute molecular mass, average temperature, etc., are all different. In Figure 5 the solute CaClz in two different solvents, water and ethylene glycol, evidences the great influence that the nature of the dispersing phase has on the observed phenomenology. From experimental data such as those of Figures 3-5 it can be clearly shown that net solute flux occurs in the experiments. To do quantitative measurements, this set of equalities can be used:
Af
where A[N(f)]/Af are the rates of change of the number of moles of solute in the warm (w) and in the cold (c) compartment; these are deduced from compartment volumes, Vw and Vc, and measured concentrations [C(t)lw and [C(f)Ic;SMis the effective membrane cross section. The experimentally determined third and fifth terms of expression 16 must be equal and opposite at all times for an experiment to be valid. Application of this equation to data such as those reported in Figures 3-5 shows that in each case net solute fluxes occurred through the membrane, from the cold to the warm side, in the absence of net volume transport. While the equality of the terms second to fdth in eq 16 holds good at all times, the deduction of solute flux from them is possible only in the fmt stages of the experiment, where a linear variation of concentration is observed in both chambers. Particularly the initial flux Js(t=O) is an important parameter than can be easily determined.
The graphical procedure adopted to this end is illustrated in Figure 6 and rests on the observed exponential time dependence of J,(t) that is also to be expected on the basis of both theoretical approaches. The graphs reported in Figure 6 confirm these expectations, besides yielding the values of [J(0)lifor each solute. At this point, if one wants to express the experimental results by means of phenomenological coefficients, reference should be made to the equations developed in section 2 on the basis of the two alternative approaches mentioned in the Introduction. According to membrane distillation (MD), solute transport occurs only by back-flow of solution, as prescribed by eq 3. Measurements of hydraulic permeability, to be described in the following, are needed to determine the phenomenological parameter A S figuring in the equation, therefore, we shall return later on to the evaluation of solute fluxes J i b by this approach. The radiation-pressure theory introduces the phenomenological parameters D,D’,and s to describe solute fluxes and component’s fractionation. Equation 16 allows us to calculate the [J8(t)liat any time f and by extrapolation, at f = 0; upon substitution in eqs 11 and 12, the values of D,D’, and s listed in Table I1 have been obtained. These are representative of the behavior of various solutes in the solvents water and ethylene glycol and refer to the membranes TF-1000 and TF-200 but can be considered also generally representative of the phenomenology of transport through hydrophobic porous membranes. ISOtbdandNooliootherrml Meesuremeats WithRwsdveata The apparatus represented in Figure 2 can be also employed to measure isothermal and nonisothermal permeabilities, as well as for the determination of the value of steady-state pressures. We performed all these measurements with pure water or ethylene glycol, and with some of the various dilute solutions used in this study. We have recently published the results of experiments of thermoosmosi~,’*~~ with water and aqueous solutions; it will thus be sufficient to summarize this part, concentrating only on some new aspects of the present work. Isothermal fluxes of the liquids contained in the semicells were produced by applying hydrastatic pressure across the membrane, playing on the levels of the reservoir R and flow pipe P. Volume fluxes were measured with the help of a stopwatch. A linear relationship was found in all cases between hydraulic flux J,hyd and driving pressure hp:
JVhyd= A,AP (17) Ai (mol dyn-I s-I) being the hydraulic permeability of the liquids
The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6349
Nonisothermal Matter Transport in Porous Membranes
r
loo
BSA
1
I
0.5
1 ,5
1
time x 10.'
2
2.5
3
(sa
Figure 6. Time dependence of drift velocities [U(t)li of various solutes through the nonisothermal channels of a TF-200 membrane. Solute fluxes [Jb(t)li were divided by solute concentrations to obtain the [U(t)]? Note the linearity of the semi-log plots, during the early part of the experiments, allowing extrapolation to t = 0.
