pubs.acs.org/Langmuir © 2009 American Chemical Society
Collective Thermal Diffusion of Silica Colloids Studied by Nonlinear Optics Neda Ghofraniha,*,† Giancarlo Ruocco,†,‡ and Claudio Conti‡ †
Dipartimento di Fisica - Universita’ “La Sapienza”, Piazzale Aldo Moro 5, 00185, Rome, Italy, and ‡ SOFT-INFM-CNR, c/o Universita’ “La Sapienza”, Piazzale Aldo Moro 5, 00185, Rome, Italy Received May 15, 2009. Revised Manuscript Received September 16, 2009
We investigate the collective thermal diffusion of silica charged spheres in Sulpho-Rhodamine B/water solution at different concentrations by measuring time-dependent thermal and Soret lensings. We show a significant concentrationdependence of the thermal diffusion coefficient DT, not previously reported. Moreover, the results clearly show that both mass diffusion and Soret coefficient are collective quantities being strongly dependent on the particles’ packing fraction. Our Z-scan setup allows us to investigate the dynamics of the system at low wave vector, which addresses the influence of the interparticle interactions on the thermal diffusion of the colloids.
I. Introduction Thermal diffusion, thermophoresis, or Soret effect is commonly referred to as matter flow in temperature gradient. Although such a phenomenon has been extensively studied since the 19th century, its comprehension is well-developed only in some limited cases such as as gas mixtures1 and aerosol particles, where the heavier component diffuses toward low temperature regions and, for equal mass species, the larger component moves toward the colder region.2-4 Thermodiffusion in liquid mixtures has been widely experimentally and numerically investigated, as reviewed in ref 5. In polymeric and colloidal dispersions, thermophoresis is typically characterized by Soret (ST) and thermal diffusion (DT) coefficients; the latter is proportional to the ratio of the particle concentration gradient and temperature gradient, once the system is at stationary state, and the former is the product of ST and the mass diffusion coefficient. The sign of ST is related to the direction of migration: for ST > 0, particles move from hot to cold regions, and the reverse happens when ST < 0. The amplitude and the sign of Soret and thermodiffusion coefficients have been investigated as functions of salinity of the
solution,6-9 temperature,10-13 particle size,14-16 and concentration.13,17-20 Despite several theoretical descriptions,21-25 a generally accepted theory for thermally driven particle diffusion has not yet been established. In this paper, we investigate the thermodiffusion of charged silica spheres suspended in dye doped water by means of a nonlinear optics technique: the time-dependent Z-scan.26,27 This allows us to accurately determine both the collective diffusion and the Soret coefficients. Moreover, it overcomes the geometrical limitations of beam deflection methods28,29 and the complicated optical design of the thermal diffusion forced Rayleigh scattering method.30 We analyze the time-dependent nonlinear optical response of different colloidal dispersions at room temperature by changing the volume fraction. We observe that in all cases particles move toward the hot region and the absolute value of ST (i.e., the magnitude of the thermal diffusion in steady state) decreases by increasing the colloidal concentration, and this signals that particle mutual interaction affects the thermal diffusion. We also find that the absolute value of DT (related to the dynamics of the particles in a thermal field) significantly decreases by increasing the colloidal concentration.
*Corresponding author. E-mail:
[email protected]. (1) Chapman, S. Philos. Trans. R. Soc., A 1912, 211, 433. (2) Tyndall, J. V. Diffusion and Heat Flow in Liquids; Buttherworths: London, 1961. (3) Waldmann, K. H.; Schmitt, L. Thermophoresis and diffusio-phoresis of aerosols. In Aeosols science; Davies, C. N., Ed.; Academic Press: New York, 1966. (4) Grew, K. E.; Ibbs, T. L. Thermal Diffusion in Gases; Cambridge University Press: Cambridge, UK, 1952. (5) Wiegand, S. J. Phys.: Condens. Matter 2004, 16, 357. (6) Putnam, S. A.; Cahill, D. G. Langmuir 2005, 21, 5317. (7) Ning, H.; Dhont, J. K. G.; Wiegand, S. Langmuir 2008, 24, 2426. (8) Rasuli, S. N.; Golestanian, R. Phys. Rev. Lett. 2008, 101, 108301. (9) W€urger, A. Phys. Rev. Lett. 2008, 101, 108302. (10) Putnam, S. A.; Cahill, D. G. Langmuir 2007, 23, 9221. (11) Duhr, S.; Braun, D. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 19678. (12) Iacopini, S.; Rusconi, R.; Piazza, R. Eur. Phys. J. E 2006, 19, 59. (13) Ning, H.; Buitenhuis, J.; Dhont, J. K. G.; Wiegand, S. J. Chem. Phys. 2006, 125, 204911. (14) Spill, R.; K€ohler, W.; Lindenblatt, G.; Schaertl, W. Phys. Rev. E. 2000, 62, 8361. (15) Braibanti, M.; Vigolo, D.; Piazza, R. Phys. Rev. Lett. 2008, 100, 108303. (16) Duhr, S.; Braun, D. Phys. Rev. Lett. 2006, 96, 168301. (17) Zhang, K. J.; Briggs, M. E.; Gammon, R. W.; Sengers, J. V.; Douglas, J. F. J. Chem. Phys. 1999, 111, 2270. (18) Rauch, J.; K€ohler, W. Phys. Rev. Lett. 2002, 88, 1859011. (19) Fayolle, S.; Bickel, T.; Le Boiteux, S.; W€urger, A. Phys. Rev. Lett. 2005, 95, 208301. (20) Piazza, R.; Guarino, A. Phys. Rev. Lett. 2002, 88, 208.
