Non-isothermal mass transport in porous media

Journal of Membrane Science, 3 (1975) 191-214 m Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

191

NON-ISOTHERMAL MASS TRANSPORT IN POROUS MEDIA

F .S . GAETA and D .G . MITA International Institute of Genetics and Biophysics, Via Marconi 10, Naples (Italy)

Summary

Propagation of acoustic waves in non-isothermal dense media produces radiationpressures whose nature is analyzed here . The case of a flux of thermal - rather than acoustic - energy can be analogously treated, and a molecular theory of thermal diffusion is accordingly formulated . The special case of thermal diffusion occurring within the liquid-filled channels of a thin partition - "thermodialysis" - is quantitatively analyzed within the frame of reference of the radiation-pressure approach . Some apparatuses developed for the experimental study of thermodialysis are described . The results obtained are discussed and shown to be in accord with the theoretical expectations.

1 . Introduction Coupling of mass transport to a flux of thermal energy is known to occur in a variety of physical situations . Most interesting - and least understood are phenomena of this class taking place in dense fluids . We have developed a fundamental approach to the problem, based on the notion of radiation forces generated by the flux of thermal excitations, consisting in liquids mainly of high-frequency elastic waves (thermal waves) . Having demonstrated [1-4] the validity of this approach in the case of thermal diffusion in the bulk liquid, we have proceeded to extend the theory to the case of nonisothermal transport through porous partitions . The existence of liquid-filled pores crossed by a flux of heat, each equivalent to a microscopic Soret cell, makes the radiation-pressure theory of thermal diffusion applicable to this case . Because of the perturbation introduced by the partition, some modifications are to be expected . This straightforward approach leads to quantitative predictions which have been substantiated by experiment . Curiously enough, non-isothermal membrane transport - amply investigated in the more complicated case of selective partitions - has been relatively neglected for the simple situation treated by us . We demonstrate here the possibility of "uphill" transport powered by the radiation forces produced by the flow of heat, and also show the importance of modifications in solvent structure induced by the interactions with the



1 92

walls of the pores . The relevance and specificity of the observed effects to the nature of solute, solvent, and partition opens interesting perspectives of practical applications . 2 . Theory (a) Acoustic radiation pressure in non-isothermal liquids Let us consider here longitudinal plane waves propagating in a dense medium. We suppose that the medium is macroscopically at rest and that a temperature gradient dT/dy is superimposed in the direction of propagation of the acoustic waves, so that elastic parameters - and hence the velocity of propagation v of the disturbance - are position-dependent. The propagation equation becomes rather complicated, but its principal features are simple, as we shall describe in the following . In the case of a plane compressional wave in a liquid, the propagation velocity is v = -,1_ 01p, where $ is the compressibility coefficient of the liquid . In the presence of a temperature gradient, v will be both temperature- and position-dependent and the forward and backward particle displacements along the y axis, coincident with the direction of wave propagation, will no longer be symmetrical ; each plane will oscillate anharmonically around its rest position . The displacement of an anti-nodal plane of cross-section a o gives rise to forces resulting from the combination of a parabolic response (dp dT f2--CIO

ldTdy )A/4

plus a linear contribution f, = aoA[i /(A/4)] . (See Fig. 1 .) A more detailed analysis, accounting for the reaction of the liquid on both sides of the oscillating plane gives, for the resulting restoring force F, the value : 2

F- (+ Asf3

2A dTd YT t2 , ao

(1)

which is asymmetric in the forward and backward displacements of the plane . Here X is the wave-length of the perturbation in the medium . The resulting motion of the oscillating mass, which for unit cross-section is p(A/2), yields the equation of motion : 82E

47T 2ir 2 d$ dT _+_00t+_ ate Y(

2

= 0

(2)

which is the well-known equation of the anharmonic oscillator . Approximate solutions to eqn . (2) extended to terms of any desired order are easily obtained by an iterative method [5] . Since in general the amplitude ~o is very much smaller than X, we can neglect all terms of orders higher than



193

Fig . 1 . Asymmetrical curve showing combination of linear (f,) and parabolic (f,) factors affecting the restoring force on an anti-nodal plane in the presence of a temperature gradient in the liquid .

the second . Analytical expressions are thus obtained for the displacement resulting in the following approximate expression for the instantaneous particle velocity in the sonic field : 2102

r

4 o ~ R a I P Lf

-

5g o

2

r d13

dT l =

4813 2 , dT dy ,

1 J +

d13 dT

3713131dT dy 5 02 d13 d1T }2 l

4813 2

sin w 0 t +

(dT dy I J

2to '



(T do dy dT)

31s 7 + ) LI 480 2 dy Z J

sin 2w o t+ . . . .

(3)

Ca

where 4 R

r

5E02

d$ dT 2

WO=-

p L1- 4813 2 CdTdy~ At present we want to focus our attention on the calculation of the average rate of flow of momentum per unit area in the acoustic beam . This is easily done as follows :



194

O(P) _ d(me) _ dt

di

dm

m dt +

.

(4)

dt

where m is the mass per unit area and the bar over the expressions denotes time-averaging. Since dm /dt = p o i (P, being the unperturbed density of the medium) one has : (5)

~(p) = m + Po(€) 2 = P o(~) 2 at Introducing the expression for (r,) 2 derived from eqn . (3) the approximate expression is obtained : 2arg o2 p 27x 2 t o 3

do dT

a2

dT dy

3

A2

( )

The first term in the second member of eqn . (6) represents the average rate of flow of momentum in an ordinary beam of plane acoustic waves propagating in the medium in its isothermal condition . The second term (together with others of smaller magnitude, discarded here) represents the effect of the anharmonocity produced by the existence of the temperature gradient . Just as the first term is responsible for the production of the well known Rayleigh [6,7] radiation-pressure on surfaces emitting or absorbing the acoustic energy of the beam, the second term will be responsible in turn for the appearance of a further radiation-pressure-effect due to the anharmonicity of the oscillations . While the first term is a scalar, it is interesting to observe that the second term has a vectorial character, owing to the presence in it of the temperature gradient . Its magnitude, however, will be less than that of the first term, since to is in general a very small quantity . Substituting in place of T2 the value

u2/ V 2 = ? a V2

P

where v is the phase velocity of sound and v is the frequency, and remembering the definition of the density of acoustic energy E = 2a 2 pv 2 Eo one can write eqn . (6) in the form : -

