Langmuir 1993,9, 2225-2231
2225
Absorption of Sparingly Soluble Vapor in Microdroplets: A Study Based on Light Scattering Asit K.Ray* and James L. Huckaby Department of Chemical Engineering, University of Kentucky, Lexington, Kentucky 40506-0045 Received March 25,1993. In Final Form: May 28, 1993 A technique, based on intensities of light scattered by small droplets in the planes parallel and perpendicular to the polarization plane of the incident light, has been developed for the determination smallsize and refractive index changes that occur due to the absorption of minute amounts of vapor. The techniquehas been appliedto study absorptionof nearly-immisciblewater vapor in singledioctyl phthalate droplets suspended in an electrodynamic balance under controlledhumidities. Wavelengths of transverse magnetic (TM) and transverse electric (TE)mode resonances in the scattered light were measured independentlyby scanninga ring dye laser and used to determine the size and refractive index of a droplet in dry air. Shifts in the TE and TM mode resonating wavelengths, after an exposure of the droplet to a humid environment, were interpreted to obtain minute size and refractive index changes due to the absorption of water vapor. The results show that the absolute size and size change can be detected with a resolution of one molecular layer. Moreover, the technique can be used to detect the formation of an adsorbed monolayer on the surface of a micrometer size spherical particle. The results on the absorption water vapor show that the concentrationof water in a droplet increases as the water vapor saturation ratio increases up to the saturation limit. The data have been analyzed to obtain vapor-liquid equilibriumdata over a wide range of compositions.
Introduction Many atmospheric, indoor air, and industrial processes involve physical interaction between small droplets and surrounding vapor phase molecules. Such an interaction results in absorption and adsorption of molecules from the vapor phase. When a vapor phase species in the liquid state is infinitely miscible with the droplet phase, droplets can remove that species from the vapor only by absorption, and the concentration of the absorbed species increases as the partial pressure of the species increases. Rubell and Ray et have experimentallyexamined the problem of absorption of miscible water vapor on single phosphoric acid and glycerol droplets. However, when a vapor component in the liquid state is partially miscible or nearlyimmiscible with the droplet phase, which is of interest in the present study, a droplet only absorb vapor as long as the partial pressure of the vapor is below a certain critical limit. Above the critical limit the vapor molecules can form an immiscible phase on the droplet by heterogeneous nucleation which may be preceded by the formation of an adsorbed m~nolayer.~ The absorption and subsequent formation of an immiscible phase on liquid droplets exposed to an immiscible vapor have not been examined previously. These phenomena may be examined theoretically using the bulk-phase equilibrium data. Even though there are accurate techniques available for the determination of vapor-liquid equilibrium data for miscible binary systems using bulk4 as well as microdroplet systems,2*6**current techniques for partially miscible systems are time-consuming and prone to large errors for systems with low miscibility limit^.^ I n most
* To whom correspondence should be sent.
(1) Rubel, G. 0.J. Aerosol Sci. 1981,2, 551. (2) Ray, A. K.; Johnson, R. D.; Souyri, A. Langmuir 1989,5, 133. (3) Van Der Hage, J. C. H. J. Colloid Interface Sci. 1984, 101, 10. (4)Waals, S. M. Phase Equilibrium in Chemical Engineering; Buttarworth Boeton, MA, 1985. (6) Allen, T. M.; Taflin, D. C.; Davis, E. J. Ind. Eng. Chem. Rea. 1990, 29, 682. (6) Tang, I. N.; Munkelwitz, H. R. Aerosol Sci. Technol. 1991,15,201.
cases, data on the miscibility limits in conjunction with a two-parameter activity coefficient model (e.g. van Laar equation) are used to predict vapor-liquid equilibrium compositions. The purpose of the present study is to develop a technique for the determination of minute size and composition changes of a homogeneous droplet and for the detection of an adsorbed layer on the surface of a droplet with an objective to examine the interaction between a droplet and vapor molecules of an immiscible system. In this paper, we also show that the experimental data on single droplets obtained over a relatively short time period can be interpreted to determine vapor-liquid equilibrium data of partially miscible systems over a wide range of compositions. The technique used here is based on resonances (intensity peaks or troughs) observed in the light scattered by a droplet. In the following sections, we will briefly review light scattering theory to explain the rationale behind our experimental technique and then present our results on dioctyl phthalate droplets exposed to water vapor at various humidities.