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II. Experimental Section Materials. Silica colloidal spheres with diameter of 70 nm are dispersed in a 0.03 mM solution of Sulpho-RhodamineB (S-RhB) and deionized water. Dye molecules absorb visible light that locally heats the medium, and the induced thermal gradient influences both water density (thermal effect)31 and the colloidal particle concentration (thermodiffusive or Soret effect).32 Both phenomena lead to a nonlinear optical response characterized by (21) Dhont, J. K. G. J. Chem. Phys. 2004, 120, 1632. (22) Dhont, J. K. G. J. Chem. Phys. 2004, 120, 1642. (23) Ruckenstein, E. J. Colloid Interface Sci. 1981, 83, 77. (24) Dhont, J. K. G.; Wiegand, S.; Duhr, S.; Braun, D. Langmuir 2007, 23, 1674. (25) Parola, A.; Piazza, R. Eur. Phys. J. E. 2004, 15, 255. (26) Bahae, M. S.; Said, A. A.; Wei, Y. H.; Hagan, D. J.; van Stryland, E. W. IEEE J. Quantum Electron. 1990, 26, 760. (27) Ghofraniha, N.; Conti, C.; G. Ruocco, G.; Zamponi, F. Phys. Rev. Lett. 2009, 102, 038303. (28) Giglio, M.; Vendramini, A. Appl. Phys. Lett. 1974, 25, 555. (29) Giglio, M.; Vendramini, A. Phys. Rev. Lett. 1975, 34, 561. (30) K€ohler, W. J. Chem. Phys. 1993, 98, 660. (31) Sinha, S.; Ray, A.; Dasgupta, K. J. App. Phys. 2000, 87, 3222. (32) Rusconi, R.; Isa, L.; Piazza, R. J. Opt. Soc. Am. B 2004, 21, 605.
Published on Web 10/09/2009
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distinct time scales: ∼ms and ∼s, respectively. The thermal effect is responsible for local refraction changes (thermal lens), while the Soret effect can in principle produce either absorption or refraction variations. In the reported experiments, there is evidence that the colloidal diffusion does not influence light absorption (as shown below), probably due to the fact that S-RhB molecules are not adsorbed on the silica particle’s surface. In other words, as the colloidal particles move apart from the beam, the absorption coefficient is not relevantly affected, as we verify by direct measurements (see Figure 3). Nine samples at different colloidal volume fractions (φ = 0.005, 0.01, 0.02, 0.03, 0.05, 0.08, 0.11, 0.17, and 0.34) are considered. Experimental Setup. In the adopted Z-scan approach, the light source is a CW pumped diode laser operating at wavelength λ = 532 nm with 500 mW maximum power. For large time scales (0.1-1000 s), the laser beam radiation is modulated by a mechanical shutter, with aperture 6 mm and opening time 700 μs. The beam is focused by a 75 mm focal length lens producing a 20 μm beam waist 1/e radius w0 at the focal plane (z = 0). A photodetector with rise time 14 ns and an angular acceptance of 0.27° was used to probe the light power. With the beam spot on the detector being larger than the photodiode surface, only the light transmitted along the beam axis is collected, giving access to the nonlinear phase shift.26,33 The output signal is then amplified by a homemade amplifier with variable gain (104-107) and rise time 200 ns. The detection apparatus is connected to a computer-based data acquisition (DAQ) system directly triggered from the shutter controller, which allows one to observe the light intensity I(t) transmitted from the sample in time. For measuring light absorption, we use the same setup with a 25 mm focal length lens before the detector, thus collecting the whole transmitted beam. The colloidal dispersion is syringed between a standard glass window (on the bottom) and a coverslip (on the top), sealed by double-adhesive gene-frame (Thermo Scientific) with 0.25 mm thickness and 15 mm 16 mm longitudinal lengths. Samples are positioned horizontally and impinged from below by the laser beam. Convective effects are largely moderated because the smallest edge of the sample is parallel to gravity. The samples are positioned on two motorized translator stages with 0.1 μm minimum incremental motion and are moved along the two directions perpendicular to the beam axis. Each single measurement is taken at a different point of