2270
The Journal of Physical Chemistry, Vol. 83, No. 17, 1979
capture mechanism and a selective capture scheme both proposed previ~usly;~ the latter should be replaced by the successive tunneling scheme.& However, the results do not exclude thoroughly the possibility of the partial contribution of mobile holes in cation formation in glasses. Acknowledgment. We thank Dr. M. Tachiya for suggestions concerning theoretical aspects.
References and Notes (1) (a) Shida, T.; Hamlll, W . H. J. Chem. Phys. 1900, 44, 2369; (b) ibid., 1900, 44, 2375; (c) ibid., 1900, 44, 4372. (2) Gallivan, J, B.; Hamill, W . H, d. Chem. Phys. 1966, 4 4 , 2378.
F. S.Gaeta
and D. G.Mita
(3) Skelly, D. W.; Hamill, W . H . J . Phys. Chem. 1900, 70, 1630. (4) (a) Arai, S.; Kira, A,; Irnarnura, M. J . Phys. Chem. 1970, 80, 1968; (b) Arai, S.; Irnarnura, M. J. Phys. Chern. 1979, 83, 337. (5) (a) Kira, A.; Nakamura, T.; Irnarnura, M. J . Phys. Chem. 1978, 82, 1961; (b) ;bid., 1977, 87, 591. (6) Rice, S. A.; Piliing, M. J . Prog. React. Kinet. 1978, 9 , 93. (7) Miller, J. R. J . Chem. Phys. 1972, 56, 5173. (8) Kira, A.; Imamura, M. J . Phys. Chem. 1978, 82, 1966. (9) (a) Miller, J. R. J. Phys. Chern. 1975, 79, 1070; (b) /bid., 1978, 82, 767. (10) Mlller, J. R. Science 1975, 789, 221. (11) Hamill, W. H.; Funabashi, K. Phys. Rev. B 1977, 76, 5523. (12) Webrnan, I.; Kesbner, N. R. J . Phys. Chern. 1979, 83, 451. (13) Shida, T.; Iwata, S. J . Am. Chem. SOC.1973, 95, 3473. (14) Kira, A.; Imamura, M. J. Phys. Chern. preceding article In this issue. (15) Grand, D.; Bemas, A. J . Phys. Chem. 1977, 87, 1209.
Thermal Diffusion across Porous Partitions. The Process of Thermodialysis F. S. Gaeta” and D. G. Mita Intefnatbnal Institute of Genetics and Biophysics of C.N.R., 80 125 Naples, Italy (Received August 8, 1978; Revised Manuscript Received March 28, 1979) Publlcatlon costs assisted by the Italian National Research Council
Coupling of mass flow to a flux of thermal energy is a well-known occurrence in various experimental situations. We have shown that in condensed matter these transport phenomena are produced by the radiation pressure developed by the flow of heat. We extend here this approach to nonisothermal transport through porous partitions characterized by a reflection coefficient nearly equal to zero. Analytical expressions, which allow qualitative and quantitative predictions for the mass flow occurring through the partition, can be derived from the theoretical approach. The phenomenon, which we call “thermodialysis”, is theoretically and experimentally proved to be capable of producing matter flow proceeding “uphill”, that is against gradients of hydrostatic pressure and/or of electrochemical potential. Also experimental evidence lending support to other fundamental predictions of the theory is presented.
Introduction The flow of heat through a liquid solution can cause a reduction of the entropy of the fluid mixture by inducing selective migration of its components. Such an ordering effect takes place in ionic, molecular, and macromolecular solutions of any degree of complexity. This phenomenon, called thermal diffusion or the Soret-Ludwig effect, from the names of its discoverers, has been known for nearly a ~entury.l-~ When the flow of thermal energy takes place across dense membranes permeated by the solution, the associated transport of matter has been called thermoosmosis.5-27 Nonequilibrium thermodynamics has provided a theoretical framework for the treatment of these processes. Such a general and powerful theoretical approach, however, has the shortcoming of being unable to yield a molecular description of the phenomena. The coupling between the heat flow and the associated fluxes of matter is expressed by means of coefficients whose dependence on the nature of the substances involved in the process is not specified. Experimental studies of thermoosmosis, on the other hand, have generally been concerned with semipermeable, selective membranes. Much less attention has been devoted to the case of porous partitions, having a Staverman reflection coefficient equal to zero, that is to partitions utterly unable to discriminate between solute and solvent. Perhaps this has been due to the feeling that the investigation would not be worth trying, owing to the lack of osmotic and/or ultrafiltration effects across such partitions. Let us, however, try to consider the problem from an unconventional point of view. Imagine a porous partition sandwiched between two solutions maintained at different 0022-3654/79/2083-2276$01 .OO/O
temperatures. All components of the solutions can freely diffuse across the existing pores. The pore diameter, however, is supposed to be small enough to quench hydrodynamical convection and turbulent motions inside each channel. Under these conditions each liquid-filled pore constitutes a microscopic Soret cell in which thermal diffusion will take place at a very fast rate, when the thickness of the partition (i.e., the height of each equivalent Soret cell) is small. The “characteristic time” for the establishment of equilibrium is given by 0 = d2/n2D,where d is the height of the thermodiffusive column (in our case the partition thickness) and D (cm2-s-l) is the ordinary coefficient of diffusion. We have proposed a theory of thermal diffusion in liquids which attributes the redistribution of the components of a solution in a temperature gradient to radiation pressure generated by the flow of thermal These excitations are supposed to consist of very highfrequency elastic waves (Debye waves). We have published much experimental evidence supporting the radiationpressure a p p r ~ a c h . ~ l - ~ ~ Without going into details concerning the genesis of thermal forces produced by a flux for thermal energy in condensed phases, which we have extensively treated e l ~ e w h e r e ,we ~ ~shall - ~ ~recall ~ ~ ~here that, on the grounds of the radiation-pressure theory of thermal diffusion, the force acting upon a molecule of cross section up contained in a nonisothermal liquid will be given by
where the K s are the thermal conductivities and the v’s 0 1979 American Chemical Society
The Journal of Physlcal Chemistry, Vol. 83,No. 17, 1979 2277
Thermal Diffusion across Porous Partitions