TABLE II: Measured Values of Glycol" membrane system Gelman TF-1000
Gelman TF-200
and si and of the Ratio D:/si = Di in Solutions of Nonvolatile Solutes, in the Solvents Water and Ethylene
solution (molarity, M) KCl in H 2 0 (2.5 X NaCl in H20(2.5 X Phe" (salt sol) (5.0 X IO-*) PEG-2000 in H 2 0 (5.0 X lo-)) BSAb (salt sol) (1.6 X KCI in H 2 0 (0.1) NaCl in H 2 0 (0.1) CaClz in H 2 0 (3.0 X Phe" (salt sol) (5.3 X PEG-2000 in H 2 0 (1.0 X BSAO (salt sol) (1.6 X H b (salt sol) (4.0 X CaC1, in glycol (3.0 X KCI in glycol (0.1)
T,,, OC 25 25 35 17.5 17.5 30 30 25 30 30 17.5 17.5 25 25
AT, O 30 30 25 25 25 30 30 35 25 30 25 25 35 20
C
Di, cm2 s-I 1.70 X IOd 1.05 X 10" 5.5 x 10-7 1.7 X lod 4.6 x 10-7 5.1 X 4.0 X 3.7 x 10-7 2.4 X 2.4 x 10-7 5.5 x 10-7 5.0 X 1.5 X 1.9 x
Ofi,cm2 s-l K-I
5.3 X 10" 4.4 X 4.0 x 10-8 2.6 X 12.0 x 10-8 2.7 X 2.4 X 2.04 x 10-8 1.3 X 1.4 x 10-8 3.1 x 10-8 6.8 X 10" 7.3 X 6.45 x
(2/AT)(CW - Cc)/ (C, + Cc),K-I 3.12 X 4.19 X 7.27 X lo-* 1.53 X lo-' 2.6 X 5.3 x 10-2 5.9 x 10-2 5.5 x 10-2 5.6 X lo-' 6.0 X lo-* 5.6 X 1.36 X 10-lc 4.8 X 3.4 x 10-2
"Thesedata were obtained with Gellman, TF-1000 and TF-200 membranes, at the T, and AT values specified in columns 3 and 4. Phenylalanine in phosphate buffer solution, pH = 7.0. bBovine serum albumin (BSA) in universal buffer solution; its molecular mass is 68.000 daltons. CSteady-stateseparation has not been attained (see Figure 3 for BSA in TF-200 membrane). employed. These experiments were performed at various temperatures; the pressure and temperature dependence found by us shows that in every case hydraulic flow is visosity-controlled. When in place of a pressure difference a transmembrane temperature difference AT was applied, nonisothermal volume fluxes Jvn-isoth were observed: Jpisoth = BiA T (18)
Bi (mol cm-2 s-l OC-l) being the respective nonisothermal permeabilities. Experimentally determined values of Ai and Bi are reported in Tables I11 and IV. The data in the first one refer to water and to dilute aqueous solutions of electrolytes or of amino acids, the differences of isothermal and nonisothermal permeabilities among pure solvent and dilute solutions being negligible (separation effects have little influence on the volume fluxes, owing to the short time needed to do the measurements with water). In Table IV the data instead refer to pure ethylene glycol only, owing to the long time needed to measure Bi during which solute concentration gradients are produced in the cells. In the third column
of these two tables the ratios Bi/Aiin the same liquids are reported. These values should be equal to the steady-state pressures per unit temperature difference,22from eqs 17 and 18 indeed we have [(fl)Fisothlst= (B i / A i )AT
(19) With our apparatus we can also perform direct measurements of the steady-state pressure. Such data, reported in the fourth column of Tables I11 and IV, can be compared with those derived from eq 19. For the two liquids studied by us, the two sets of results agree reasonably well among them and exhibit a similar dependence on Tav. But, it can be seen that the steady-state pressures directly measured are systematically higher than (Bi/Ai)ATin the case of water, while on the other hand, for ethylene glycol the values of steady-state pressure are always lower than the corresponding (Bi/Ai)hT. We are not yet able to offer an explanation of this contradictory experimental behavior in two cases similarly involving a simple one-component liquid. The dependence of J v ~on- the~ temperature ~ ~ difference AT applied transmembrane has been also determined. The results obtained with the membranes TF-1000, TF-450, and TF-200 and
6350 The Journal of Physical Chemistry, Vol. 96, No. 15, 1992
Gaeta et al.