E E } o ( do dT 1

O(P) = a +

3 R

`dT dy /

(6 )

from which it is easily seen that both radiation pressures are proportional to the energy density . For reasons which will appear in the following, we prefer to write the above expression in terms of the acoustic intensity I = 0(E) = E -v rather than of E, obtaining in this way :

1 95 4(P) = 0(E) + O(E) t o

av harmonic

r df dT

3u (i 1dTdy )

(7)

anhatmonic

(b) Radiation pressure of thermal waves and thermal diffusion At this point a short digression on the physical nature of thermal excitations in liquids is necessary . It is reasonable to assume that thermal energy in liquids consists mostly of very high frequency elastic waves, probably distributed in a Debye spectrum up to frequencies of the order of 10 12 Hz [1] . In the light of this last observation and of what we have written above, one is led to conclude that the application of a temperature gradient to a liquid, besides provoking a flux of thermal energy, also creates the condition in which this flux of high frequency waves couples with a flux of momentum . It is well known that the density of flux of thermal energy 0(Q) is equal to K dT/dy, where K is the thermal conductivity . By analogy with eqn . (7) 0(Q) being equivalent to 0(E) - the density of momentum flux coupled to the flow of heat will be obtained by dividing 0(Q) by v, this now being the propagation velocity of thermal waves . Brillouin light scattering experiments have shown that v has values very close to the velocity of acoustic waves, and therefore the density of momentum flux in our case is given by : 0(p)=H

K dT dy

v

(S)

which is - apart from the proportionality constant H - identical to the expression previously derived by us [1,2] from purely dimensional considerations . The proportionality constant H in the present case must be derived experimentally, because of the lack of a sufficient knowledge of the energy distribution in the spectrum of thermal waves . Both terms of eqn . (8) have the physical dimensions of a pressure . Therefore, wherever the density of momentum flux is altered, for whatever reason, radiation pressures will appear of magnitude equal - in the proper units - to the variation of 0(p). A flux of thermal energy is expected therefore to give rise to radiation pressures on every material surface limiting two regions characterized by different values of K and/or v . For instance, at the interface between two different media A and B, a pressure 7r(A a) will appear, equal to the variation of 0(p) : fK dT rK dT )A dy (9) n(A,B) ` [~0(P)](A .a~ =H[ lv dy `v )aJ



196 From this last expression it is evident that radiation forces will act on the interface unless H = 0. The actual value of H can be determined by experiment, as we shall see . The vectorial character of the terms appearing in eqn . (9) gives rise to a clear-cut difference between the behavior of our radiation pressure and Rayleigh's pressure, the difference being in the scalar quantity E . When a slab of a solid material is immersed in an isothermal liquid and elastic waves are propagated through the system, the pressure is always directed from the medium in which the energy density E is higher to the medium in which it is smaller . In either case the flux of acoustic energy would generate equal and opposite forces on the opposite faces of the slab [8] . If a temperature gradient is applied in the direction of propagation of the waves, the anharmonic term of eqn . (7) would instead contribute a nonvanishing force. Analogously, a flux of thermal energy producted by a temperature gradient superimposed on the liquid and on a slab of solid material immersed in it will generate measurable forces on the system constraining the immersed slab . These forces in the case of a solid, flat disc are given by : Fs = [('sl,s)sup -

=-H

=-2H

( nl,s)inf ]as

R-dy - `Kdy)BJ ` K ay)1 \----J

(Kay~ s -

( v dy) l ]~ a $

I a8

where subscripts `sup" and "inf" stand for the upper and lower face of the disc, I and s denote liquid and solid, respectively, while a 8 is the crosssection of the disc . The minus signs of the second and third terms derive from our convention that the y axis points in the direction of wave propagation, while the temperature gradient is opposite to this direction . Interestingly enough, the sense of the force F will be equal or opposite to that of grad T, depending upon whether \K \a dy's V dy/1 § We have obtained ample experimental verification of all these theoretical expectations in [9,10] . Let us now consider a solution or a suspension of particles contained in a cell of height a whose top and bottom consist of two plane, parallel, horizontal metallic plates, maintained at temperatures T, and T,, respectively . The particles will drift toward one of the two plates along the direction of the temperature gradient, because of forces developed by thermal waves. The solution will consequently concentrate at one end of the non-isothermal



19 7

liquid column . This is the well-known Soret-Ludwig effect (or thermal diffusion), whose driving force in our approach is given by : F=-2H

I (K d) 1 - Kay )p

lap

(10')

where subscript p now stands for "disperse particle" of section u p . In the phenomenological theory of thermal diffusion the concentration ratio Cc/Cw of the components in the cold and warm part of the solution can be expressed through phenomenological coefficients called "Soret coefficient" S ['K- '] and a "coefficient of thermal diffusion" D' [cm' sec - ' ° K''] In our theoretical approach it is possible to represent these coefficients as well as the concentration ratios in terms of the radiation force F of eqn. (10') . In the case of a dilute solution, after a running time long enough to approach steady-state, in a Soret cell of height a the concentration ratio will be given by [1] CC

C.