are the (phase) velocities of propagation of high-frequency acoustic waves; ~ ~ , pis2the coefficient of transmission of acoustic energy between liquid and particle and dT/dx is the temperature gradient. Quantities with subscript 1are ones characteristic of the solvent, subscript p instead refers to quantities whose character is more complex, being related to solvated particles. The magnitude and the sense of the drift motion of the various molecular species in the temperature gradient will depend, according to eq 1, on collective properties such as thermal conductivity and velocity of propagation of elastic high-frequency waves. From the above expression for the radiation force, other equations can be easily derived fro two phenomenological coefficients called, respectively, the coefficient of thermal diffusion, D’ ( c m 2 d o C - l ) ,and the Soret coefficient, S (OC-l).Z8-30 The interposition of the porous partition between a hot and a cold solution introduces two major differences between thermodiffusive transport in an ordinary Soret cell and in a membrane pore. In the first place the value of the temperature gradient will be increased very much when 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 kinetics of the effect of thermal diffusion will be accelerated, since the force acting on the suspended particles is increased (eq 1). In the second place, an effect, subtler than the one just described, will manifest itself whenever the diameter of the pores 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 bulk, through the action of the forces of a d h e ~ i o n . ~Many ~ - ~ ~physical properties of the liquid are appreciably modified through the action of van der Waals and London dispersion forces, especially the transport properties. Thermal conductivity of water, for instance, has been reported to increase 100-fold when enclosed between nearby mica surfaces.43 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 increment of thermal conductivity, per se, entails greater radiation forces, in so much as the density of the flux of thermal energy is increased. The variation of v1anyway affects the ([(K/u)(dT/dx)]i- [(K/u)(dT/dx),]) term, thus altering the value of F. As a consequence of the influence of wall proximity on the physical characteristics of the medium, eq 1 must be rewritten to describe thermal diffusion in narrow pores as follows:
( Eu E) dx - ( Eu
“) dx
p(*)
(2)
where subscript 1” for the liquid indicates that the corresponding value of the quantity is the one relative to the modified liquid, while the star within parentheses on quantities refers to solute particles, indicates that these quantities might also be affected (though generally to a much lesser extent) through the influence which the altered state of the liquid contained in the pores has on the degree of solvation. From eq 2 analytical expressions can be derived for the phenomenological coefficients S* and D **,equivalent, for our case, to the ordinary Soret and thermal diffusion coefficients. Solute flux and concentration ratios C,(t)/ C&t) in the warm (C,) and cold container (e,)accordingly can he predicted, when all quantities in eq 2 were known.
Unfortunately, thermal conductivity and propagation velocity of high-frequency elastic waves in the modified liquid phase adjacent to pore walls are not known. Anyway, two qualitative, but clearcut predictions can be made, on the basis of our approach, and subjected to unambiguous experimental verification. One is that particles may be pushed in a sense opposite to the one in which they should drift following the gradient of electrochemical potential and hydrostatic pressure. The extent to which this kind of “uphill” transport may continue against opposing gradients will depend solely on the rate of working of the externally driven energy flux. The second, more specific, prediction concerns the behavior of thermal diffusion across porous septa on the basis of a critical examination of the nucleus of eq 2. Indeed, the quantity (3) determines the sense of thermodiffusive drift in the pores, relative to the sense of heat flow. We have seen that the thermal conductivity of a liquid is drastically increased by interaction with a solid surface;43it is likely therefore that the ( K / v )ratio will also be increased relative to the liquid in bulk. In the case of a solution in which solute particles have a characteristic (K/u),which is smaller than the one of the liquid in bulk, they will thermodiffuse at a higher rate from warm to cold across the channels owing to an increase of the first term in expression 3. In the case, however, in which ( K / v ) ,is higher than ( K / U and ) ~ the particles drift against the heat flow in the liquid in bulk, they will do so at a decreased rate in the pores because
There is a possible special case, however, when the increase of the ratio ( K / u ) of the liquid is so great that ( K / V ) ~ , becomes greater than (Klu),, and
in such case the following striking behavior is expected: the particles will drift in opposite senses in the bulk liquid and in the fluid contained in the channels. Concluding, our approach implies, for nonisothermal transport through porous partitions, 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 unselective porous partitions. As we have seen, new effects are foreseen, some of these never observed before in thermoosmosis. If substantiated by experiments, there are sufficient reasons to consider this form of transport an independent phenomenon on its own right, for which we have therefore proposed the name of thermodialysis,as distinct from both thermoosmosis and ordinary thermal diffusion. Experimental Section Apparatus, Materials, and Methods. The present investigation was undertaken with the purpose of ascertaining the existence and the characteristics of uphill
F. S. Gaeta
and D. G. MRa
Ftpm 1. Exp(oded view of the apparatus. Parts are indicated by f&whg mmberr: (1) metal cashg: (2) semicab (3)paws partllbm (4) lhsnnomuples: (5) motu and propelk: (6) Vertex: (7) Ushaped glass lube lor circulation of thsfmostating Ruld: (8) -meter. The dimensions are as loliows: a = 35 mm: b = 115 mm: c = 145 mm: d = 15 mm; e = 60 mm; I = 70 mm. The perspex walls are 1 cm
thick.