TABLE III: Hydraulic and Thermoosmotic Permeabilities Measured with Pure Water, in Six Types of Hydrophobic Membranes (Third and Fourth ~~"(AI)H mol dyn-? 09
membrane type TF- 1000
T,",
TF-450
TF-200
FGLP
Celgard-2400
Celgard-2402
O C
S-I
10 20 30 40 50 60 70 10 20 30 40 50 60 70 10 20 30 40 50 60 70 10 20 30 40 50 60 70 20 30 40 50 20 30 40 50
1.105 1.380 1.845 2.400 2.860 3.570 4.250 0.967 1.040 1.320 1.770 2.145 2.417 2.895 0.525 0.825 1.240 1.792 2.337 2.662 3.150 0.688 1.102 1.620 2.328 3.020 3.47 1 3.890 1.032 1.365 1.640 1.850 0.212 0.262 0.295 0.317
108(BI)H mol cm-y s-I K-l 4.05 5.90 8.01 11.21 15.36 19.62 23.16 4.64 5.23 8.64 13.28 18.78 26.85 39.20 8.87 14.80 22.39 30.70 40.88 46.37 25.90 7.98 13.05 20.25 27.00 37.10 41.75 48.50 1.91 2.71 3.54 4.46 1.41 2.02 2.60 3.15
9
(BI/A1)H dyn cm9 K-1 3665 4275 4329 467 1 5370 5496 5449 4798 5028 6545 7502 8755 11109 13540 1685 17939 18056 17132 17492 17419 17751 11598 11842 12500 11597 12284 12028 12468 1850 2029 2158 241 1 6651 7710 8813 9937
9
[(W*H,OI"/AT, dyn K-I 5120 5950 6035 6540 7 500 7970 7730 9336 9573 12320 14530 15894 21080 24986 24980 26400 27100 26300 26750 26100 26700 16238 16579 17500 16237 17198 16839 17455 3130 3509 3880 4050 11100 12800 14300 16250
OThe ratios of these permeabilities are given in the fifth column, while the steady-state pressures per degree of transmembrane temperature difference can be found in the last column.
TABLE I V As in Table UI. but for Pure Ethylene Glycol, in the Membranes TF-450 and TF-200
TF-450
TF-200
20 30 40 50 60 70 20 30 40 50 60 70
9.25 13.16 17.37 22.60 28.90 35.77 1.53 1.99 2.42 3.03 3.73 6.15
pure water are reported in Figure 7;a linear dependence appears as RP and MD theories predict. 5. Discmion The new experimental results obtained by this investigation will be now concisely discussed in terms of the alternative approaches of membrane distillation and thermal radiation pressure. We shall start by considering the evidence brought forward by the measurements of solute flux in the absence of volume flow. These findings cannot be treated by means of the original formulation of the membrane distillation theory but must be discussed in terms of the hybrid approach. In the closed-cell system the evaporation of solvent from the warm cell results in the establishment of a pressure gradient that pushes the solution back into the warm chamber through the liquid-filled channels. The
1.47 3.94 8.13 14.72 25.42 48.49 2.23 6.76 11.60 21.71 40.74 71.36
158 300 467 650 891 1356 1457 3396 4793 7161 10900 11598
127 145 190 247 317 489 670 923 1300 1770 2480 3190
transmembrane pressure difference [P h t h I s t causes hydraulic flow of the solution in bulk. Equation 3 in which A* has been replaced by the value derived from eq 17 then represents the resulting solute flux,expected to proceed from the cold to the warm chamber as experimentally observed. The only phenomenological parameter figuring in eq 3 is the hydraulic permeability A of the membrane. The measured values of this quantity for each membrane type and at all temperatures employed are given in Tables I11 and IV for solutions in water and ethylene glycol. By introducing these values together with the appropriate ([C(t)]& and A?' into eq 3, the expected solute fluxes may be calculated. The correct order of magnitude is obtained in the case of solutions in both water and ethylene glycol. The theory also forecasts the final solute concentration ratios to be produced by the process. These should turn out to be infinite, insomuch as the conveyor-belt
Nonisothermal Matter Transport in Porous Membranes
m I
I
I
,
I I ,
I
AT ( O C ) Figure 7. Nonisothermal water fluxes, as a function of transmembrane temperature difference AT, in the hydrophobic porous membranes TF-1000(A), TF-450(0)and , TF-200( 0 ) . Average temperature: T,, = 30 OC (-) and empty symbols; T,, = 45 OC (---) and full symbols. mechanism constituted by solvent vapor flow and solution counterflow continues to transfer solutes from the cold to the warm side until the concentration on the former drops to zero. However two further considerations can be made concerning this point. The first is that a term accounting for diffusive flow down the concentration gradient may be added to eq 3. This allows for a nonzero steady-state residual concentration in the cold chamber. The second is that the rate of diffusive transport is opposed by liquid flow within membrane channels; this will reduce or even obliterate the effect of diffusion. Solute transport observed in the closed-cell experiments can be also compared with the predictions of the radiation pressure theory. According to this approach, solute transport through the nonisothermal membrane is described by eqs 11 and 12 insomuch as it consists of thermal diffusion within solution-filled channels. It is expected that by introducing the experimental data into these equations, the resulting values of Di, D’,, and si should turn out to be comparable to the corresponding ones measured in ordinary thermal diffusion in the liquid in bulk. An examination of Table I1 shows indeed that thermal diffusion coefficients calculated from eq 11 for the initial solute fluxes through membrane channels are of the same order of magnitude as the corresponding ones in the liquid in bulk, ranging from 1 W to lO-’ (cm2s-’ K-I). Agreement of Soret coefficients with literature data instead is not as good, our values being generally larger than those measured by experiments in the liquid in bulk, that are of the order of K-l for similar aqueous solutions; no reference data are available for solutions in ethylene glycol. We proceed now to calculate the diffusion coefficients Di characteristic of the process. This can be done either by using the ratio D’,/s, or by applying eq 11 to couples of measurements effected in successive instants tl and t2 within the linear range. In either way similar results are obtained, that turn out to be systematically smaller than literature data. Evidently only if a slower process of diffusion occurs within membrane pores, can the high s values experimentally found be justified. Considering