K

= exp

dT

K

dT ,f

ao (11)

2H[

'u dy ) 1 - ~a dy'p

`XTav

where `X is the Boltzmann constant, 'X T av expressing the randomizing effect of thermal agitation . Any inversion in the sense of the radiation force, therefore, causes an inversion in the sense of migration of the disperse particles . Our experimental investigations of thermal diffusion have given convincing proof of the above theory [10-14] . (c) Thermal diffusion across porous partitions Let us now consider the case of a non-selective porous partition sandwiched between two solutions kept at different temperatures . The partition is assumed to be completely permeable to all components of the solution, that is, utterly unable to discriminate between solute and solvent or among different solutes . Our operative definition of such a porous partition is that of a (thin) sheet of any solid material containing anastomizing cavities or "pores" in contact with both faces of the sheet ; sometimes these might even be straight channels across the thickness of the partition . If the dimensions of the pores are small enough to effectively quench convective turbulence within the liquid filling the cavities, then each pore constitutes a microscopic Soret cell. It is interesting to observe that thermal diffusion in this situation will very quickly bring about separations of the components . Indeed, for one thing, the temperature gradient will be high when the thickness of the partitions is small ; also, the characteristic time B for the establishment of equilibrium will be short, because of the very small distance along which diffusion must take place . It is well known indeed that in a Soret cell B = a'/fr 2 D, where D (cm = sec - ') is the ordinary diffusion coefficient and a is the height of the liquid column . Even if D in the channel has a value different from that in the bulk liquid because of the tortuosity effect, the above argument still holds .

1 98

The interposition of the porous partition between a hot and a cold solution will modify in various ways the ordinary process of thermal diffusion . Precisely : (i) As already mentioned above, the value of the temperature gradient will be greatly increased if the partition is very thin . The whole temperature difference, indeed, will be confined to the thickness of the partition, if vigorous stirring is provided . Correspondingly the thermal diffusion effect (which according to eqns. (10') and (11) increases with the temperature gradient) will be enhanced . (ii) The flux of thermal energy will be divided between the solid material of which the partition is made and the liquid contained within its pores . The density of heat flow will be higher within the medium of higher thermal conductivity . Thermodiffusive transport, accordingly, is expected to be particularly relevant across very thin septa and - everything else being equal - increasing with increasing difference between the thermal conductivity of the liquid and that of the medium constituting the partition . (iii) Another effect, subtler than the ones just described, will manifest itself whenever the pore diameter is small enough for the entire liquid volume contained in each channel to be affected by the interactions with the surrounding solid wall . The structure of the fluid in this case is altered relative to the liquid in the bulk, through the action of the forces of adhesion [15-18] (effect of epitaxis) . Many physical properties of the epitaxed liquid are appreciably modified, especially the transport properties . The thermal conductivity of water, for instance, has been reported to increase more than one-hundredfold when enclosed between nearby mica surfaces [ 18] . It is likely - even if not experimentally ascertained - that the velocity of high-frequency waves in the liquid is also affected by the proximity of the surface . The increase of thermal conductivity - per se - entails greater radiation forces because the density of the thermal energy flux is increased . The variation of ul also affects the v L ku

ay ~l - l

dy'p

term and thus influences the value of F. In this connection it is important to observe that, while the molecular rearrangement taking place in epitaxis in general only involves a few molecular layers immediately adjacent to the solid surface, the modifications of the spectrum of thermal waves that such a rearrangement entails may have longrange effects . Indeed, thermal excitations travel over distances ranging from a few to some tens of wavelengths, before being modified by inelastic interactions with the disordered liquid lattice [3] . This means in our case that throughout molecular layers at least some 10 3 A thick adjacent to the solid surface the spectrum of thermal waves will be modified with respect to the spectrum characteristic of the bulk liquid . It is therefore very reasonable to



1 99

3

assume that within channels having diameters of the order of 10 A, the liquid is in a state altogether different from that of the bulk liquid . Such effects will also be felt in larger pores to a certain extent . Because of the influence that epitaxis has on the physical characteristics of the medium, eqn . (10') must be re-written to describe thermal diffusion in narrow pores as follows :

K dT rK dT (12) (u I e - , ) , a p c+> u dy PM dY where the star on the subscripts of K/u and dT/dy indicates that the correF = -2H

sponding value of the quantity is the one relative to the epitaxed liquid, while the star in brackets indicates that these quantities might also be affected (though generally to a much lesser extent) by epitaxis, through the influence of the altered state of the liquid in the pores on the degree of solvation . It is evident that expressions analogous to eqn . (11) can be derived for the present case from eqn . (12)' . From the above, various predictions can be made concerning the behavior of material transport across porous partitions in which thermal energy is flowing. Let us then state explicitly some consequences of the theoretical approach discussed above . In the first place, the existence of a driving force produced by the temperature gradient, while not surprising in itself, being a well-known tenet of nonequilibrium thermodynamics, results in the present context from the analysis of a precisely defined molecular mechanism . The existence of such a driving force in turn entails the possibility of transport proceeding "uphill", that is against gradients of electrochemical potential and hydrostatic pressure . The extent to which this kind of transport may continue against opposing gradients will depend solely on the magnitude of the externally driven flux of thermal energy . On the other hand, more specific predictions can also be made concerning the behavior of thermal diffusion across porous septa, on the basis of a critical examination of the nucleus of eqn . (12) . Indeed, the quantity : [{

)1*-

luf ay ) pc.)J

(13)

determines the sense of thermodiffusive drift in the pores relative to the `The possibility that a single kind of equation, like eqns . (10') or (12), might describe transport in wide as well as in narrow channels, is also established by the thermodynamic theory (see for instance the treatment of "thermal effusion" in ref . 24) . The reasons for both the similarities and the differences between the two cases are essentially the same in our treatment as in the phenomenological theory . These consist in the modifications of the modalities of transport induced by the interactions of the permeating medium with the pore walls . In our approach, however, a well-defined molecular mechanism is proposed for such interaction .