transport produced by the flow of heat across some simple porous partitions. Furthermore, evidence was sought which could substantiate the supposed role of liquid-pore walls interactions in thermodialysis. Schematically our experimental devices consist of two containers separated by, and communicating through, a porous partition sandwiched between them. The two liquid volumes are maintained a t different temperatures, constant throughout the experiment. In this investigation we used two different types of apparatuses. The first one consists of two containers, separated by a vertical porous septum sandwiched between them, with two rectangular corresponding coincident openings (Figure 1). A metal structure holds the parts together. Egual liquid volumes are initially introduced to fill both containers u p to the same level. Heating and cooling are provided through circulation of externally thennostated fluids. Thermocouples introduced near each face of the partition give a continuous reading of the temperature difference across the system. Continuous stirring is provided 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. The containers are open above, so that whenever there is volume flow across the partition the level of the liquid in each container varies and hydrostatic pressure differences are generated. Sufficient space is left above the upper limit of the septum to allow conspicuous volume flow without producing spilling of the liquid from the top of the vessel. During the run, samples of the solutions can be extracted for analysis of the concentrations of the ionic or molecular solutes. Another type of apparatus used is represented in Figure 2. It consists of two cylindric containers made of Pyrex glass, each one ending with a flat flange. The horizontal partition is held tight between these two flanges. A metal structure holds the two sections together. The solution contained in the upper section is heated, and the one in the lower is cooled. The flat bottom of the gold-plated heater can be adjusted a t regulable distances from the partition; a temperature sensing device ensures automatic regdation 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
M2.
One v w m 01 w cparcal hodzomahemtrane apparatus cmvbte wah Vertcx a d W n n o c w m .but &LXNW 01 lower codhg system. tR plate. arm regslerlcq unns. The dOuble deComeler 6 a m 10 give an mea 01 the real dmension. This apparatus allows measurements 01 the hydrostatic pressure prcducea In the p~ocess.by opening Ihs stopcock a1 Ihe oonom 01 the manometric pope. When mis stopcock IS closed, instead. constant-volme measuemenls can be effected
external to the container. A thermocouple system monitors 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. This last arrangement was found to ensure sufficient convection on each side of the septum so as to eliminate concentration and temperature gradients within the bulk of the solutions, even if some unstirred (Nernst) layers of solution adjacent to the membrane always remained. These layers are of a thickness generally smaller than the one of our membranes, but not completely negligible in the calculation of the temperature gradient or in the evaluation of the concentration gradients. In some case we preferred to stir the lower part of the apparatus by a magnetic device, while we agitated the upper part from above with a toy electric motor. As can be seen in Figure 2 the lower container is connected through a glass stopcock, to a capillary manometric pipe. With the stopcock closed the transport phenomenon can be studied in absence of volume flow; when it is open. the pressure across the porous partition can be measured hy observation of the level of the liquid in the manometric pipe and ita variations in time can be followed. The net volume flow in this case is limited to the small manometric volume. Coming now to the procedures inherent to the preparation and performance of a run, we have found that a lot
The Journal of Physical Chemistry, Vol. 83,No. 17, 1979 2279
Thermal Diffusion across Porous Partitions
of care must be paid to certain apparently trivial details in order to obtain good reproducibility of the experimental results. In general, previous bathing of the porous element in the solvent to be used in the run greatly helped to obtain more reproducible results; when possible we boiled the membranes in the solvent or in the solution. When the solvent was water, we used twice-distilled water; a few experiments with distilled, deionized water gave results indistinguishable from those obtained with common distilled water. When the solvents were others than water we always used first-grade materials of purity pro-analysis; we did the same for the materials used as solutes. The macromolecular solutes poly(vinylpyrro1idones) were purchased, in batches of various molecular weights, from Fluka, Switzerland. We also used a variety of porous septa; the glass-fiber ones, of two different thicknesses, were purchased from Millipore; Millipore HAWP filters having pores of 0.45 wm are made of mixtures of cellulose esters. Nuclepore membrane filters were obtained from Nucleopore Corp., Pleasanton, Calif., having a wide choice of calibrated pores in the range from 300 to 80 000 8, diameter sizes. Other filters also used by us were Metricel (made of cellulose triacetate as, for instances, GA4), Versapor epossi, polypropylene, and Acropor, all made by Gellman, USA. The concentration variations of both the enriched and depleted solution, on the two sides of the porous partition, were assessed by us in various ways, depending on the properties of the disperse and dispersing phases. Whenever possible, we used spectrophotometric measurements, performed generally with a recording Beckman DB/GT spectrophotometer and occasionally with a flame spectrophotometer. When this kind of measurement was not possible, or other measurements were preferable, we used alternative kinds of quantitative analysis, namely pH determinations and conductometric assays. In the case of ionic solutions, whenever possible, we coupled chemical titration with activity determinations made with an Orion 404 pH meter equipped with ion-specific electrodes. At the same time, during each run (with the obvious exception of those carried out at fixed volume) we continuously registered the hydrostatic pressure differences produced, determining their time dependence. Thermodialysis as a Process of “Uphill” Transport. From our theoretical approach it follows that the transport of ions and molecules across a porous partition mediated by a flow of thermal energy may be a process of “uphill” transport. Namely, we expect such transport to take place also against gradients of electrochemical potential and/or differences of hydrostatic pressure, since in our treatment the temperature gradient actually produces thermal forces of well-defined characteristics. We shall start by describing some experiments of a semiquantitative kind, confirming thermodialysis as a process of uphill transport akin to thermal diffusion. These are just sample cases out of many similar results, some of which, of industrial interest, are dealt with else~here.~~*~~ If an aqueous solution of methyl violet (8 mg/L) is placed in two containers separated by a partition of glass fiber AP-20 (0.30 mm thick after swelling), and the contents of the containers are brought and maintained a t a few degrees centigrade respectively above and below room temperature (20 “C), in an apparatus of the type represented in Figure 1, a flow of water takes place from the hot to the cold container, and a flow of solute takes place at the same time in the opposite sense. The concentration ratio C,/C, = 1.8 of methyl violet in the warm and cold
1
1-
ti
30
:”‘
2 - 2,o
/AA‘
10
/ x / ~ x
’,5
-1, time,(min)
or
20/x 0
100 ,
200 ,
300 ,
400
500
600
2280
The Journal of Physical Chemlsfv, Vol. 83, No. 17, 7979
F. S. Gaeta and D. G. Mita
4
7
Figure 4. Evolution of hydrostatic pressure head (0)and of concentration ratios C,/C, (X), in a run similar to the one of Figure 3, but followed for a longer time and employing aqueous 0.1 1 m solution of acetic acid. The average temperature was taken equal to 23.5 " C and a A T = 17 O C was applied across the AP-20 porous partition.