The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6351 for a moment Table I1 and Figures 3 and 4, it may be also observed that the highest values of s are those relative to the macromolecular solutes, bovine serum albumin and hemoglobin. This agrees with the results obtained in ordinary thermal diffusion with macromolecular compounds and also corresponds to the theoretical predictions of the radiation pressure theory for the dependence of Soret coefficient on solute molecular mas^.^^,^ We can now proceed from the phenomenological equations inspired by the radiation-pressure approach to the heart of its physical reasoning. Inserting the experimentally determined D’, and vr into eq 14, one gets values of [(K/u)/ - (K/u),]; these differences result in the order of magnitude that would be expected by looking at the values of K/u for various common substances in Table V. However, it should be pointed out that the observed sense of solute flux is reversed with respect to the one obtained in a Soret cell. Indeed in our experiments the solutes were generally observed to flow from the cold to the warm compartment. It is interesting to observe that [(K/u),(Klu),] for a given solute depends on the nature of the membrane, Le., on pore size. On the other hand, the changes of this quantity in solutions of different solutes in the same solvent suggest that the K/u of the solutes stays constant in different membranes while those of the solvent depend on pore size. This circumstance, already pointed out by us,16 is also congruous with the hypothesis of Tasaka et al.,34 according to whom nonisothermal transport in porous membranes correlates with the “entropy of transport” of the liquid in the pores. This quantity in turn depends on the kind and degree of local (short-range) order in the liquid-which also determines the colligative properties of the material. Thus the K/u ratios are themselves expected to depend strongly on the type of pore walls-hydrophilic or hydrophobic-and within each of these two classes should also depend on channel radius. It should be finally mentioned that eq 11 can be written in the form of a differential equation, which upon integration gives an exponential time dependence of concentration in each chamber, the solute flux vanishing for t -. The conclusion is thus reached that both theoretical approaches correctly predict various aspects of solute transport, as shown by this study. Both theories however fail to give a rigorously quantitative interpretation of all the experimental features. This discussion has the heuristic merit of calling attention to the fact that the diffusion coefficient of solutes in liquid-filled membrane channels might have a value different from that measured in the solution in bulk. A higher value would make the RP theory not applicable to nonisothermal transport of solute in hydrophobic porous membranes; lower values would make the MD approach untenable. Measurements of isothermal diffusion in the membranes employed for the nonisothermal experiments are presently under way in our laboratory. Nonisothermal volume transport of liquids is now discussed. The MD model assumes that volume flux occurs through a process of evaporation at the warm side of the membrane and condensation at the cold one. Thus the force responsible for the process will be the difference of vapor pressure in the warm and cold chambers. The knowledge of the temperature dependence of vapor pressure will allow us to obtain the variation with average temperature and transmembrane AT of the nonisothermal flux. This can be done to a first approximation by using the Clausius-Clapeyron equat i ~ n We . ~ ~are now going to check the correspondence of eq 2 to our present results, reported in Tables I11 and IV, second column. This was done by fitting these values to an Arrhenius form using a computer nonlinear minimization program. Applying thii method to the data of Table 111, we have obtained good results for the TF-1000 and TF-450 membranes, where the x2 values at the minima were 3.67 X lo-’ and 3.61 X respectively and the activation energies 303 cal g-1 for membrane TF-1000 and 436 cal g-l for membrane TF-450. On the other hand, the results for the membranes TF-200 and FGLP were poor, with x2 values always greater than 10; the activation energies in these cases were 263 and 282 cal g-I. The fitting to a straight line of the data of the last two membranes has given better results: the x2 values being 1.28 for TF-200 and 1.22 for FGLP; then, in these cases,
-
6352 The Journal of Physical Chemistry, Vol. 96, No. 15, 1992
:
Gaeta et al.