200 sense of heat flow, as in the case of the bulk liquid . Because epitaxis alters the values of both K and v of the continuous phase (and to some extent perhaps also the values in the disperse phase) drastic changes in the phenomenon might be observed in thermal diffusion in narrow pores relative to thermal diffusion in bulk . We have seen that the thermal conductivity of a liquid is drastically increased by epitaxis [18 ) . It is reasonable to assume that the ratio (K/v)I is also correspondingly increased, which means that (K )I* > (K

)t

Let us now consider the case of a solution in which the disperse particles have a characteristic (Klv)p which is smaller than the value in the bulk liquid . In such a case the particles will thermodiffuse at a higher rate from warm to cold across the channels filled with the epitaxed liquid because

[(Kay ) I * -

(Kd~)p(*)~ >

)I - (K

[(K

)p 1 >0

(14)

In the case, however, in which (K/u)p is greater than (K/v)I and the particles drift against the heat flow in the bulk liquid, they will do so at a decreased rate in the epitaxed fluid if : L( K v

ay)I*

(v dy) p t*,J <

a

L(K y )I V

\v dy)p

]<

0

(15)

There is a special case possible, however, when the increase of the ratio (Klu) of the liquid produced by epitaxis is so great that (Klv)I* becomes greater than (K/u)p : (I),* >

(K)p > (K) I

(16)

In such a case, the following striking behavior is expected : the particles will drift in opposite senses in the liquid in bulk and in the channels filled with the epitaxed fluid . All the above predictions are clear-cut and can be easily submitted to experimental verification . In conclusion, our theoretical approach implies aspects which are distinct from the case of thermal diffusion in the bulk liquid and also utterly different from what could be expected by extrapolating current theories of thermoosmosis to non-selective porous partitions . As we have shown, new effects are foreseen, some of these never observed before in thermoosmosis . These are sufficient reasons to consider this form of transport as an independent phenomenon, for which we propose the name of thermodialysis, as distinct from both thermoosmosis and ordinary thermal diffusion . It is essential now to see whether the theoretical approach is substantiated by experiment .

201 3. Experimental (a) Apparatus The present investigation was undertaken with the purpose of ascertaining the existence and the characteristics of uphill transport produced by the flow of heat across some porous partitions . Furthermore, evidence was sought which could substantiate the supposed role of epitaxis in thermodialysis . Schematically our experimental devices consist of two containers separated by, and communicating through, a porous partition . The two liquid volumes are maintained at different temperatures, each held constant throughout the experiment. In this investigation we used three different types of apparatuses . The first one consists of two containers, separated by a vertical porous septum sandwiched between them (Fig . 2) . A metal structure holds the parts together . Heating and cooling are provided through circulation of externally thermostated fluids . Thermocouples introduced near each face of the partition give continuous readings of the temperature difference across the system . Continuous stirring is provided by magnetic stirrers, or by other suitable methods, to ensure as much as possible the uniformity of temperature within each container. This same stirring also avoids the formation of concentration gradients in the free liquid . Volume flow across the partition alters the levels of the liquid in each container so that hydrostatic pressure differences are generated . During the run samples of the solutions can be extracted for analysis of the concentrations of the solutes . Another type of apparatus used by us is represented in Fig . 3 . It consists of two cylindrical containers made of pyrex glass, each one having a flat flange . The horizontal partition is held tight between these two flanges . A metal

Fig. 2 . "Open top" apparatus for thermodialysis, employing porous septa positioned in a vertical plane . This very simple version is better employed for semi-quantitative work . It allows large volume flow and gives easy access to solutions on both sides .

20 2

Fig . 3 . One version of our cylindrical horizontal membrane apparatus complete with Vertex and thermocouples . The lower cooling system, tilt-plate, and registering units are not shown. The scale is added to give an idea of the real dimensions . This apparatus allows measurements of the hydrostatic pressure produced in the process, by opening the stopcock at the bottom of the manometric pipe . When this stop-cock is closed, constantvolume measurements can be effected .

structure holds the two sections together . The solution contained in the upper section is heated, and the one in the lower section is cooled . The flat bottom of the gold-plated heater can be adjusted at fixed distances from the partition ; a "Vertex" ensures the regulation of the temperature . Cooling of the lower section was in some cases obtained by immersion in a thermostated bath, and in others by a Peltier cooling element external to the container . A thermocouple system is used to monitor the temperature difference on the two sides of the septum . The whole apparatus rests on a tilt-plate allowing the partition to be set horizontal, or to be slightly inclined, to ensure sufficient convection on each side of the septum so as to eliminate concentration and temperature gradients . This apparatus can be used in two ways . As can be seen in Fig. 3 the lower container is connected through a glass stop-cock to a capillary manometric pipe. With the stop-cock closed, the transport phenomenon can be studied in the absence of volume flow . When it is open, the pressure across the porous partition can be measured as a function of time by observing the level of the liquid in the manometer . The net volume flow in this case is limited to the small manometer volume.

2 03

The third kind of apparatus used in this investigation consists of an adaptation of a thermogravitational apparatus. This apparatus was originally developed by Clusius and Dickel [19,20] with the aim of improving thermal diffusion in liquids. It consists of two parallel flat plates between which the solution to be subjected to the temperature gradient is contained . Because of the fact that the two surfaces are nearly vertical - (the apparatus is slightly tilted, with the hot surface higher than the cold one), - a regular convective flow develops, the hot solution moving upward counter-current to the cold solution . This counter-current system very much improves the separations obtained by thermal diffusion . Such an apparatus has been described at length in our papers on thermal diffusion [10-12] . To adapt the same principle to thermodialysis, we have built an apparatus which we shall call here "double thermogravitational" (Fig . 4). The temperatures of the two halves of the apparatus are held constant by circulating a liquid (externally thermostated) through some 45 channels machined in the back of each plate . Two cylindrical reservoirs are provided in each plate, from which the enriched and depleted solutions can be drawn . This apparatus was found to work best when positioned almost horizontally, the hot plate being above . An inclination of 5 ° to the horizontal was found to give the best results . The system of convective currents developed in the apparatus is represented in Fig . 5, where the mass transport produced by thermodialysis and thermal diffusion are also indicated . The descending liquid current on the hot side of the septum exchanges matter by thermodialysis with the cooler ascend . ing liquid current on the other side of the partition . This counter-current system ensures a constant value of the temperature difference across the membrane . It also causes a continuous substitution of the solution enriched or depleted by thermodialysis with new fluid drawn from the reservoirs . Con-

MM side

7

cold

out

in

Fig. 4 . Section of the double thermogravitational apparatus . The porous partition MM' separates the cold from the warm half, each one equipped with semi-cylindrical reservoirs and quartz windows designed to allow continuous optical measurement of the variation of the concentrations with time .