aqueous solution of acetic acid, 0.11 mol/L. The running time here was considerably extended, having taken special care to prevent evaporation losses. The substantial similarity of these results represented in Figure 4 with those of methyl violet can be appreciated. It is interesting to observe that a nearly constant rate of increase of the solute separation ratio is maintained even 2.4 (corresponding after the rather high value of C,/C, to a percent separation (C, - C,)/Co N 90%) has been reached. By using the same apparatus and membrane, and employing an aqueous 1-m solution of sodium acetate, the following set of runs has been performed: (A) The cell is filled on one side of the membrane with a cold (+15 "C) aqueous solution of CH,COONa and with an equal volume of warm (+32 "C) distilled water on the other side; (B) warm (+32 "C) solution against cold (+15 "C) distilled water; (C) solution a t 23.5 "C against water at the same temperature. The results of one series of such runs are displayed in graphical form in Figures 5 and 6. Volume flow from the warm into the cold container takes place in both nonisothermal runs (Figure 5) though at a different rate. It is interesting to see that the relevant concentration gradient, favorable in case A to the observed matter transport, adverse to this transport in case B, has only a moderate effect on transmembrane volume flow. Very interesting are also the results of the isothermal run displayed in the same figure. Volume increase in the solution compartment is observed in this case, though at a much slower rate (curve C in Figure 5 ) . This result tells us that the reflection coefficient of the AP-20 membrane, even if small, has a nonzero value, probably due to the
Figure 5. One molal aqueous solution of sodium acetate. Rates of formation of a hydroastatic pressure-head adverse to volume transport. Curves A, B, and C, respectively, refer to the situations described in the text under headings A, B, and C. Analytical expression for each process are given in the figure in the form A h ( t ) = a t .
,
0
1
2
3
4
5
6
7
time (hours1 8
Figure 6. Rates of increase of solute concentrationin the compartment initially containing distilled water. Curves A and B refer to the corresponding situations described in the text. The points in this figure are the measured concentrations corrected for the volume changes in the compartment.
modified state of water in the pores. An osmotic pressure therefore is indeed developed by the concentration gradient, and this osmotic pressure in case A favors the volume transport due to thermal radiation pressure, while in case B it opposes this transport. Also interesting is the observation that the solute diffuses into the pure water
The Journal of Physical Chemistry, Vol. 83,No. 17, 1979 2281
Thermal Diffusion across Porous Partitions
TABLE I: Results of Experiments of Thermal Diffusion Performed with a Thermograviational Column Compared with Those Obtained in Experiments of Thermodialysisa sep ratios and Soret caeff ~~
thermal diffusion aq solution of
~ , , bmg/L
CJC,
thermodialysis s, O C - ’
s, O C - 1
CJCW
x
partitionC
10 10
1.15 1.15
1.1x 10-3 1.1 x 10-3
1.94 1.62
3.3
2.4 X
HAWP collodiond
methylene blue
2 2
1.20 1.20
1.4 x 10-3 1.4 x 10-3
1.52 1.22
2.1 x 0.9 x
cellulosed HAWP
propylthiouracil
8 8 8
1.07 1.07
1.21 1.52 1.12
0.9 x 2.1 x
1.07
0.2 x 10-3 0.2 x 10-3 0.2 x 10-3
0.6
X
HAWP GA4 collodiond
10 10
1.12 1.12
0.7 x 10-3 0.7 x 10-3
1.73
2.7
x
1.15
0.7 X
theophylline
uridine
lo-’ 2.3 X lo-’ 1.1 x lo-*
HAWP collodiond
HAWP 1.60 10 1.16 1.15 x 10-3 collodiond 1.24 10 1.16 1.15 x 10-3 a The membrane apparatus was of the type represented in Figure 2, with the manometric pipe excluded. It is to be remembered that runs of both thermal diffusion and thermodialysis were executed applying a AT of 20 ‘C, at the average temperature of 30 ‘(2, the duration of runs being in all cases 4 h. The temperature gradients in the thermogravitational Type of porous partition used Initial concentration. column and across the porous partitions are of similar magnitude. in thermodialysis. Holes, produced in these membranes by electric discharge, were evaluated to have diameters in the range from 0.8 to 2.2 qm. caffeine
I;
compartment in both nonisothermal experiments and that is does so at a higher rate in case A, that is, in the case in E which a smaller rate would be instead expected because of solvent drag. When aqueous solutions of sodium acetate of equal concentrations are introduced in the warm and in the cold compartment of the cell both solvent and solute transports are observed.48 In Figure 7 results obtained with an aqueous 0.2 m solution of CH,COONa are graphically displayed. The results of two runs a t the same temperature (Tav= 23 OC) but with a temperature difference of AT = 17 “C in one case and AT = 27 O C in the other are reported. This behavior is analogous to the case of methyl violet. Also the effect of an increased temperature difference is as expected. Summing up, the above results are strongly indicative of the existence of a thermal radiation force independently acting on solvent and solute particles, in full accord with theoretical expectations. We will presently proceed to discuss evidence concerning the role of adhesion forces in thermodialysis. Influence of Adhesion Forces on Thermodialysis. We have performed many experiments with the aim of detime (min.) 1 tecting the influence of adhesion forces on transport 0 2bo 360 through nonisothermal porous partitions. Flgure 7. Initial content in both containers: 0.2 m aqueous solution In the case of disperse particles which in the bulk liquid of sodium acetate. Rates of formation of pressure and concentration thermodiffuse to the cold regions of the solution such ratios in two runs at the same average temperature of 23.5 O C and behavior should be retained and enhanced in narrow pores temperature differences of 17 and 27 O C , respectively. because of the increase in (K/u),. In Table I we compare some results of experiments of thermal diffusion with comparable with the ones obtained in the case of the results of thermodialysis obtained by the apparatus of porous partition (columns 5 and 6). Figure 2 and various types of solutes and porous partitions. Coming now to the situation described by expression 4 These results can be directly compared, also because of in the particularly significant ease of inequality 5 , we have the practically equal distance over which the