L
*"I 3100
TPC)
Figure 9. Temperature dependence. of the ratio (k/lOHfl: ( 0 ) and (0) left-hand ordinate temperature dependence of [(l/T)(K/u)]Hp: and right-hand ordinate axis.
IEEE Trans. Son. Ultrason, SU-32,381 (1985)).] 0.28
0.30
0.32
0.34
0.36
Mean Temperature. l d 3 ( K - ' ) Figure 8. Top (a): nonisothennal permeabilities of water, plotted against 1/TaV(K-I) for membranes TF-200 ( 0 )and TF-450 (0). Bottom (b): as above, but for ethylene glycol; membrane TF-200 (0) and TF-450
(0).
as reported in other w0rks,22-34a linear behavior is established. The results obtained with the Celgard membranes also give a linear dependnece of Jv on Tav;they have not been inserted in our computer program however, owing to the small number of points available per membrane. The calculated activation energies are in all cases much lower than the heat of vaporization of water (540 cal g-I at 100 "C). The results for the first two membranes are somewhat better than all others. In Figure 8 we represent the experimental points (nonisothermal flux vs mean temperature) and the curves resulting from the statistical fitting procedure for water and membranes TF-450 and TF-200; an eye inspection of the figure confirms the facts previously commented. The data of Table IV,with pure ethylene glycol, show a nonlinear behavior. The fitting procedure yielded very good results: the xz value at the minimum is 2.77 X lo-' for the TF-200 membrane and 5.27 X 10-' for the TF-450 membrane; the number of freedom degrees is four for these two cases. On the other hand, the obtained activation energis were 209 cal g-l for the membrane TF-200 and 215 cal g-' for the membrane TF-450. These values may be compared with the value of 191 cal g-l of the latent heat of vaporization for ethylene glycol at 198 "C, its boiling point. In Figure 8 we have also drawn the experimental data with the fitting curves for ethylene-glycol and these two membranes; the good fit is evident. The MD theory does not explicitely predict the dependence of [(AP),"-hth]ston transmembrane average temperature; for the present time however this aspect will not be discussed. We now proceed to consider thermoosmotic volume flow Jvth and transmembrane pressure [(AP)lthIst within the frame of reference of the RP approach. Equation 9 (with x l and x2 taken on opposite membrane faces in each liquid-filled pore) assigns the steady-state pressure difference. The expected temperature-dependence of the pressure produced by the drifting phonons can be deduced from that of the quantities figuring in eq 9. The quantity (K/u)HP is plotted in Figure 9 and the constant H should be only slightly affected by temperature. In the case of ethylene glycol unfortunately only one value of K/u is available, at T = 25 OC,and thus no similar plot can be drawn. [Sound velocity of ethylene glycol at +25 OC = 1658 m PI (from A. R. Selfridge
Thermoosmoticwater flux on the other hand, is determined by the driving pressure [(AP/'-isoth]st as well as by viscous resistence, the latter exponentially decreasing with temperature. on T,, therefore is expected by The dependence of [(J ) RP to have the same t;e3'as the quantity [(1/v)(K/U)]H,O that is also plotted againt Tin Figure 9. As can be seen, the nonisothermal fluxes in membranes TF-1000, TF-200, and FGLP exhibit the expected trend; those in the TF-450 membranes, however, do not. The behaviour of [(Jv)HO]n-hth as a function of the transmembrane temperature difference AT, fully agrees with the expectations of the RP approach, as seen by the results in various membranes, reported in Figure 7. 6. Conclusion