Light Scattering by Spherical Particles For a linearly polarized plane electromagneticradiation of intensity Ii and wavelength A, incident on a sphere, the far-field scattered intensities 1 1 and I2 in the planes perpendicular and parallel to the plane of polarization are, respectively, given b ~ 7 - ~
and ~~
(7) van de Hulst, H. C. Light Scattering by Small Particles; Dover: New York, 1981. (8) Kerker, M. The Scattering of Light and Other Electromagnetic Radiation; Academic Press: New York, 1983. (9) Bohren, C. F.;Huffman, D. R. Absorption and Scattering of Light by S m l l Particles; Interscience: New York, 1983.
0 1993 American Chemical Society 0143-1463/93/2409-2225$04.~~/0
2226 Langmuir, Vol. 9, No. 8,1993
where r (>>A) is the distance from the center of the sphere and t9 is the angle measured from the backside of the sphere. The angle-dependent functions are defined by
where P,l(cos 8) is the associated first-order Legendre function of first kind of degree n. For a homogeneous sphere of radius a, the scattering coefficients a, and 6 , of the transverse electric (TE) and transverse magnetic (TM) modes are functions of the size parameter x = 2raIX and refractive index of the sphere m,relative to the surrounding medium. The intensity of scattered light as a function of size parameter shows a series of peaks or troughs called resonances. A resonance occurs when the denominator of a scattering coefficient attains a minimum value, and the coefficient itself attains a maximum value of 1. A resonance due to a coefficient, a, or b,, is called a TM or TE resonance of mode n and order 1, where the resonance corresponding to the lowest value of x is labeled 1 = 1.The position of a resonance is interrelated to the size parameter x , and refractive index m,and low order resonances have extremely narrow widths. Thus, experimentallyobserved low order resonances can be used for accurate size and refractive index determination. Whether a particular resonance due to a scattering coefficient appears in a scattering spectrumor not depends on the contribution of the resonating term compared to the total contribution of nonresonating terms as well as on the angle-dependentfunctions T , and T,, as indicated by eqs 1and 2. Recently, Huckaby and Raylohave shown that in the neighborhood of e = 90°, T , >> T , for resonances that appear for x > 50. Thus, the terms due to TE electric modes (i.e., b, coefficients) dominate over the terms due to TM modes (i.e., a, coefficients) in eq 1,while a reverse situation occurs for eq 2. Under these conditions, intensities of scattered light in the planes perpendicular and parallel to the plane of polarization of the incident beam can be approximated by
Thus, only TE mode resonances are observed by a detector recording 11(8=90°) and only TM mode resonances by a detector recording 12(8=9O0). Equations 4 and 5 form the basis of our unambiguous, independent determination of TE and TM resonances. Two photomultiplier tubes (PMT's) placed at 6 H 90°, in the planes perpendicular and parallel to the plane of polarization of the incident beam, are used to detect independently TE and TM resonances, respectively. On the basis of the theory, an experimental scattered intensity versus size parameter spectrum at a fixed angle, showing multiple resonances, can be interpreted to obtain particle size and refractive index. Intensity spectra can be measured either by varying the wavelength of the incident (10)Huckaby, J. L.;Ray, A. K. Submitted for publication in Appl.
opt.