temperature sought experimental evidence supporting such a rather gradient is applied in the two cases. This distance is also surprising theoretical prediction. the length of the path along which thermodiffusional drift To this end, an experimental situation must be found occurs. It was possible to obtain such a situation by using in which a drastic change in the effect of transport is a “thermogravitational column” in place of a conventional observed in correspondence to the variation of a single Soret cell. This rather sophisticated apparatus has been parameter affecting the structure of the liquid. According described in detail in many of our preceding ~ o r k s , ~ ~to~our ~ hypothesis ~ , ~ ~ concerning the role of liquid-pore wall it will not be necessary to repeat this description here. interaction the pore diameter can be such a parameter. Its Suffice it to say that results reported in the third and decrement indeed entails increased interaction of the fluid fourth column of Table I are representative of the thercontained in the channels with the walls and hence a modiffusive behavior of the respective substances in the stronger modification of the structure of this liquid relative bulk liquid, at values of the temperature gradient well to the liquid in bulk. Since a variation of pore size, anyway, -4
2282
The Journal of Physlcd Chemistry, Vol. 83,No. 17, 1979
F. S. Gaeta and D. G. Mita
Plgure 8. Concentration ratios produced by thermodialysis in solutions of poly(vinylpyrro0done)K-25 in l-butanol, plotted against membrane pore size. Duration of runs 5 h; temperatures of warm and cold container 4-40 and 4-20 OC,respectively; membrane thickness lo-* mm (except for membranes wilh pores of 800, 500, and 300 A, where it was 5 X lo3 mm thick); C, final concentration in the warm container; C,,final concentration in the cold container.
also affects the hydrodynamic hindrance opposing the fluxes crossing the channels, a decrease of transport produced by the decrease of pore diameter would not be unequivocably attributable to the hypothized increase of the K/u of the liquid. Only an inversion in the sense of transport followed by an increase of its entity and rate, with decreasing pore diameter, if observed, would be conclusive. We have studied, with the apparatus of Figure 2, the thermodialysis of solutions of poly(vinylpyrro1idone)K-25 of 24 000 m u in l-butanol across Nuclepore polycarbonate membranes. These membranes are available in pore sizes variable from 80 000 to 300 A. Each type has cylindrical pores of a given diameter extending from face to face. The concentration of the macromolecular solute (16mg/L), the average temperature T,, = 30 "C,and the temperature difference AT = 20 "C,etc., were kept constant in all runs. It is to be remembered that poly(vinylpyrro1idone) macromolecules in 1-butanol drift from the cold toward the warm part of the solution in ordinary thermal diff ~ s i o n Therefore, . ~ ~ ~ ~ ~according to eq 1, the value [ ( K / u)(dT/dx)], relative to the liquid in bulk will be smaller than the one relative to the suspended particles; the quantity ([(K/u)(dT/dx)],- [(K/u)(dT/dx)],,) is therefore negative. When this same solution is run in the apparatus of Figure 2, employing Nuclepore membranes of decreasing pore sizes, the results represented in graphical form in Figure 8 are obtained. To better understand these results it should also be remembered that all membrane properties are identical throughout the series, except for pore density. In the case of the membranes with pores of 300-800 A, thickness also is halved. These differences in thickness and pore density obviously affect in turn the values of the observed separation ratios C,/C,, but their influence can be easily accounted for. Therefore, from the experimental results, the calculation of the effective drift velocities across the partition and the evaluation of the coefficient of thermal diffusion, DW, can be performed. If V is the volume of the solution contained in each of the two containers and A is the total surface of the pores in the working area of the membrane, A t the running time, and
Co the initial concentration of the solution, the drift velocity udrift will be given by ccv1 cov1 Udrift = - - - = - - C, A A t Co A A t For the fixed-volume apparatus of Figure 2 (with manometric pipe excluded) the above expression is rigorous. If variations of volume are allowed, more complicate expressions apply.48 Since Udrift = DW(dT/dx),also the coefficient of thermal diffusion of the poly(vinylpyrro1idone) molecules in 1-butanol alcohol can be calculated, once a rough value of the temperature gradient in the channels is assigned on the basis of the applied temperature difference and thickness of the partition. These data are summarized in Table I1 and Figure 9. If the values of D**obtained in this way from experiment are to be compared with the theoretical e x p r e s ~ i o nmodified , ~ ~ ~ ~ ~for the present case, then difficulties arise. Such difficulties stem from the lack of a clear definition of the value of dT/dx in the pores. In the first place, the membrane thickness should include the immobilized layers of solution adjacent to each one of its faces. In the second place, the variation in the intensity of the temperature gradient connected with the variation of thermal conductivity of the liquid has to be accounted for. Finally the repartition of the heat flux between the solid material of the partition and the liquid in the channels should be evaluated. A t present the knowledge of the value of the K / u of the adsorbed liquid is unknown. Quantitative comparison between theory and experimental results therefore still has to wait until much more information on the system will be available. What is interesting anyway is the fact that the values of D Y calculated from Udrift and the rough value for dT/dx compare reasonably well with those which can be found in the literature for experiments of thermal diffusion of macromolecules of similar shape and mass.@52
Discussion and Conclusions For the discussion of the above to be conclusive, it is useful to assess (a) how much these results are consistent with the idea that thermodialysis is a (modified) kind of
The Journal of Physical Chemistry, Vol. 83, No. 77, 1979 2283
Thermal Diffusion across Porous Partitions o"x1d(cm2,*&~."I
)
+2
3m
F w
+1
0
hld 6.d 2.10'
3.'lo4
4 2
id
6:10'
zid
'
4
8.10
(ANGSTROMS)
-1
-2
-3 -4
-5
-6 -7
-6
-9 -10 -11
Figure
9.