This investigation of nonisothermal matter transport in hydrophobic porous membranes was undertaken with the aim of collecting experimental data that could be used to bring out the respective weak and strong points of the theories proposed to interpret the observed phenomenology. The original formulation of the capillary distillation approach must now be changed, insomuch as the proof is in our hands, that net transfer of nonvolatile solutes occurs through the channels of nonisothermal hydrophobic porous membranes. True enough, a pressure gradient is applied simultaneously to the temperature gradient, creating a different initial condition, to which the theory should be adapted, as indicated in the new a p p r ~ a c h . ~The ~~~' approach of Tasaka is compatible with all the experimental evidence gathered 50 far. However, being a thermodynamic theory, it cannot be of much help in the identification of the physical mechanisms causing this class of membrane phenomena. The modified membrane distillation theory and the phonon radiation pressure approach both survive after this work. However weak and strong points of each one are now clearer: the theoretical inadequacies or incompletenesses emerge and some key expcriments specified below can be proposed, that may falsify either one of the two theories. Let us then conclude with some short comments: Mute Fhme and Separations. The conveyor-belt mechanism of the MD theory correctly predicts the order of magnitude of solute fluxes but would lead to unlimited solute fractionation. A diffusive term should be added to eq 3 to account for the finite fractionation experimentally found; on the other hand, the diffusion coefficient of solutes should be measured in the membrane channels, to prove that it attains values high enough to account for diffusive fluxes that must proceed opposite to volume flow. The RP theory tries to explain thermal diffusion in the bulk liquid and nonisothermal matter transport in permeable mem-
The Journal of Physical Chemistry, Vol. 96, No. 15, 1992 6353
Nonisothermal Matter Transport in Porous Membranes TABLE V: Thermal Conductivities, Sound Velocities at Ultrasonic -tities for various md a Frequencies, and Ratios of Few Solids 104K,
substance water
methanol ethanol butyl alcohol propyl alcohol
acetone benzene
chloroform
n-pentane n- hexane
n-heptane
n-octane n-nonane n-decane carbon tetrachloride m-xylene o-xylene nitrobenzene
toluene cyclohexane chlorobenzene bromobenzene polystyrene plexiglass polyethylene glass (crown)
erg cm-l
lo%,
s-I K-' 5.818 8.084 6.387 6.587 6.743 2.078 2.022 1.965 1.719 1.660 1.601 1.531 1.530 1.688 1.594 1.520 1.386 1.470 1.560 1.640 1.209 1.169 1.128 1.240 1.180 1.110 1.280 1.230 1.182 1.394 1.403 1.407 1.440 1.370 1.320 1.260 1.420 1.366 1.295 1.447 1.381 1.316 1.074 1.036 0.967 1.341 1.297 1.363 1.315 1.523 1.494 1.433 1.605 1.248 1.265 1.138 -0.795 -2.093 -2.930 -4.200
cm s-I
Klu, erg cm-2 K-I
14.33 15.02 15.39 15.54 15.54 11.90 10.82 9.75 12.26 11.21 10.16 12.63 11.70 12.46 11.60 10.74 13.88 12.90 1 1.94 11.20 10.45 9.78 9.08 10.94 9.94 8.94 11.74 10.66 9.98 12.07 11.25 10.18 9.90 12.50 11.70 10.90 12.64 12.06 11.30 13.04 12.30 11.56 9.82 9.16 8.545 13.96 13.14 14.13 13.38 15.19 14.51 13.24 12.96 12.77 12.89 11.70 -23.50 -28.80 -21.20 -56.30
0.362 0.405 0.4 14 0.424 0.433 0.1749 0.1875 0.2020 0.140 0.148 0.158 0.1210 0.1308 0.1340 0.1375 0.1413 0.0998 0.1140 0.1320 0.1466 0.1 160 0.1196 0.1240 0.113 0.1 19 0.124 0.109 0.113 0.118 0.1168 0.1245 0.1350 0.1426 0.109 0.112 0.115 0.112 0.112 0.114 0.1 11 0.1 12 0.1 14 0.109 0.113 0.117 0.0968 0.0987 0.0965 0.0983 0.1003 0.1030 0.1028 0.1160 0.0978 0.0990 0.0973 -0.034 -0.078 -0.140 -0.075
T, K 280 300 320 340 360 280 300 320 280 300 320 293 293 280 300 320 280 300 320 336 280 300 320 280 300 320 280 300 320 280 300 320 330 280 300 320 280 300 320 280 300 320 280 300 320 280 300 280 300 280 300 280 300 293 290 290 290 290 290 290
branes of all kinds by a single radiation-pressure mechanism. Accordingly, reasonably similar values of the characteristic b e m ~ u s i v eparameters D;,and are in such merent systems. Indeed we have seen that .eq 11 yields the correct order of magnitude for D;, thus supporting this assumption. The fractionation of solutes experimentally observed is higher than expected, this however unless direct measurements of the isothermal diffusion coefficient in the solution-permeatedmembrane would prove that the latter is 1 or 2 orders of magnitude smaller than in the solution in bulk. On the other hand, the RP theory should also convincingly explain why solute fluxes in membrane