204

Calibrated teflon gasket

T s^

M R,

e

a© n nnir n a _as aee _ as _

Fig . 5 . The "core" of the double thermogravitational apparatus is schematically represented here (with the various dimensions not in scale) so as to allow visualization of the flow of convective currents at the two sides of the porous partition MM' . The flow of the solute is also represented (thick arrows) in the simple case where the direction of migration of the disperse phase is from the warm to the cold region, both in thermal diffusion (thick solid arrows) and in thermodialysia (thick unshaded arrows) . Temperature T, is higher than T, . This fact produces in the actual experimental situation a system of laminar convective currents like the ones represented in the figure . The whole apparatus rests on a tilt-plate ; the optimal angle of tilt is =5' . centration gradients forming in the liquid layers adjacent to the porous partition are promptly removed, and the process of thermodialysis is thus strongly enhanced . At the same time, the process of ordinary thermal diffusion which takes place in each of the two halves of the apparatus concentrates or dilutes the solution while it is flowing parallel to the porous partition . The ratio of the concentrations in the two reservoirs of each half of the apparatus C w e /C w i and Cc,s/Cc,i are thus produced by thermal diffusion in each thermogravitational column, while the ratio between the average of the concentrations in each pair of reservoirs _ (Cw,s + Cw,i)

(Cc,s + Cc,i)

represents the effect of thermodialysis . Here the subscripts "w" and "c" stand for "warm" and "cold" side, and "i" and "s" for "inferior" and "superior", respectively . (b) Materials and methods In the preparation and performance of a run, we found that much care must be paid to certain apparently trivial details, in order to obtain good reproducibility of the experimental results . The handling and preparation of the partitions or membranes before a run is critical, and in general previous bathing of the porous element in the solvent to be used in the run produced much more reproducible results . When the solvent was water, we used double distilled water . When the solvent was other than water pure "pro-analysis" grade was used . Solutes were of similar grade . The macromolecular solute polyvinylpyrrolidone was purchased from FLUKA . Switzerland .

205

We also used a variety of porous septa . The glass-fiber ones were AP Millipore ; HAWP mixed cellulose ester filters having pores of 0 .45 ,urn were also obtained from Millipore corp ., Bedford, Massachusetts, U .S .A . Nuclepore membrane filters were obtained from Nuclepore Corporation, Pleasanton, California, U .S .A . with a wide range of calibrated pores in the range from 300-80,000 A diameter . "Metricel" membranes such as grade GA4 (made of cellulose triacetate) and ` versapor epoxy" membranes were obtained from the Gelman Co ., Ann Arbor, Michigan, U .S .A . The concentration variations of both the enriched and depleted solutions at the two sides of the porous partition were assessed in various ways, depending on the properties of the disperse and continuous phases . Whenever possible, we used spectrophotometric measurements, performed generally with a recording Beckman DB/GT spectrophotometer and occasionally with a flame spectrophotometer . Occasionally chemical titration or refractometric methods were also employed . During each run (with the obvious exception of those carried out at fixed volume) we also continuously recorded the hydrostatic pressure difference produced . 4. Results (a) Thermodialysis as a process of "uphill" transport From our theoretical approach it follows that material transport across a porous partition mediated by a flow of thermal energy may be a process of "uphill" transport . Thus we also expect such transport to take place against gradients of electrochemical potential and/or differences of hydrostatic pressure, since in our treatment the temperature gradient actually produces forces of well-defined characteristics . We shall start by describing an experiment of a semi-quantitative kind confirming that thermodialysis is a process of uphill transport akin to thermal diffusion . If a water solution of methyl violet (10 mg/1) is placed in two containers separated by a glass fiber partition and the contents of the two containers are brought and maintained at a few degrees above and below room temperature (25°C), in an apparah of the type represented in Fig . 2, a flow of water takes place from the hot into the cold container, and a flow of solute takes place at the same time in the opposite sense (Fig . 6) . The final concentration ratios, CW IC,, of methyl violet in the warm and cold solutions are higher than the ones which would be expected from the observed flow of water into the cold container . The volume flow, if constituted by a current of pure water extracted from the warm solution, would give C,/CC = 1 .6 after 100 min in the experiment in which the AP-20 septum was used . The Ca,ICc actually observed is 2 .2 . Similar considerations apply to the experiment with an AP-25 septum . The two types of partitions differ only in thickness (respectively about 0 .30 and 0 .85 mm after swelling) . The flow of water



206

a = Values of AP (lest-hand scale) 60

x = Values of concentration ratios (right-hand scale)

3.5

3.00

25 a

0

.5

100

200

300 400 Time ( min)

500

Fig . 6 . An aqueous solution of methyl violet is run in an apparatus of the type shown in Fig . 2, using either a Millipore AP-20 or AP-25 glass fiber porous septum . These two septa differ only in thickness (about 1 to 3) . In either case the average temperature and temperature difference are the same : T 4„ = 30°C, aT = 25°C. Plotted in the figure are the hydrostatic pressure difference and solute concentration ratio vs . time .

also produces a pressure head against which transport continues . The solvent is transported in this case against a strong concentration gradient . The solute, moving in the opposite sense, also proceeds from the dilute into the concentrated, hot solution . In Fig . 6 the time dependence of the concentration ratio and of the observed hydrostatic pressure are reported . In the run with the AP-20 septum after some 100 min the solvent flow is reversed (and the pressure head decreases) until, after 200 more min, a stationary pressure difference is established . The flow of solute from the cold into the warm container, however, continues throughout the wholwrun . The onset of the final solute transport rate (after 300 min) coincides with the end of solvent transport . It is interesting to note the relative independence of the rate of solute transport from the rate of volume transport (the rate of increase in time of the pressure head is obviously proportional to the velocity of volume flow) . Only slight inflections in the Cw 1Cc curve appear at the point of reversal of the volume flow and at the end of volume flow . Analogous results were obtained with the other septum . An alternative to this experiment is obtained by putting the solution of methyl violet into the hot container and cold distilled water on the other side of the membrane . Even with a relatively small temperature difference of 15 °C it is possible to extract pure water from the solution against both the