Ray and Huckaby
beam or by varying the size of a droplet illuminated by a fixed wavelength of light. When an evaporatingor a growing droplet is illuminated by a fixed wavelength laser beam, the intensity as a function of time shows a series of resonances due to the variation of the size parameter. By matchingsuch spectra with the theory, several investigators have obtained size and refractive index of homogeneous droplets as functions of time.25J1-13 Fixed wavelength varying size intensity spectra can be used to determine size and refractive index with high accuracy if the functional relation for the variation of size with time is known a priori or determined independentlyfrom other measurements. In the absence of such information, the uniquenessof size and refractive index determination cannot be guaranteed. Moreover, the technique can only be applied to a situation where the droplet size changes significantly to produce multiple resonances in the intensity spectrum. For the present problem where minute size and refractive index changes that occur due to absorption, atechnique based on varying wavelength only, deserves consideration. An intensity spectrum obtained by varying the incident wavelength shows a series of resonances at various incident wavelengths. It should be noted that unlike in fixed wavelength intensity spectra, the refractive index of a droplet varies with wavelength due to the dispersion. In previous studies, variable-wavelength intensity spectra were analyzed by two distinct ways. Ch$lek et al." used the shape of such a spectrum to obtain the size and refractive index of a droplet. However, they did not consider the dispersion in the droplet, thus sacrificing accuracy in their estimations. Hill et al.16 used a technique to determine the size of a particle of known dispersion from the resonances observed in a fluorescence scattering spectrum. Their technique is based on the alignment of observed resonating wavelengths with theoretical values, without considering the shape of the intensity spectra which depends on the scattering angle. It should be noted that the position of a resonance is slightly altered by the scatteringangle which can only be determinedby matching the shape of an intensity spectrum. In the present study, wavelengths at which resonances occur along with the shape of the intensity spectrum are used for the absolute size and refractive index determination. Positions of resonating wavelengths depend on the size and refractive index of a droplet, and the positions change when the size and refractive index of the droplet change due to the absorption of a vapor. The data on the shifts of the resonating wavelengths are utilized to determine minutesize and refractive index changes. Sincethe present study involves absorption of a vapor which is nearly immiscible in the liquid state with the droplet-phase component,we need to make sure that a droplet remains homogeneous when exposed to the vapor, that is, an absence of a second phase in the form of a layer on the surface. The data of the present experimental scheme can precisely discern the presence of a layer on the surface of a droplet. The method for the discrimination between minute size changes due to absorption and adsorption is based on the (11)Richardson,C.B.;Hightower, R. L.; Pigg, A. L.Appl. Opt. 1986, 25,1226. (12)Taflin, D. C.; Zhang, S. H.; Allen, T.;Davis, E. J. AICHE J.1988, 34,1310. (13)Ray, A. K.;Souvi, A.; Davis, E. J.; Allen, T. M. Appl. Opt. 1991, 30,3974-3983. (14)Chjrlek, P.;Ramaswamy, V.; Ashkin, A.; Dziedzic, J. M. Appl. Opt. 1983,22,2302. (16)Hill, S . C.;Rushforth, C. K.; Benner, R. E.; Conwell, P. R. Appl. Opt. 1985,24,2380.
Langmuir, Vol. 9, No. 8, 1993 2227
Absorption of Vapor in Microdroplets
theory of scattering by a coated sphere.ls Equations 4 and 5 also apply to coated spheres but the expressions for the scattering coefficients a, and bn depend on the size parameters x = 2uaIX and xc = 2uaJ X , based on the outer radius a, and the core radius a,, respectively, and the relative refractive indices m and m,, of the shell and core, respectively. The scattering coefficients for a coated sphere show resonances similar to a homogeneous sphere, and Hightower et al." and Ray et al.l8 have used intensity versus time data to determine core and/or outer radii of evaporating coated droplets. However, for two droplets, one homogeneous and one layered, having identical amounta of two components, the position of a resonance of the homogeneous droplet differs with the position of the identical mode and order number resonance of the coated droplet. On this basis, we have developed a method to determinewhether the addition of a speciesfrom the vapor phase to a microdroplet has resulted in the formation of an adsorbed layer or the species has dissolved homogeneously in the droplet. To understand the method, let us consider a droplet of pure A having a refractive index mA = 1.485. If the droplet radius is a = 19.0 pm, the detectors arranged to measure the TE and TM mode scattered light will observe the TE and TM resonance peaks due to the coefficientsbn=21+16 and a214,15, at the incident light wavenumbers VTE = 17 032.82 cm-l (i.e., x = 203.3387 or ATE = 587.1018 nm) and VTM = 17 055.21 cm-l (Le., x = 203.6060),respectively. If some arbitrary amount of pure B having a refractive index mB = 1.333 is added to the droplet, the spectral positions of these two resonances will change depending on whether B homogeneously mixes in the droplet or forms a layer. When B completely dissolves in the droplet, we assume that the volume change of the droplet can be obtained from the ideal solution law and that the refractive index of the homogeneous solution droplet follows the relation m = mAVA + m&-
uA)
(6)
where V A is the volume fraction of A. When the addition of B results in the formation of a layer on the surface of the droplet, we assume that the mutual solubilities of A and B are negligible. Thus, the refractive indices of the core and the layer of the resulting droplet are m, = mA and m = mB, respectively, and the volume of the layer is equal to the volume of B added. Under the above assumptions, the calculated shifts (Le., initial position before the addition of B - final position after the addition) of the two peaks versus the change in the droplet radius for the two possible scenarios are shown in Figure 1. Results show that as long as the droplet remains homogeneous, both peaks shift almost identically and the shift increases as the size increases. But when a layer of lower refractive index than that of the core forms, the TM resonance shifts more than the TE resonance, and the difference in the shift increases as the layer thickness increases. For example, for a size increase of 0.050 pm due to a homogeneous mixing of B, the TE peak shifts by A m = 31.26 cm-l and the TM peak by A m = 31.48 cm-1, and the shifts differ only by 0.22 cm-l. On the other hand, when a layer of B of 0.050 pm thickness forms, the calculated shifts of A m = 27.86 cm-l and A m = (16) Aden, A. L.;Kerker, M. J. Appl. Phys. 1961,22,1242. (17) Hightower, R. L.;Richardson,C.B.;Lin,H.-B.; Eversole, J. D.; Campillo, A. J. Opt. Lett. 1988,13, 946. (18) Ray, A. K.; Devakottai, B.; Souyri,A.; Huckaby,J. L. hngmuir 1991,7,525-531.