the sample not previously illuminated. This allows to reduce the time between measurements necessary for the system to return to equilibrium. Especially, each chosen point of the sample is separated by a 0.15 mm step (much larger than the beam waist). In addition, this approach ensures that the observed susceptibility is not affected by the history of the system under radiation. For high concentration samples, the shutter frequency is 0.01 Hz (duty cycle at 80%), the DAQ sample rate and channels are, respectively, 103 Hz and 9 104, and each output transmitted intensity profile is the average of 65 single measurements. For low concentrations, the shutter frequency is 0.001 Hz (duty cycle at 80%), the DAQ sample rate and channels are, respectively, 102 Hz and 9 104, and each output transmitted intensity profile is the average of 80 single measurements. All experiments are performed at room temperature.
III. Results and Discussion Thermal Lens (TL). When the laser beam is focalized through light absorbing materials (e.g., Rhodamine-dye solutions), it locally heats up the sample, generating an inhomogeneous density profile. The induced radial refractive index gradient acts as a diverging lens, and it changes the central-beam transmitted intensity. In Figure 1, we show the dynamics of normalized transmitted light intensity In(t) over the incident power P for 0.03 mM S-RhB aqueous solution for three different incident (33) Ghofraniha, N.; Conti, C.; Ruocco, G. Phys. Rev. B 2007, 75, 224203.
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Figure 1. Normalized transmitted light intensity divided by incident laser power of 0.03 mM S-RhB-water solution for three incident powers ((O) 4 mW, (0) 7 mW, (]) 12 mW) and two different sample positions (z0, full symbols). Solid lines are thermal lens theoretical curves (see text). Inset: sketch of z-scan profile; the full circles correspond to two different sample positions (z=-6.8 mm and z=+6.2 mm).
powers and two different sample positions with respect to the lens focus (z=0). The negative thermal-lens behavior and the linear dependence on the power P are evident. The normalized transmitted intensity In(t) is defined as IðtÞ - Iðt ¼ 0Þ ð1Þ In ðtÞ ¼ Iðt ¼ 0Þ The value I(t = 0) of the intensity at the starting time t=0 of the thermal effect is calculated by using the analytical expression for In(t) (see below), as derived from refs 34 and 35. The relation between the gradient of the refractive index Δn and the temperature ΔT gives the following radial beam phase variation
Dn ΔΦTL ðr, z, tÞ ¼ Leff k DT
½ΔTðr, z, tÞ - ΔTð0, z, tÞ
ð2Þ
T
where the effective length is defined as Leff = [1 - exp(-R0L)]/R0, with R0 being the absorption coefficient and L being the thickness of the sample; r is the transversal radial coordinate, k = 2π/λ is the vacuum wave vector and, T is the sample temperature. The temperature gradient ΔT(r,z,t) is given by the solution of the heat equation36,37 D χ ΔTðr, z, tÞ ¼ χr2 ΔTðr, z, tÞ þ Iðr, zÞ ð3Þ Dt K with χ and K being the thermal diffusivity and conductivity, respectively, of the solution and PLR0 -2ðr=wÞ2 Iðr, zÞ ¼ e ð4Þ πw2 the radial radiation intensity, with w=w(z) being the beam radius waist at the sample position z. The solution of eq 3 is reported in ref 34 with τTL = w2/4χ being the characteristic thermal diffusion time over the beam spot size together with the Fresnel diffraction analysis,35 that leads to the theoretical expression of the TL transmitted intensity ITL(t) as written in eq 8 of ref 34. In Figure 1, solid line gives In(t)/ P =[ITL(t) - ITL(0)]/[ITL(0)P] with zD = 38 cm, R0 = 0.3 mm-1, Leff = 0.24 mm, K = 0.58 W/m K (water thermal conductivity), (34) Sheldon, S. J.; Knight, L. V.; Thorne, J. M. Appl. Opt. 1982, 21, 1663. (35) J€urgensen, F.; Schr€oer, W. Appl. Phys. 1995, 34, 41. (36) Gordon, J. P.; Leite, R. C. C.; Moore, R. S.; Porto, S. P. S.; Whinnery, J. R. J. Appl. Phys. 1965, 36, 3. (37) Groot, S. R. D.; Mazur, P. Nonequilibrium Thermodynamics; North Holland: Amsterdam, 1962.