Values of the coefficient of thermal diffusion D'", calculated from the data reported in Figure 8.
TABLE 11: Data for Each Membrane Studied pore diam, W 80000 50 000 30 000 1 0 000 6 000 2 000 800
500 300
A,a cm2 19. 29.8 53.6 59.2 32.2 35.8 11.5
1~
lo-' X X lo-' X lo-' x 10.' X lo-' 4.4, X lo-' 1.6, X lo-' X
cm 52.0 33.0 18.6 16.9 31.0 28.6 86.9 223.7 621.0
concn ratiosc
1.80
udrietrdpecm.s-1
lo-' lo-' lo-'
0.66 0.62
13.70 X t2.15 X t1.10 X -1.04 X -1.92 X -1.81 X -5.70X -15.20 X
0.58
-43.40 X lo-'
1.42 1.15
0.80 0.79 0.75
low3 lo-' lo-' lo-'
D'*,e,f cmZ.s-'.OC-1 11.85 X t1.07 X lom7 t0.55 X l o T 7
-0.52 x 10-7
-0.96 x 10-7 -0.90 x 10-7 -1.42 x 10-7 -3.80 X 10" -10.80 X l o m 7
s*,g
"C-1
D*,h cmz.s-1
2.9 X lo-' 1.7 X l o - ' 0.7 X lo-'
6.3 X 6.2 x 7.9 x
-2.1 x io-' -2.4 X IO-' - 2 . 7 X lo-'
8.2 X 6.3 X 6.8 X 15.8 X 39.7 X
-1.1 x i o - 2 - 1 . 2 x IO-' -1.4 x
4.7 x
lom6
Concentration Total pore area in the working surface, Ratio of volume of solution contained in each vessel to A. Drift velocity, Udrift = ( C w / C , ) ( V / A ) ( l / ~ t ) .e Calculated from experimental ratios c,/c, obtained in At = 5 h runs. Coefficient of thermal diffusion of solute particles, P Soret coefficients, are obtained from the Cw/C,ratios. data. Italicked values are used to indicate a reValues of the isothermal diffusion coefficient D'*/S* in the modified liquid. versed sense of drift of solute in the temperature gradient, relative to ordinary thermal diffusion in the liquid in bulk. a
thermal diffusion; (b) to what extent they provide information on the existence of an altered state of the liquid in the pores and on its influence on the process; (e) in which way they may be considered as supporting evidence for the proposed radiation-pressure theory of this kind of thermal transport. To proceed to the discussion of points a and b, values of Soret coefficients for thermal diffusion in the pores, S*, should be obtained and compared with Soret coefficients for the same solutions in ordinary thermal diffusion cells. From data such as those of Figure 7, however, it is very difficult to proceed to the calculation of the values of Soret coefficient in the pore. This is because saturation of solute transport is not reached there and because of the complications due to the accompanying volume flow. Anyway, if the concentration ratio obtained after 1400 min in the aqueous solution of acetic acid of Figure 4 is considered, one can derive from it a value which is an acceptable underestimation of S*. The calculation of S* is possible in this case because the solvent flow has practically stopped, as can be seen from Figure 4; the steady-state separation (Cw/Cc)m,if reached, would be higher than the value 2.4 reached after 1400 min. From this concentration ratio the value S* = 5.1 X O C - l is obtained. Mea-
surement of S for the same solution in an ordinary thermodiffusive apparatus gives S = 4 X O C - l at the same average temperature. The altered state of the liquid in the pores accordingly entails a more than tenfold increase of Soret coefficient. A more accurate evaluation of S* is possible by using results obtained with the constant-volume apparatus of Figure 2, where a more favorable volume-to-surface ratio allows to reach steady-state separations in about 5 h. Aqueous solutions of acetic acid were run in this apparatus, using AP-20 partitions. These mixtures also yielded interesting results when run47p48in the open-top apparatus of Figure 1. Calculated values of S* for the present experiments are displayed in graphical form in Figure 10. The coherence of these points with the one derived from Figure 4 is evident, remembering that this corresponds to separations anticipated to be lower than those obtained nearer to equilibrium. It is interesting to observe the variation with concentration of S*; the minimum in the curve of Figure 10 corresponds to the azeotrope. It will also be interesting to study the behavior of the ordinary Soret coefficient throughout the same concentration range. Calculation of S* can be also made for each of the runs reported in Table I and compared with the values of S
2284
F. S. Gaeta and 5.G. Mita
The Journal of Physical Chemistry, Vol. 83, No. 17, 1979
a
colmolcs
/It
1
Figure 10. Soret coefficients, S“, for aqueous solutions of acetic acid at various concentrations, run in an apparatus of the type of Figure 2, equipped with an AP-20 porous septum. All these runs were performed at a T,, = 26.0 O C and with a AT = 10 O C . The point (0) at Co = 0.11 mol/L represents the value of S”calculated from the data of Figure 4. The point (0)represents the value of S calculated from the results obtained by running the same solution in a thermogravitational column.