Figure 1. Calculatedshiftsof TE and TM mode resonance peaks versus size increase of a droplet due to homogeneous growth and due to growth of a surface layer.
34.44 cm-l are substantially different from each other and from the previous shifts for the homogeneous addition. Moreover, the results show that a measurable difference (i.e., about 0.11 cm-l) in the shifts of TE and TM mode resonances exists even for the formation of a monolayer (i.e., Aa N 10 A). Since in the present experimental scheme, described in the following section, TE and TM resonances are detected individually by two PMT's, we can precisely differentiate and detect small size changes due to the absorption or adsorption of a vapor of unknown refractive index.
Experimental Seation Experiments were performed in an electrodynamic balance where single DOP droplets were suspended in an air stream of controlled humidity. The balance consists of two central ring electrodesand two endcapelectrodesabove and below the central ring electrodes. A variable ac potential drives the central ring electrodeswhile a bipolar dc potential is applied across the endcap electrodes. A detailed description of the balance hae been provided by Dhariwalet al.'* On a charge particle, satisfyingthe stability criteria, the a.c. field creates a net time averaged force directed toward the center of the balance. At the null point (i-e., center) of the balance, the time varying force vanishes, and all vertical (i.e., gravitational and drag) forces are balanced by the force due to the dc potential across the endcap electrodes. A schematicof the experimental system is shown in Figure 2. The balance is housed in a cylindrical chamber. A constant temperature fluid circulates through the chamber. The thick wall and large mass of the chamber provide high thermal stability. Typically the temperature of the chamber varied lesa than 0.03 O C during the course of an experiment. To prevent evaporation of a suspended droplet, the chamber is saturated by DOP vapor by coating the wall with liquid DOP. An air stream, either dry or humid, enters through the bottom of the chamber. The humid air stream is generated by passing a dry air stream through an evaporator-condensersystem. The stream leaving the condenser is saturated with water vapor at the outlet temperature of the condenser. The dew points of the streamat the entranceand at the exit of the chamber are measured by chilled mirror dew point monitors. A tunable ring dye laser beam entering vertically through a hole in the bottom electrode is used to illuminate a droplet at the null point. The wavelength of the laser beam can be varied continuously in the range of 565 nm (Le., -17 700 cm-l) to 606 nm (i.e., 16 500 cm-1). The laser beam has a line width of less than 10" nm. The laser is computer controlled and has a
-
(19)Dhariwal, V.; Hall, P. G.;Ray, A. K. J. Aerosol Sci. 1993, 24, 197-209.