pores proceed from cold to warm, while thermodiffusive fluxes in the solution in bulk are more often observed to proceed in the opposite sense, with only a few documented exceptions.4348 Evidently one should determine experimentally the isothermal diffusion coefficient in solution-permeated hydrophobic membranes. Further stringent evidence should be sought on the mechanism that causes inversions in the sense of thermodiffusive fluxes. Nonisothermal Volume Fluxes. The MD theory predicts an exponential dependence of Jvn-isoth on average temperature owing to the exponential increase of satured vapour pressure. A linear dependence of Jvn-isoth on transmembrane temperature AT is claimed even if the difference of satured vapor pressure between warm and cold chamber exponentially depends on AT. The RP theory holds that the transmembrane pressure is given ). for water, according by eq 9, with (dT/dx) = ( A T / c ~ ~Hence to the data reported in Table V and Figure 9, the trend of [(AP)jn-isoth]st against T,, should be similar to that of (k/u)H20 and the trend of (Jyn-isoth)Hp analogous to the one of [ ( l / v ) ( k / 1()]H20. As for the dependence of thermoosmotic pressure and volume flux on transmembrane temperature difference, these should be definetely linear in every kind of membrane. The measurements of thermal fluxes in ethylene glycol coflirm the expectations of the MD theory; the measurements in water of J P w t hhowever confirm the expectations of the RP theory; furthermore, the dependence of these quantities on AT fully supports this last approach. There is an evident need to obtain further, precise measurements of steady-state pressures and of volume fluxes in dependence of T,, and AT. If possible these measurements should be done in the same porous membrane, treated to revert from a water-repelling to a water-wettable condition. Before closing, three more general arguments should still be concisely considered: (1) In the RP theory eqs 9 and 10, which should be employed for the calculation of thennwmotic pressures and fluxes, contain the quantity H,Le., a reflection coefficient governing phononparticle interactions; its definition in the present state of the theory is insufficient to allow rigorous calculations to be made. Theoretical and/or experimental progress is essential on this point if the theory has to make quantitative deductions from first principles. (2) Azeotropic mixtures have been fractionated by nonisothermal membranes. We have been the first to prove this to be feasible employing hydrophilic membra ne^;^^ it was thus shown that it is pcssible to separate by thennodialysis components having the same vapour pressure. Other authors extended our results to other azeotropic mixtures, employing hydrophobic membranes; these findings cannot be explained by MD theory. Some weight should also be given to the fact that the RP theory is quite general, its field of application ranging from thermal diffusion in the homogeneous liquid to nonisothermal transport in hydrophobic and hydrophilic membranes. The MD theory has a field of application strictly limited to membranes possessing hydrophobic channels. Solute transport owing to temperature gradients has been observed in permeable membranes of both kinds and in Soret cells, the drift velocity of solutes in a unitary temperature gradient being very much the same in all these cases. It appears quite unlikely that completely unrelated mechanisms, acting in different material systems could produce such qualitatively and quantitatively comparable effects.
Acknowledgment. We gladly acknowledge the invaluable assistance of Dr. Anna Maria Aliperti for revising the manuscript. Registry No. PTFE, 9002-84-0; PEG, 25322-68-3; Phe, 63-91-2; KCI, 7447-40-7; NaC1, 7647-14-5; CaC12, 10043-52-4;polypropylene, 900307-0; ethylene glycol, 107-21-1.
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