2 07

osmotic gradient and the hydrostatic pressure head which gradually builds up . No trace of methyl violet is found in the cold container, within the limits of detectability of our spectrophotometric analysis. (b) The effect of epitaxis We have performed many experiments with the aim of detecting the influence of epitaxis on transport through non-isothermal porous partitions . In the case of disperse particles, which in the bulk liquid thermodiffuse to the cold regions of the solution, expression (14) predicts that such behavior should be enhanced in the presence of epitaxis . In Table 1 we compare some results of experiments of thermal diffusion with results of thermodialysis made employing various types of solutes (second column) and of porous partitions (fifth column) . These results can be directly compared because of the nearly equal distance over which the temperature gradient is applied in the two cases. This distance is also the length of the path along which thermodiffusional transport occurs . It was possible to obtain such a situation by using a "thermogravitational column" in place of a conventional Soret cell . TABLE 1 Results of experiments of thermal diffusion, performed with a thermogravitational column, are compared with those obtained in experiments of thermodialysis . The membrane apparatus was of the type represented in Fig . 3, with the manometric pipe excluded . These runs of both thermal diffusion and thermodialysis were executed applying a A T of 20° C at an average temperature of 30 °C . The duration of runs was in all cases 4 h . The temperature gradients in the thermogravitational column and across the porous partitions are of similar magnitude Continuous phase

Disperse phase

Water

Theophylline

Water

Methylene blue

Water

Propylthiouracil

Water

Uridine

Water

Caffeine

Thermogravitational column : separation ratios Cc/C w

Separation ratios in thermodialysis Porous partition

Separation ratios

10 10

1 .15 1 .15

H .A.W.P . Collodion°

2 2

1 .20 1 .20 1 .07 1 .07 1 .07 1 .12 1 .12

Cellulose° MANY. H .A.W .P. GA .4 Collodion° H.A .W .P. Collodiona H .A.W .P. Collodion°

1 .94 1 .62 1 .52 1 .22

Initial concentration C° (mg/1)

8 8 8 10 10 10 10

1 .16 1 .16

1 .21 1 .52 1 .12 1 .73 1 .15 1 .00 1 .24

a Holes produced in these membranes by electric discharge were found to have diameters in the range from 0 .8 to 2 .2 µm .



208

The results reported in the fourth column of Table 1 are representative of the thermodiffusive behavior of the respective substances in the bulk liquid, at values of the temperature gradient comparable to the ones obtained in the case of the porous partiton (sixth column) . Coming now to the situation described by relation (15), in the particularly significant case to which the inequality (16) applies, we tried to obtain unambiguous experimental evidence supporting such a rather surprising theoretical prediction . We require an experimental situation in which a large change in the transport rate is observed with a variation of a single parameter affecting the structure of the liquid . According to our hypothesis concerning the role of epitaxis, the pore diameter can be such a parameter . With decreasing pore diameter there is a closer interaction of the fluid contained in the channels with the walls and hence a change in the structure of this liquid relative to the bulk liquid . Since a variation of pore size also affects the hydrodynamic resistance to the fluxes across the channels, a decrease of transport produced by a decrease of pore diameter would not be unequivocably attributable to the hypothized increase of the K/v of the liquid induced by epitaxis . Only the observation of a reversal in the transport direction followed by an increase of its rate, with decreasing pore diameter would be conclusive . We have studied with the apparatus of Fig . 3 the thermodialysis of solutions of polyvinylpyrrolidone K-25 of 24,000 molecular weight in n-butanol across Nuclepore polycarbonate membranes . These membranes are available in pore sizes from 80,000 A down to 300 A . Each type has cylindrical pores of a

r r 20

16 aB lr, si OR

4

4610' I1e r,,

3IOP

410 .

610 4

6M''

6104

PORE DIAMETER (ANGSTEOEG)

x`> >

v

20

`.

c .. Fig . 7 . Concentration ratios produced by thermodialysis of solutions of polyvinylpyrrolidone K-25 in n-butanol, plotted against membrane pore size . Duration of runs 6 h ; temperatures of warm and cold containers +40 ° C and +20 °C, respectively ; membrane thickness 10"' mm (except for membranes with pores of 800 A, 500 A, and 300 A, those being 5 X 10' mm thick) ; Cw. final concentration in the warm container ; C, final concentration in the cold container .

209 single diameter, extending from face to face . The concentration of the macromolecular solute (16 mg/1), the average temperature Ta. = 30°C, and the temperature difference AT= 20°C, were kept constant in all runs . Polyvinylpyrrolidone macromolecules in n-butanol move from the cold toward the warm part of the solution in ordinary thermal diffusion [10,13] . Therefore, according to eqn . (10') the value KdT\

T W11 relative to the bulk liquid will be smaller than the one for the suspended particles ; the quantity

ay

- `v

Vp

~

is therefore negative . When this same solution is run in the apparatus shown in Fig . 3, employing Nuclepore membranes of decreasing pore sizes, the results represented in graphical form in Fig . 7 are obtained . All membrane properties were identical throughout the series, except for pore density and - in the case of the membranes with pores of 300 to 1000 A - also for thickness. These differences of thickness and of pore density obviously affect the values of the observed separation ratios, but their influence can be easily accounted for. The observed behavior is exactly what we expect to find in the presence of epitaxis . (c) Experiments with the double thermogravitational apparatus