2228 Langmuir, Vol. 9, No. 8,1993
Ray and Huckaby Table I. A Comparison between the Observed and Calculated Peak Positions observed resonance calculated calculated droplet peak position, type mode and Peak position, rhc radius, a, pm vi,o, cm-' order (n,1) 16 788.862 TE(281,21) 274.198 57 25.9935 25.9936 16 878.637 TE(287,21) 275.666 49 16 968.512 TE(289,21) 277.133 02 25.9935 25.9935 TE(295,21) 281.522 99 17 237.266 17 322.150 TE(293,22) 282.908 32 25.9935 17 501.830 TE(297,22) 285.842 35 25.9934 16 938.033 TM(288,21) 276.634 32 25.9934 17 028.078 TM(290,21) 278.106 24 25.9935 25.9935 17 207.948 TM(294,21) 281.044 01 286.802 85 25.9936 17 560.536 TM(298,22) 17 650.561 TM(300,22) 288.272 73 25.9935 mean radius, U = 25.9935 pm standard deviation, u. = 0.63 A
W AR I
I
DRY AM
t
I
Figure 2. Schematic of the experimental system. wavemeter which measures the wavenumber of the beam with
a resolution of 5 parts in 108.
Two fixed-position photomultiplier tubes (PMT's) are positioned in the planes parallel and perpendicular to the plane of polarization of the laser beam to measure intensities of light scattered by a dropet at about 6 = 90°. Each PMT is placed behind a precision air slit to detect scattered light over A6 = 0.05O. As discussed before, the PMT in the plane parallel to the plane of polarization of the incident beam detectsonly TM mode resonances while the other PMT detects only TE mode resonances. In a typical experiment, a DOP droplet was suspended in a steady stream of dry air, and the laser was scanned from 16 700 to 17 700 cm-1while collectingintensitydata at intervals of 5000 MHz (0.1667 cm-I) to establish the positions of sharp resonances with well-defined peaks. The regionsaround the observed sharp resonances from both the detedors were scanned again with a data intervalof 260 MHz (0.00833cm-*)to determinethe positions of the resonances more accurately. After determining the resonance locations in the dry air stream, a humid stream was introducedby switchingoff the dry stream. The positions of two resonances, one from each detector, were monitored until the droplet growth due to the change in humidity terminated. After new positions were established for the resonances, the humidity level of the air stream was increased and then decreased in steps while monitoring the peak positions.
Data Analysis Light scattering theory is basis for the interpretation of our experimental data. To determine the size of a droplet, we identified a number of sharp resonance peaks in the intensity spectra. These peak positions were measured with high resolution scans. A typical example of such peak positions, obtained from a DOP droplet in dry air at T = 23.4 'C, is given in Table I. We will describe the size determination procedure on the basis of this example droplet.
To estimate an approximate size of the droplet,*I2l we used the following relation M-'(m2 - 1)1/2 a= (7) 217Av,(m2 - 1)'l2 where Aum is the spacing in wavenumbers between two resonances of successivemodes having the same order and polarization. From the peak positions listed in Table I, we find that a number of TE and TM peaks are separated by multiples of about 90 cm-l. This separation distance is the spacing between two same order resonances whose mode numbers differ by two. Using the mode separation distance of 45 cm-l, and an approximate refractive index index of m N 1.485, we calculate an approximate droplet size of a N 26.8 pm. Plots of theoretical scattering intensity as a function size parameter, for values in the neighborhood of a N 26.8 pm, m N 1.485, O m N 90°, and OTM N 90') were visually compared with the experimental intensity versus wavenumber curves from the two detectors. It was found that theoretical spectra for the TE and TM mode detector angles of O m = 90.30' and t 9 = ~91.18' ~ showed good visual agreementswith corresponding experimentalspectra over a size range of 25.90 pm Ia I26.10pm and a corresponding refractive index range of 1.483 Im I1.488. In this range, a change in the refractive index shifted the theoretically calculated spectra to a higher or a lower size without affecting the visual appearances as well as the mode and order numbers of the individual resonances. Thus, the visual matching provides an acceptable range for the refractive index and a corresponding range for the droplet size for the observed spectra as well as mode and order numbers of the observed resonances. Although the theoretically generated spectra for over a range of wavelength-independentrefractive index showed good visual agreements with the experimental spectra, the positions of all the resonances in the observed spectra could not be aligned perfectly with those calculated from the theory due to the presence of dispersion. In this study we used the following three-parameter Cauchy dispersion formula for the alignment purposes m(u) = A + Bv2 + Cv4 (8) To determine the droplet radius and the constants of the dispersion formula, we used an alignment procedure as follows. For an assumed droplet size a, the size parameter Xi,o = 2 ~ a v i , at ~ , each observed resonating wavenumber vi,o, is known. Since the mode and order ~~
~~~
~
(20) Probert-Jones, J. R. J. Opt. SOC.Am. A. 1984,1,822. (21)Chglek, P.J. Opt. SOC.Am. A. 1990, 7, 1609.