obtained from the corresponding runs in the thermogravitational column. As one can see, S* turns out to be systematically greater than S. Altogether these results are consistent with the assumption that thermodialysis is a modified form of thermal diffusion, and that the state of solvent in the pores, entailing a greater density of heat flux, produces more intense radiation forces and higher values of Soret coefficients. Some tentative conclusions can be drawn from the above: In the first place, since the observed fluxes of matter are definitely “uphill” an external energy source is required to keep such processes going. In our approach the way in which the flux of thermal energy crossing the system produces radiation forces which do the work required for the observed transport is indeed well defined. The amount of work which such forces can perform is quite adequate to account for the observed effects. Also the relative independence of solute and solvent fluxes evidenced in Figures 3 and 4 is easily interpretable. Water in those cases is observed to be transported from warm to cold while the solute migrates in the opposite sense. The principal advantage of the radiation-pressure theory of thermal diffusion consists in the fact that is assigns completely symmetrical roles to disperse and dispersing phases28p30~34 consistently with the observed independence of solvents and solute fluxes. In the second place, upon inspection of the data reported in Figures 8 and 9 and in Table 11, another fact catches the eye, namely, the inversion in the sense of drift of the poly(vinylpyrro1idone) particles in the temperature gradient. This inversion occurs at a value for pore diameter of about 2.0 X lo4A, i.e., when all the liquid contained in the channels is within 1 pm of the solid wall in good agreement with what is known about the range within
which interaction with a solid wall appreciably affects the properties of a liquid medium. In terms of our approach to the problem, the inversion in the sense of solute transport is unambiguously indicative of the change of sign of the term ([(K/u)(dT/d~)]~, - [(K/u)(dT/dx)],w), since all other quantities in eq 2 are positive. However, not all these quantities are necessarily independent of pore size. It is, at present, very difficult to discriminate among effects upon each one of the parameters in the equation. Within the limitations set by these difficulties, anyway, the data of Figures 8 and 9 allow still another observation to be made: since the dependence of the concentration ratios is found to be approximately linear throughout the whole range of pore sizes from 8 X lo4 to 2 X lo3 A, it is evident that hydrodynamical hindrance does not play an important role here. To conclude, the inversion in the sense of transport proves, in our opinion, the dependence of the effect of thermodialysis on the nature and intensity of the interactions between the liquid and the walls of the channels. Very interesting is also the observation of the rather abrupt change of slope of the curve of Figures 8 and 9 for pore diameters below 2000 A. This is not an effect which can be ascribed just to the increased value of the temperature gradient, owing to the smaller thickness of the membranes with pores of 800, 500, and 300 A. The huge increase of D Min these membranes shows indeed that a new factor starts to play a role in sufficiently thin channels. Such a case could either consist in the variation of the degree of solvation of the macromolecules, or be due to the different influence of pore size on ul and KI, respectively. One of these two parameters could be affected already above 2000 A, and the other only much below this value of pore diameter. On the other hand, the possibility that it is the degree of solvation which is affected is apparently indicated by the abruptness of the transition between the two slopes in the curve of D&. The stripping of the macromolecules could be the consequence of competition between the interactions of the solvent with the charges on the polycarbonate wall, and the comparatively much weaker influence exherted on the l-butanol by the poly(viny1pyrrolidone) particles. What is difficult to evaluate is the relative influence of such a stripping event on the C T ~ ( * ) , up(*), and T ~ ~ ,of~ the ( * macromolecules ~ ~ in the solution. Soret coefficients S* can be obtained from the concentration ratios, and this evaluation of S* is not affected by some of the approximations introduced in our calculation of D&. It is, therefore, interesting to compare the values of each of these parameters (columns 6 and 7 of Table 11). The ratio D e / S * gives the isothermal diffusion coefficient D* of the liquid contained in the pores. In the last column of Table I1 values of D* calculated in this way are reported, Apart from the fluctuations in these values in the neighborhood of the inversion points which are probably not significative, there is a marked increase of D* in the narrowest channels, in general keeping with the foregoing discussion. To conclude, all the experimental evidence gathered SO far lends, in our opinion, strong support to the theoretical approach proposed above. This approach in turn appears as a suitable guideline for further experimentation. Acknowledgment. The authors are very grateful to Professor A. Klemm for enlightening suggestions and constructive criticism. References and Notes (1) C.Ludwig, S - 8 Akad. Wiss. Wien, 20, 539 (1856). (2) Ch. Soret, Arch. Sci. Phys. Nat., [3],2, 48 (1879).
Deconvolution in Single Photon Counting (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) (29)
Ch. Soret, C. R . Acad. Sci., faris, 91, 289 (1880). Ch. Soret, Ann. Cbim. fbys. [5], 22, 293 (1888). G. Lipmann, Compt. Rend., 145, 104 (1907). M. Aubert, Ann. Cbim. fbys. [a], 20, 145, 551 (1912). H. Freundlich, "Kapillarchemie", 2nd ed, Akademische Verlagsgesellschaft, Leipzig, 1922 p 371; ibid., Vol. 1, 3rd ed., 1930. B. V. Derjarguin and G. Sidorenkov, Doki. Acad. Sci. USSR, 32, 622 (1941). H. P. Hutchinson, I. S.Nixon, and K. G. Denbigh, Discuss. Faraday Soc., 3, 86 (1948). H. F. Winterkorn, "Proceedings of the 27th Annual Meeting of the Highway Research Board", 1947, p 433. R. Haase, Z. Pbys. Chem. (Frankfurt am Main), 21, 244 (1959). R. Haase and C. Steinert, Z. fbys, Cbem. (Frankfurtam Main), 21, 270 (1959). R. Haase and H. J. De Greiff, Z. fbys. Cbem. (Frankfurt am Main), 44, 301 (1965). R. Haase, Z. fbys. Cbem. (Frankfurt am Main), 51, 315 (1966). R. Haase, H. J. De Greiff, and P. Buchner, Z. Naturforscb. A , 25, 1080 (1970). R. Haase and H. J. De Greiff, Z . Naturforsch. A , 20, 1773 (1971). C. W. Carr and K. Sollner, J . Electrochem. SOC.,109, 616 (1962). R. P. Rastogi, Blokhra, and R. K. Agarwal, Trans. Faraday Soc., 60, 1386 (1964). R. P. Rastogl and K. Singh, Trans. Faraday Soc., 62, 1754 (1966). R. P. Rastogi, K. Singh, and P. C. Skukla, Proc. Int. Symp. Fresh Water Sea, 3rd, 4, 203 (1970). R. P. Rastogl, K. Singh, and P. C. Skukla, Indian J. Cbem., 9, 1372 (1971). Y. Kobatake and H. Fujita, J. Cbem. fbys., 41, 2963 (1964). N. Riehl, Z. Elektrochem., 49, 306 (1943). H. Voellmy and P. Lauger, Ber. Bunsenges. Phys. Cbem., 70, 165 (1966). M. S. Dariel and 0.Kedem, J . fbys. Cbem., 79, 336 (1975). W. E. Goldstein and F. H. Verhoff, AICbE J . , 21, 229 (1975). H. Vink and S.A. A. Chishti, J . Membr. Sci., 1, 149 (1976). F. S.Gaeta, Pbys. Rev., 182, 289 (1969). F. S.Gaeta, Sommerschule "Physik des flussigen Zustandes", St. Georgen, Sept 18-29, 1972, F. Kohler and P. Weinzierl, Ed., Universitiit Wien, IV, pp 9-29.