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Figure 2. Normalized transmitted light intensities of 0.03 mM S-RhB-water solution (φ = 0) and 0.03 mM S-RhB-water + silica colloids at two different volume fractions (φ = 0.02 and φ= 0.08), for incident power P=12 mW. Solid line is the theoretical curve for thermal lensing (see text).
Figure 3. Normalized transmitted light intensities of S-RhBwater + silica colloids for two different volume fractions, for measurements of absorption and refractive index perturbation. Open symbols, z = -6.8 mm; full symbols, z = 6.2 mm. (O) P=7 mW and (0) P=12 mW.
z = -6.8 mm (top line) and z = +6.2 mm (bottom line), and zc = 2.3 mm and using (∂n/∂T)T as a fit parameter. τTL = 4.4 ms is obtained by using w = 40 μm and the thermal diffusion coefficient of water χ = K/(FcP) = 14 10-8 m2, with the water density F=1 g/cm3 and specific heat capacity cP=4.18 J/(g K). In Figure 2, we show the normalized transmitted intensity of a dye-water solution in comparison with those of dye-watersilica suspensions for two distinct volume fractions. At short times, the thermal lensing is not affected by the presence of colloids, being only determined by the water density variation. At long time scales, the thermal diffusion of the silica spheres alters the profile of the transmitted light in a way that is strongly dependent on the colloid concentration. It is to stress that in Figure 2 the decoupling of the two effects (thermal and Soret) is evident (the temperature gradient is stationary at t = t* = 0.22 s). Soret Lens (SL). We investigate the collective thermal diffusion of colloidal silica (Figure 2) by analyzing the self-action of laser radiation propagating inside the sample. Figure 3 shows the normalized transmitted light intensity I(t)/I(0) for two volume fractions and incident laser power. The focusing Soret lens behavior of the system is evident at long times, because the corresponding perturbation to the refractive index has opposite sign to that due to the thermal effect. The silica index of refraction is higher than that of the solvent, and hence, light focusing implies that silica beads are attracted from high temperature regions and their diffusion results into an increase of the local index of refraction. Moreover, the absence of nonlinear absorption (flat curves in Figure 3) indicates that the motion of the spheres does Langmuir 2009, 25(21), 12495–12500
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Figure 4. Normalized transmitted light intensities over the laser power of S-RhB-water + silica colloids at three incident powers ((O) 4 mW, (0) 7 mW, and (]) 12 mW) and for different volume fractions.
not influence absorptive properties as the dye molecules are not adsorbed on the surface of colloids. In Figure 4, we show the normalized transmitted intensities (eq 1) over the incident laser power In(t)/P (P = 4, 7, and 12 mW) for four different colloidal packing fractions. When increasing the concentration, the build-up time of the Soret lens decreases, and this signals that the collective diffusion coefficient grows with the packing of the interacting spheres. At high concentrations, the deviation from the linear dependence on the solicitation at long times (t>20-30 s) may be caused by higher order nonlinear effects and probably by convective modes that are more evident at long time scales being the Soret effect at saturation, while at shorter times they are overwhelmed by the thermal and the thermal-diffusive signals. The analysis of the colloidal Soret lens can be performed by means of the diffraction integral formula from the HuygensFresnel principle, as for the TL. The beam phase perturbation is written as Dn ½Δcðr, z, tÞ - Δcð0, z, tÞ ð5Þ ΔΦSL ðr, z, tÞ ¼ Leff k Dc c where Δc(r,z,t) is the change of the radial particle mass fraction due to their thermal diffusion and c is its average value. Δc(r,z,t) is given by the solution of the mass flux equation37 D Δcðr, z, tÞ ¼ Dc r2 Δcðr, z, tÞ Dt þ Dc ST cð1 - cÞr2 ΔTðr, tÞ
ð6Þ
where Dc is the particle collective diffusion coefficient and ST is the Soret coefficient defined as 1 jrcj ð7Þ ST ¼ cð1 - cÞ jrTj As shown in Figures 2 and 4, the particle concentration gradient builds up, and after that the temperature profile reaches a stationary state; this allows to us take r2ΔT(r,t) in eq 6 from the stationary solution of eq 3. By using eq 4, the solution of eq 6 is hence given by 2 ! !3 R0 PST 4 2r2 2r2 1 5 Ei - 2 - Ei - 2 Δcðr, z, tÞ ¼ 4πK w w 1 þ 2t=τSL ð8Þ with τSL=w /4Dc being the characteristic collective mass diffusion time over the beam spot size. By using the Fresnel diffraction 2