Langmuir, Vol. 9, No. 8, 1993 2229
Absorption of Vapor in Microdroplets
w
0
9=0.016 cm-' h
0
a t a=25.9935 p m
T E mode, observed
I
TE mode. 0 = 90.30'. a=25.9935 p m . m from E q . ( l O )
I
TU mode. obierved
I
E
TM mode,
95.90
25.95
28.00 26.05 Droplet radius, a ( p m )
26.10
Figure 3. Peak alignment error versus droplet radius. numbers of the observed peak at vi,o have been established from the visual spectra matching procedure, we can calculate a refractive index mi, for the theoretical peak position at zip On this basis, for an assumed size in the acceptable range, we calculated a refractive index at each observed resonating wavenumber by aligning the experimental peak with the theoretical peak position. Then, by use of a regression routine, the calculated refractive index versus observed wavenumber data were fitted to eq 8 to obtain the best estimates of parameters A, B, and C. For each observed peak at vi,o, we calculated a theoretical peak position vi,c (i.e., Xi,c = 2 ~ a v i ,for ~ ) a refractive index m ( @ , obtained from the dispersion formula, using an iterative procedure. An estimate of alignment error between the observed and calculated peak positions was obtained from the following relation
where NpFis the number of observed resonances and Y i p [m(ubc),alis the theoretically calculated resonance position for a refractive index m(vi,J at wavenumber vi,c, obtained from the dispersion formula associated with the radius a. Using a computer program with a double precision accuracy, we repeated this procedure over the entire acceptable size range by changing the assumed droplet radius in incrementa of 1 A. The calculated alignment error 4 is a smooth function of droplet radius and has one minimum which is associated the best estimates of the droplet radius and ita dispersion parameters. Figure 3 showsthe calculated alignment error versus radius for the droplet under consideration. The only minimum in the alignment error occurs at the radius of a = 25.9935 pm, and the corresponding dispersion formula is mmP(v) = 1.4627815 + (9.5402029 X 10-")v2 (4.88811802 X 10-20)v4(10) The theoretical intensity spectra calculated for the estimated size and dispersion formula are compared with the experimental spectra in Figure 4. Except for the absence of very sharp resonances in the experimental spectra, the experimental spectra show excellent agreementa with the theoretical spectra. The absence of sharp resonances is due to the presence of a slight imaginary component in the refractive index which has been neglected
e =
91.18'.
a=25.9935 p m . m from E q . ( l O )
I
Wavenumber (cm-')
Figure 4. Comparison between experimentally observed and theoretically calculated scattering spectra for a DOP droplet. in the present study. Table I shows the experimental resonance peak positions in wavenumbers and corresponding theoretical peak positions in terms of size parameters, as well as the calculated radius at each peak position. The'resulta show that the mean value of the radius is 25.9935 pm, with a standard deviation of 0.63 A. All the calculated radius values lie within a = 25.9935 pm & 1 A. Thus, the size is estimated with a resolution of 4 parts in 10s. However, it should be noted that a roughness on the order of a molecular layer always exists on the surface of a dropet; thus, a size resolution of than one monolayer is physically meaningless. The resolution value indicated here is based on an average size as interpreted from the light-scattering data.