The separation ratios achieved with this apparatus are much higher than those obtained with the same materials and similar conditions in other apparatuses. This is not surprising, since in the double thermogravitational column the ratio of the membrane surface area to the volume of each reservoir is one order of magnitude higher than in the other experimental set-ups and the rates at which the concentrations vary are proportionally enhanced . Furthermore, the counter-current circulation of fluid on the two sides of the porous partition is rigorously uniform in space and time . In the fifth column of Table 2 the average separation ratios are reported . These are given by [(C W s + Ca ;)/(C, s + C~,;)] and represent the separations produced by thermodialysis across the membrane . In the sixth column of the same table the maximum separation ratios are given . These numbers represent in each case the ratio of the concentration of the solution in the reservoir in which the solution becomes most concentrated to the value in the reservoir most depleted of solute . That is, the values reported in the sixth column of Table 2 represent the overall effect of a cascade process made up of a thermodiffusive process, a thermodialysis process, and a second thermodiffusive separation. For reasons of simplicity, we have not reported the concentrations

210

TABLE 2 Results of runs performed with the double thermogravitational column . Most of these experiments were carried out at T av = 30° C, AT = 30°C and plate distanced = 1 .2 mm . The three runs in benzene were effected at a Tav = 30° C and AT= 20°C . Negative separation ratios refer to solute transport occurring in the same sense as the heat flux Continuous Phase

Disperse Phase

Porous partition

Water

Theophylline Caffeine Uridine Polyvinylpyrrolidone 360,000 m .w . Polyethylene glycol 6000 m.w . Propylthiouracd Theobromine Methylene blue Theophylline Polyvinylpyrrolidone 360,000 m .w .

Glass fiber AP-20

Benzene

Theophylline Caffeine Theophylline

Glass fiber AP-20

sec-butyl alcohol Ethyl alcohol

Theophylline Theophyaine Caffeine

Glass fiber AP-20

Versapor (Gelman)

Versapor

Average separation ratio

Maximum separation ratio

10 10 10 10

+20 .0 +8 .2 +10 .3 +4 .2

+28 .7 +11 .2 +19 .8 +7 .7

2000

+3 .8

+5 .3

8 10 2 10 10

+8 .4 +14 .0 +13 .0 +6 .5 +2 .3

+10 .7 +18 .5 +25 .5 +10 .3 +2 .8

Initial concentration C° (mg/q

10 10 10

-1 .12 -1 .25 -1 .14

-1 .14 -1 .40 -1 .31

10 10 10

+1 .13 1 .00 1 .00

+1 .21 1 .00 1 .00

in all four reservoirs, in general . We give however the results for one run in Fig. 8 . Here, the spectrophotometric analysis of the contents of all four reservoirs at the end of a run, together with the spectrum of the original solution is presented for the system 15 mg/I theophylline-water . What makes these results particularly interesting is the fact that in this apparatus the temperature gradients in each of the two thermogravitational columns and across the porous septum are approximately the same . Therefore, the confrontation of the separation ratios in the thermodiffusive process and in the simultaneous thermodialysis process, allows a direct comparison between the two types of transport . 5. Conclusions From the above, the following conclusions can be derived : (a) The separations induced by thermodialysis in the components of liquid solutions are large enough to stimulate interest in the phenomenon for practical applications . This is also true because the apparatuses and methods employed are simple in design and operation . (b) The concentration of heat and matter transport in the pores evidently entails a great increase in the effectiveness of the transport process compared to thermal diffusion in the liquid in bulk, in full agreement with theoretical expectations.



211

(c) "Uphill" material fluxes have been unquestionably obtained in various cases, consistent with a mechanism leading to the production of radiation forces capable of doing the required work . The "modus operandi" of the postulated mechanism justifies the relative independence of solute and solvent fluxes in the channels of these unselective membranes . (d) The striking reversal observed in the sense of transport of polyvinylpyrrolidone in n-butanol with decreasing pore size, unambiguously demonstrates the influence of epitaxis on thermodialysis . The observed increase in the transport produced by epitaxis in the cases in which the reversal is not observed (eqn . 14) may be less conclusive as a check of the theory, but it may be of great practical significance . (e) The examination of the results of Fig . 7 does not allow us to conclude whether the observed variation in slope at the smallest pore sizes is due only to the smaller thickness of these membranes, because other parameters are also altered at the same time . To investigate this last point, let us calculate the effective transport velocities of the solute in the pores, and from these attempt an evaluation of the thermal diffusion coefficient, D' . Let V be the volume of the solution contained in each of the two containers, S the total surface area of the pores,

Fig . 8 . Results of the spectrophotometric analysis of the solutions extracted from each of the four reservoirs of the double thermogravitational column at the end of a run . Original solution : theophylline in water C ° = 15 mg/l, T ai, = 30 °C, C T = 20 ° C ; porous partition Millipore AP-20 ; distanced between hot and cold plate : 1 .2 mm. After 4 h of running : w , = 1 .31 ; Cr i 0 ,, = 1 .40 ; = Cws + Cw,' = 12 .86 ; maximum separation

Cw,il C

tC

ratio : Cw,;IC0,, = 18 .4 .



Cep,+C~ i



212

A t is the running time, and C o the initial concentration of the solution . Then the transport velocity, transport will be given by : Ca V 1 Co V 1 Vtranspoxt = Cp S At Cty S At

(17)

For the fixed-volume apparatus of Fig . 3 (with the manometric pipe excluded) the above expression is rigorous . If variations of volume are allowed more complicated expressions apply . The values of utranaport are given in the fifth column of Table 3 . Since, according to our basic tenet, thermodialysis is a kind of modified thermal diffusion, the relation vt 1.811epoh = D" dT/dy should apply, where D'* [cm' sec - ' ° K - '] is the coefficient of thermal diffusion in,the liquid contained within the pores . The main difficulty here stems from the lack of a clear definition of the value of dT/dy in the pores . In the first place, the membrane thickness should include the immobilized layers of solution adjacent to both of its faces . In the second place, the variation in the magnitude of the temperature gradient due to the variation of thermal conductivity of the epitaxed liquid should be accounted for. Finally the distribution of the heat flux between the solid material of the partition and the liquid in the channels should be evaluated . At present we are unable to quantitatively account for the effect of these phenomena on the temperature gradient . By estimating this gradient on the basis of the temperature difference and the thickness of the partition, the values of D'* given in the sixth column of Table 3 are obtained . These are displayed in graphical form in Fig . 9 . It is interesting to observe that the values of D'* for the epitaxed liquid are not unreasonably different from those which can be found in the literature for thermal diffusion of macro-