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analysis as in TL and relations 5 and 8, the SL normalized transmitted light intensity results as 8 " #92 = ISL ðtÞ < θSL 2h ¼ 1atan ISL ð0Þ : ð9 þ h2 Þ τ2tSL þ 3 þ h2 ; 2 392 2 2 2 > = 1 þ þ h 1 þ2t=τSL 6 7 SL ln4 þ 5 > > 9 þ h2 ; : 4 8 > 0), some authors17,18 show that DT is independent of polymer concentration in the dilute (46) Corti, M.; Degiorgio, V.; Giglio, M.; Vendramini, A. Opt. Commun. 1977, 23, 282. (47) Piazza, R. Philos. Mag. 2003, 83, 2067. (48) G. Meyerhoff, K. N. J. Polym. Sci. 1962, 57, 227. (49) Ecenarro, O.; Madariaga, J. A.; Navarro, J. L.; Santamaria, C. M.; Carrion, J. A.; Saviron, J. M. Macromolecules 1994, 27, 4968.
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Figure 9. Collective Soret coefficient of S-RhB-water + silica colloids versus volume fraction measured by NLO technique. Solid line indicates the linear decay at low φ. Inset: Linearity of Dc measured by NLO at low φ.
Figure 10. Collective thermal diffusion coefficient DT of S-RhBwater þ silica colloids versus volume fraction measured by NLO technique.
regime and it decreases at higher concentrations approaching glass transition. On the other hand, in refs 48 and 49, the slight decay of DT when increasing the concentration is evidenced. For octadecyl coated silica beads in toluene (hard spheres),13 DT shows a weak concentration-dependence for φ e 0.1 and a pronounced decrease at higher φ, while for aqueous suspensions of charged polystyrene spheres6 DT turns out to be independent of particle concentration. On the contrary, in ref 50, the authors, while using a stochastic description, theoretically predict the decay of the thermal diffusion of charged repulsive colloids (ST > 0) with respect to the volume fraction. For micellar solutions51 at high temperature, DT increases monotonically with concentration at low concentration and it becomes concentrationindependent for the higher one, while at lower temperature a minimum is observed. To our knowledge, the results in Figure 10 show, for the first time, a significant concentration-dependence of thermal diffusion in a charged silica sphere suspension.
IV. Conclusions In conclusion, by detecting the time-dependent nonlinear optical susceptibility and by examining the Soret lens behavior of charged silica particles dispersed in dyed water, we characterized the collective dynamical response of the colloidal system. The motion of the silica particles toward high temperature regions locally increases the index of refraction of the sample. Our NLO setup has allowed us to detect the dynamics of the induced (50) Hernandez-Flores, A. O.; Mayorga, M. J. Phys.: Condens. Matter 2004, 16, 317. (51) Ning, H.; Datta, S.; Sottmann, T.; Wiegand, S. J. Phys. Chem. B 2008, 112, 10927.
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distortions in the transmitted radiation due to the refractive changes directly related to the matter flow. Such beam phase variations have been analyzed by means of the diffraction integral formula from the Huygens-Fresnel principle, and the dependence of the collective diffusion and Soret coefficients (observed at length scales much larger than particles’ size) on the packing of the particles has been clearly shown. Moreover, a significant decrease of the negative thermal diffusion coefficient DT by increasing the colloidal concentration has been reported. These results clearly show that both stationary (ST) and dynamical (DT) responses of charged silica spheres to external thermal field are
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“collective”, which means that they are strongly dependent on the interparticle interaction. The proposed nonlinear optical technique is hence a valid method to investigate the thermal diffusion of macromolecules, an issue that is still not well-understood from a microscopic point of view. Acknowledgment. We thank Roberto Piazza and Joao Carlos Filos Moraes for useful and interesting discussions. The research leading to these results has received funding from European Research Council under the European Community’s Seventh Framework Program (FP7/2007-2013)/ERC Grant Agreement No. 201766.
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