Results and Discussion In a typical experiment, a droplet was suspended in a dry air stream, and sharp TE and TM mode resonances of the droplet were measured. Then the partial pressure of water vapor in the stream was increased or decreased in steps. After each step change, positions of two peaks, one selected from the TE and another from the TM resonances, were monitored. The observed initial positions of TE and TM mode peaks in dry air and their final positions after a change in the humidity level were used to calculate the amount of absorbedwater. Figure 5 shows the initial and finalpositions of TM(294,21) and TE(295,21) resonances of the peaks listed in Table I. The data show that when the droplet growth terminated after an exposure to air with water vapor at a partial pressure pressure of 20.29 mmHg, the TE and TM peaks shifted by 5.0 and 4.97 cm-l, respectively, from their positions in the dry air. Almost identical shifta of TE and TM mode peaks indicate that the droplet remained homogeneous after the exposure to the humid air. In principle, observed shifta of two, one TE and one TM, peaks are sufficientto calculate the size and refractive index changes of a droplet. However, the presence of dispersion complicates the problem. In the present study, we used a method based on successive approximationsof the amount of water absorbed by the droplet. To implement this procedure, we assumed that the absorbed water in a DOP droplet underwent an ideal volumetric mixing and that the droplet was a single phase homogeneous solution of DOP and water. Furthermore, we assumed that the refractive index m(v) of the solution droplet can be described by the relation developed by
2230 Langmuir, Vol. 9, No. 8,1993
1
Dry a n ,
1
a = 25 9935 pm
1720798
17237 27
4
II S
Humid air pH = 2 0 2 9 m m Hg Aa=l'? 3 n m I I 1720301 1723227
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4
220
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a=29 8268 y r . T = 2 4 8 'C
-I 180 -,
/
a $1404
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4-
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/TM(294 21) Peak
2
Ray and Huckaby
*
/
(I-..i
TE(295.21) Peak
17195 17205 17215 17225 11235 17245 Wavenumber. e m -
Figure 5. Measured TE and TM resonance peak positions in dry air and in air with water at a partial pressure of ~ H , O= 20.29 mmHg.
Bruggeman22 m2(v)- mHp$(v) "H 0 3
2m2(u)- mH2-,?v)
+ (1 - uH,O)
m2(v)- mDop2(v) =O 2m2(v)- mWp2(u) (11)
where U H is~ the volume fraction of water. The above relation applies to randomly mixed two-component systems.9 For water at 25 "C, the following dispersion formula2swas used 1.17487 X lo6
+
V2
0.4345906~~ )'I2 (12) 6.742816 X lo9- v2 Assuming an initial value for the volume of water absorbed by the droplet, we estimated the droplet size and ita refractive index using eqs 10 through 12. Using the estimated size and refractive index, we calculated the theoretical shifts of the resonance peaks observed in the dry environment and compared the shifts with the observed values. The procedure was repeated until the theoretical peak shifts matched with the observed shifts, and an estimate of the amount of absorbed water was obtained. The resultant estimate of absorbed water by this successive approximation scheme was always found to be unique and unambiguous. For example,for the peak positions shown in Figure 5, TE and TM peak shifts give almost identical size changes of 11.3 and 11.2 nm, respectively. The agreement between the size changes estimated from the TE and TM peak shifts indicates that the volume change due to mixing is negligibly small, and the assumption of ideal solution behavior does not introduce any uncertainty in our results. We have also examined the effect of the solution refractive model on the calculated size change by comparing the results obtained from eq 11 with the results calculated from eq 6 and from the Maxwell-Garnett r e l a t i ~ n .For ~ small volume fractions involved in this study, the solution refractive model was found to have negligible effects on the calculated size changes. Figure 6 shows the calculated size increase of a droplet as a function of water vapor partial pressure P H ~ in , the ~~~
(22) Bruggeman, D.A.
G.Ann. Phye. (Leipzig) 1936,22, 636. (23)Tilton L.W.;Taylor, J. K.J. Res. Natl. Bur. Std. 1938,20,419.
0
5 10 15 20 25 Partial pressure of water pH,O ( m m Hg)
Figure 6. Size increase of a DOP droplet as a function of water vapor partial pressure in surrounding air.
air stream. Even at the highest relative humidity level (Le., about 99%)of the data, we observed almost identical shifts of TE and TM mode resonances; thus, the droplet remained homogeneous during the entire course of this experiment. As expected, the droplet size increases as the saturation ratio of water vapor increases. For example, the droplet size increase ranges from about 25 A at p ~ f i N 6 mmHg to about 180 A at p ~ N p23 mmHg. Moreover, the size of a droplet increases by nearly the same amount when it is subjected to the same humidity level during the increasing and decreasing humidity cycle. This indicates that a droplet attains a steady-state composition at the end of the adsorption or desorption process at a given humidity level. Ray et aL2have shown that when a nonvolatile droplet experiences an increase in the partial pressure of a volatile component in the surrounding gas phase, the droplet grows by absorbing vapor and then attains a steady-state composition at the end of the growth period. For a DOP droplet exposed to water vapor, the steady-steady composition can be expressed by where S = pH8/PHp0 is the water vapor saturation ratio or activity, P H ~is' the vapor pressure of water at the steady state temperature of the system, and ymp is the activity coefficient of DOP when the mole fraction of water in the droplet is X H ~ . The parameter 4 is defined by (14)
where D w p , ~is the diffusivity of DOP vapor in the surrounding gas mixture and Shmp is the Sherwood number for DOP vapor which depends on the droplet radius, gas stream velocity, and diffusivity. For the experimentalconditions, the value of3./ has been estimated to be less than le9; thus, from eq 13, we find that at the steady state a droplet remains nearly in equilibrium with water vapor in the gas phase and its composition is given by YHPg%O = (15) The size increase of a droplet due to the absorption at a given saturation ratio depends on its initial size. However,the steadystate composition of water in a droplet at a given saturation ratio is the equilibrium composition
Langmuir, Vol. 9, No. 8,1993 2231
Absorption of Vapor in Microdroplets 0.0020
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.-01 $
0.0000 0.0
0.6 0.8 0.4 Saturation Ratio, S
0.2
1.0
Figure 7. Droplet composition versus water vapor saturation ratio in surrounding air.