TABLE 3 Tabulated here are the pore diameters for each kind of membrane ; the total pore area 8 in the working surface of the partition ; the ratio of volume V of solution contained in each vessel to the area 8 ; the concentration ratios C w/Cc obtained in runs of 5 h in standard conditions ; the transport velocity utrwsport ; and the coefficient of thermal diffusion, D' * , of the solute particles, calculated from the experimental data . The bracketed data refer to a reversal of solute transport in the temperature gradient relative to ordinary thermal diffusion in the bulk liquid Pore diameter (A)

Total working area of pores S(cm')

V - (cm) S

ConcentraIron ratio Cw

;c.

Transport velocity of solute (em see - I ) Jew V 1 utransPortCe at

D' * (cm2 sec-' 'IC

s

80,000 50,000 30 .000 10,000 6,000 2,000 800 500 300

19 .1 X 10 -2 29 .8 X 10 -2 58 .6 X 10 - ' 69 .2 X 10' 32 .2 X 10 -2 35 .8 X 10 -2 11 .5 X 10 -2 4 .4, X 10' 1 .6, X 10 -2

52 33 .0 18 .8 16 .9 31 .0 28 .6 86 .9 223 .7 621 .0

1 .80 1 .42 1 .15 0 .80 0 .79 0 .75 0 .66 0 .62 0 .68

+3 .70 +2.15 +1 .10 -1 .04 -1 .92 -1 .81 -5.70 -15.20 -43 .40

X X X X X X X X X

10 -1 10 -1 10 -0 10 -3 10 -3 10 -1 10 -3 10 -3 10'

+1 .85 +1 .07 +0.56 -0.52 -0.96 -0.90 -1 .42 -3 .80 -10.80

X X X X X X X X X

10 -' 10 - ' 10 -1 10 -7 10 -7 10 -7 10 - ' 10 - ' 10 -1

)

213

Fig . 9 . Values of the coefficient of thermal diffusion, D' * , calculated from the data reported in Fig . 7 .

molecules [21-23] . The large increase of otransport and D" in the smallest channels clearly shows that a new effect is manifested here . It could be that Kl and yr are affected by epitaxis differently, as would be the case if, for instance, vj would depend only on the density of the liquid while Kl would be more sensitive to variations of intermolecular order . The degree of solvation of the macromolecules might also be affected, thus altering op ( , ) . The stripping of the macromolecules could be the consequence of the competition between the interactions of the solvent with the charges on the polycarbonate wall, and the comparatively much weaker influence exerted on the n-butanol by the polyvinylpyrrolidone particles . Any conclusion on this point must await further experimentation . References 1 F .S . Gaeta, Phys . Rev ., 182 (1969) 289 . 2 F .S . Gaeta and A . Di Chiara, J . Poly . Sci ., Poly . Phys . Ed., 13 (1975) 163 . 3 F .S . Gaeta, in F . Kohler and P . Weinzierl (Eds.), Sommershule "Physik des flussigen Zustandes", St . Georgen, 18-29 Sept ., 1972, Universitiit Wien, IV, pp . 9-29 . 4 F .S . Gaeta, The genesis of radiation forces in liquids and their role in transport processes . Some biological implications, Proc . IVth Inter . Winter School on the Biophysics of Membrane Transport, Wisla (Poland), Fehr . 19-28, 1977, Part III, pp . 67-106 .

214

5 N.W. McLachlan, Ordinary non-linear differential equations in engineering and physical science, Clarendon Press, Oxford, 2nd ed ., 1956 . 6 L. Rayleigh, Phil . Mag., 3 (1902) 338 . 7 L. Rayleigh, Phil . Mag., 10 (1905) 364 . 8 G . Hertz and H . Mende, Z . Physik ., 114 (1939) 354 . 9 G . Brescia, E . Grossetti and F .S . Gaeta, II Nuovo Cimento, Serie XI, 83 (1972) 329 . 10 F.S. Gaeta, G . Scala, G . Brescia and A . Di Chiara, J . Poly . Sci., Poly . Phys. Ed ., 13 (1975) 177 . 11 F.S. Gaeta and N .M . Cursio, J. Poly . Sci ., A-1, 7 (1969) 1697 . 12 F.S. Gaeta, A . Di Chiara and G . Perna, B Nuovo Cimento, Serie X, 66B (1970) 260 . 13 F.S. Gaeta, G . Perm and G . Scala, J. Poly . Sci., Poly . Phys. Ed., 13 (1975) 203 . 14 F.S. Gaeta, D .G . Mile, G . Perna and G . Scala, Il Nuovo Cimento, Serie XI, 30B (1975) 153 . 15 B.V . Derjaguin, Nature, 138 (1936) 330 . 16 B.V . Derjaguin and A.S . Titjevskaya, Proc . Second Int . Congress of Surface Activity, 1(1967)211 ;3(1957)531 . 17 E. Forslind, Acta Polytecnica, 115 (1952) 9 . 18 M .S . Metsik, Research in the Field of Surface Forces, Vol . II, Nauka Press, Moscow, 1964,p .148 . 19 K. Clusius and G . Dickel, Naturwiss ., 26 (1938) 546 . 20 K. Clusius and G . Dickel, Naturwiss ., 27 (1939) 148 . 21 G . Langhammer, Naturwiss., 41 (1954) 525 . 22 G. Langhammer and K. Quitzsch, Makromol . Chem ., 17 (1955) 74 . 23 A.H . Emery Jr . and H .G. Drickamer, J . Chem . Phys ., 23 (1955) 2252 . 24 S .R. de Groot and P . Mazur, Non-Equilibrium Thermodynamics, North Holland Publishing Co ., 2nd Ed ., 1969, pp . 426-8, eqns . 142 and 143 .