Tbe Journal of Physical Chemisfry, Vol. 83, No. 17, 7979 2285 (30) F. S.Gaeta and A. Di Chiara, J . Polym. Sci., Pbys. Ed., 13, 163 (1975). (31) F. S.Gaeta and N. M. Cursio, J. Poiym. Sci. A- 1, 7 , 1697 (1969). (32) F. S. Gaeta, A. Di Chiara, and G. Perna, Nuovo Cimento 6 , Ser. X . 06. 260 (1970). (33) G.' Brescia, E. Grossetti, and F. S. Gaeta, Nuovo Clmento 6 , Ser. XI, 8, 329 (1972). (34) F. S.Gaeta, G. Brescia, A. Di Chiara, and G. Scala, J. Polym. Sci., fbys. Ed., 13, 177 (1975). (35) F. S.Gaeta, G. Perna, and G.Scala, J. Polvm. Sci., fbys. Ed., 13, 203 (1975). (36) F. S.Gaeta, D. G. Mita, G. Perna, and G. Scala, Nuovo Cimento 6, Ser. X I , 30, 153 (1975). (37) F. S.Gaeta, "Proceedings of the IVth International Winter School on the Biophysics of Membrane Transport", WlsYa, Poland, Feb 19-28, 1977, part 111, pp 67-106. (38) B. V. Derjarguin, Nature (London), 138, 330 (1936). (39) E. Forslind, Acta Polytecbi., 115, 9 (1952). (40) E. Forslind, "Proceedings of the Second Internatlonal Congress on Rheology", Butterworths, London, 1953. (41) B. V. Derjarguinand A. S.Titjevskaya, Proc. Int. Congr. Surf. Act., 2nd. 1. 211 119571. (42) B. V.Derjarghn and V. V. Karassev, Proc. h t . Congr. Surf. Act., Znd, 3, 531 (1957). (43) . . M. S.Metsik, "Research in the Flekl of Surface Forces", Vol. 11, Nauka Press, Moscow, 1964, p 138. (44) N. V. Churaev, B. V. Derjarguin, and P. P. Zolotarev, Dokl. Acad. Nauk USSR, 183, 1139 (1968). (45) J. Clifford, Water, Compr. Treat., 5, 75-132 (1975). (46) F. S.Gaeta and D. G. Mita, J . Membr. Sci., 3, 191 (1978). (47) F. Belluccl, M. Bobik, E. Drbll, F. S. Gaeta, D. G. Mita, and G. Orlando, Can. J. Cbem. Eng., 30, 698 (1978). (48) F. Belluccl, E. Drioli, F. S.Gaeta, D. G. Mita, N. Pagliuca, and F. G. Summa, J. Cbem. Soc., Faraday Trans. 2 , 75, 247 (1979). (49) A. H. Emery, Jr., and H. G. Drickamer, J . Cbem. Phys., 23, 2252 (1955). (50) G. Langhammer, Naturwissenschafien, 41, 525 (1954). (51) 6. Langhammerand K. Quitzsch, Makromol. Cbem., 17, 74 (1955). (52) G. Langhammer,H. Pfenning, and K. Quitzsch, Z. Electrochem., 62, 458 (1955).
Applications of Fast Fourier Transform to Deeonvolution in Single Photon Countingt J. C. Andre, Laboratoire de Cbimie Generale, E.R.A. No. 136 du C.N.R.S.,E.N.S.I.C., 52042 Nancy, Cedex, France
L. M. Vincent, L.S.G. C.-E.N.S.I.C., 54042 Nancy, Cedex, France
D. O'Connor, and W. R. Ware* The Photochemistry Unit?Department of Chemistry, University of Western Ontario, London, Ontario, N6A 587. Canada (Received August 28, 1978; Revised Manuscript Received April 30, 1979)
A technique, based on the fast Fourier transform (FFT), for the deconvolution of fluorescence decay curves is proposed. To use this technique the exact functional form of the fluorescence decay law need not be assumed. The difficulties usually encountered with the Fourier transforms of noisy data are greatly reduced by choosing a limit v1 in the frequency domain and extrapolating the transformed function for v > v1 by the Fourier transform of an exponential. Inverse transformation leads in most cases to a deconvoluted decay law practically free of unwanted oscillations.
Introduction Fluorescence decay curves of electronically excited molecules that are pumped by flashes of light of short duration are distorted by the finite width of the excitation pulses and by the limited frequency response of the This is publication no. 227 of t h e Photochemistry Unit,
OO22-3654/79/2083-2285$01 .OO/O
e1ectronics.l Because of the linear nature of the fluorescent system and the detecting apparatus one can write the COnvolution equation f(t) = e(t)*d(t)*r(t)
(1)
in which f(t) is the observed decay curve, e(t) is the true time profile of the exciting pulse, r(t) is the time response 0 1979 American Chemical Society