which is independent of its size. To examine the reproducibility of our data, we have plotted in Figure 7 the observed weight fraction of water as a function of the saturation ratio of water vapor for three droplets having different initial radii. The results show that all three droplets attain almost identical water composition at a given saturation ratio, thus demonstrating the reproducibility of our experimental data. The equation for the best-fit line is given by w
~
O
(7.328 X lO-')S + (1.618 X 109)S2 -
(4.961 X 104)S3 (16) where WH& is the weight fraction of water. The data in Figure 7 show that the water concentration in a droplet increases continuously as the water vapor saturation in the gas phase increases to the saturation limit. This may appear unusual for a nearly immiscible system like water-DOP. For such a system, one may expect concentration to increase until the miscibility limit of water is reached in a droplet. The water vapor saturation ratio at the miscibility limit is given by
S m = ~30,mxq0,m
(17)
where subscript m denotes the state at the miscibility limit of water in the DOP enriched phase. Normally, a DOP droplet in contact with water vapor at S > S m can absorb water above the miscibility limit XH~O,,,,, as long as the composition of water in the droplet satisfies the following metastable equilibrium criteria"
The maximum attainable value of XH&, given by the equality, is always greater than the miscibility limit, and only above that maximum value does the system become unstable. For the present study, at the maximum saturation ratio of S = 0.994, examined the system remained either at a stable or at a metastable equilibrium. For both these situations, eq 15 applies, and thus, the results in Figure 7 represent the vapor-liquid equilibrium data of water for the DOP-water system. Conclusions We have developed an experimental scheme by which TM and TE mode resonances can be measured separately by two detectors located in the planes parallel and perpendicular to the plane of polarization of the incident beam. We have used the experimental system to obtain intensity spectra from single suspended droplets. By matching the shapes of the spectra with theoreticalspectra, and by aligning the observed resonance peaks with theoretical peak positions, we have simultaneously determined the size and wavelength-dependentrefractive index of a DOP droplet. From the observed shifts of TE and TM mode resonances due to a change in the humidity of the surroundinggas phase, we have determined minute amounts of absorbed water in a DOP droplet. The results show that the absolute size and size change due to absorption can be detected with a resolution of one molecular layer. The technique used in this study can also be used to detect the formation of a monolayer layer on the surface of a droplet. We applied the technique to study interaction between water vapor and DOP droplets. The results show that when a droplet encounters a gas phase partially saturated with water vapor, it attains a steady-state composition which is identical to the equilibrium composition corresponding to the water vapor saturation in the gas phase. The experimental data on the same droplet show the same amount of size increase when subjected to the same humidity level, during the increasing and decreasing humidity cycle. Moreover, droplets of differing size show nearly the same composition at the same humidity, indicating that the data are highly reproducible. Moreover, the results show that the experimental technique can be applied to obtain vapor-liquid equilibrium data of nearly immiscible systems over a wide range of composition, encompassing stable and metastable equilibrium regions.
Acknowledgment. The authors are grateful to the National Science Foundation (Grant No. CTS-8912282), Department of Energy, U.S.Army ChemicalRD4E Center, and Brown and Williamson Tobacco Corp. for their generous support. (24) Modell, M.;Reid, R. C. Thermodynamics and its Applicatiom, 2nd ed.; Prentice Ha& Englewd Cliffs, NJ,1980,Chapter 9.
