pubs.acs.org/Langmuir © 2009 American Chemical Society
Colloidal Attraction Induced by a Temperature Gradient R. Di Leonardo,*,† F. Ianni,‡ and G. Ruocco†,‡ †
CNR-INFM, CRS-SOFT c/o Dipartimento di Fisica and ‡Dipartimento di Fisica, Universit a di Roma “La Sapienza”, I-00185 Roma, Italy Received November 19, 2008. Revised Manuscript Received January 23, 2009
Colloidal crystals are of extreme importance for applied research and for fundamental studies in statistical mechanics. Long-range attractive interactions, such as capillary forces, can drive the spontaneous assembly of such mesoscopic ordered structures. However, long-range attractive forces are very rare in the colloidal realm. Here we report a novel strong, long-ranged attraction induced by a thermal gradient in the presence of a wall. By switching the thermal gradient on and off, we can rapidly and reversibly form stable hexagonal 2D crystals. We show that the observed attraction is hydrodynamic in nature and arises from thermally induced slip flow on particle surfaces. We used optical tweezers to measure the force law directly and compare it to an analytical prediction based on Stokes flow driven by Marangoni-like forces.
Thermophoresis, or the Soret effect, is the process of mass transport along temperature gradients. Particles ranging from single DNA molecules to micrometer-sized colloids may be manipulated, concentrated, and fractionated in nonuniform temperature environments.1 In liquids, thermophoresis has been known for a long time,2,3 but the physical mechanism driving it is poorly understood. The amplitude and direction of the phoretic effect strongly depend on the investigated system and seem to be related to the detailed microscopic nature of the particle-solvent interface.4,5 However, it is generally assumed that phoretic processes are driven by flow in the interfacial layer,6 resulting in slip boundary conditions on the particle surface. Many body effects are then to be expected when interparticle distances are so small that flow fields produced around each particle overlap. It has been predicted by Squires7 that such hydrodynamic interactions could lead to effective pseudopotentials when colloidal particles are confined to a wall by a phoretic motion. In a recent paper,8 Weinert and Braun reported direct experimental evidence for slip flow in thermophoresis. In accordance with Squires’ predictions, a “hydrodynamic attraction” was also observed when colloidal particles were confined to a wall by thermophoresis. The effective potential energy of such interaction was extracted from the distribution of relative distances between an interacting pair of particles and was compared to finite element numerical results. In this letter, we used holographic optical tweezers9 both to generate temperature gradients and to directly measure the resulting effective force between colloidal particles at a wall. The induced force is strong enough to lead to the formation of stable close-packed structures and can be efficiently exploited *Corresponding author. E-mail:
[email protected]. (1) Duhr, S.; Braun, D. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 19678. (2) Ludwig, C. Sitzungsber. Akad. Wiss. Wien, Math. Naturwiss. Kl. 1859, 20, 539. (3) Soret, C. Arch. Sci. Phys. Nat. Gen eve 1879, 2, 48. (4) Dhont, J.; Wiegand, S.; Duhr, S.; Braun, D. Langmuir 2007, 23, 1674. (5) Piazza, R.; Parola, A. J. Phys.: Condens. Matter 2008, 20, 153102. (6) Anderson, J. L. Ann. Rev. Fluid Mech. 1989, 21, 61. (7) Squires, T. J. Fluid Mech. 2001, 443, 403. (8) Weinert, F.; Braun, D. Phys. Rev. Lett. 2008, 101, 168301. (9) Curtis, J.; Koss, B.; Grier, D. Opt. Commun. 2002, 207, 169.
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to drive the quick self-assembly of a colloidal crystal. We propose an analytical expression for the effective force, which corrects the previous expression proposed by Squires, and compare it to available experimental data. A thin (∼15 μm) sample cell is obtained by sandwiching a small drop of sample solution between a coverslip and an absorbing yellow filter as the microscope slide. The sample consists of 2-μm-diameter silica beads (Bangs Laboratories SS04N) in a glycerol/water mixture (anhydrous glycerol, Fluka, and Milli-Q-quality water with an electrical conductivity of less than 10-6 Ω-1 cm-1 at room temperature). High temperature gradients (∼1 K/μm) can be produced by focusing a laser beam on the absorbing ceiling of the sample cell (Figure 1). Although temperature gradients are quite high, the overall concentration change across the cell gap for a wateralcohol mixture (ST ≈ 10-3K-1 10) is expected to be only a few percent. The same microscope objective (100 NA 1.4 in a Nikon TE2000U inverted microscope) is used both for focusing the heating beam and for bright-field imaging. Before entering the back aperture of the objective, the laser beam is expanded and reflected off of a spatial light modulator (SLM, Holoeye LCR2500). Modulating the incoming wavefront, laser light can be focused in multiple spots located in the 3D space around the objective focal plane.11 It is then possible to focus the laser heating light on one or multiple spots on the cell ceiling while the focal plane is located on the cell floor. As soon as the laser is turned on, the particles move toward colder regions under the effect of thermophoresis, following the direction of the negative thermal gradient, as expected for thermophobic behavior.5 We used a finite element method (COMSOL Multiphysics) to solve the stationary heat equation, with a heat source term modeled as a Gaussian beam that propagates in an absorbing medium (top coverslide). The distribution of the isothermal surfaces (black lines) and the temperature gradient direction (white lines) due to the absorption of a laser beam by the optical filter at the top are depicted (10) Tyrrel, J. Diffusion and Heat Flow in Liquids; Butterworths: London, 1961. (11) Di Leonardo, R.; Ianni, F.; Ruocco, G. Opt. Express 2007, 15, 1913.
Published on Web 3/5/2009
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Figure 2. Brownian motion of a single particle in the inner region of a ring of heated spots. The starting and ending points of the recorded trajectory are indicated by a circle and a square, respectively. The trajectory spans a 25 s time interval, during which the particle explores a region of linear size larger than its diameter. This observation excludes the possibility of particle close-packed clustering due to thermophoretic drift at the center of the ring.
Figure 1. Temperature field in the cell. Isothermal contours with 1 K spacing are shown as black lines, and white lines represent the temperature gradient streamlines. (a) Single beam at r = 0 and (b) same power distributed along a ring of radius 15 μm. in Figure 1a. Once the particles reach the bottom wall, a strong interparticle interaction is manifested, leading to the formation of close-packed clusters. To reduce the radial components of the temperature gradient, we focused on the top absorbing wall an array of 20 traps along a circumference of 30 μm diameter. A simulated temperature distribution for this geometry is reported in Figure 1b, showing that the temperature gradient is almost purely orthogonal to the wall, in the inner region of the heated ring. Isolated particles in the inner ring region are pushed against the bottom wall and then perform unbiased 2D Brownian motion (Figure 2). However, thermophobic particles in the outer region are prevented from entering the heated region. In this way, we can have a small number of particles in the inner region and record the dynamics of cluster formation . Each cluster was able to find the closest packing structure in a few tens of seconds. A few sample clusters are shown in Figure 3a. These clusters are completely reversible and disintegrate as soon as the temperature gradient is turned off (Figure 3b). At higher concentrations, particles assemble into stable 2D crystals (Figure 3c). The existence of a confinement-induced long-range attraction between particles of like charge has been observed and debated during the past 10 years.12-14 Such an interaction is (12) Larsen, A.; Grier, D. G. Nature 1997, 385, 230. (13) Polin, M.; Grier, D.; Han, Y. Phys. Rev. E 2007, 76, 041406. (14) Trizac, E. Phys. Rev. E 2000, 62, 1465.
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also found to increase when the like charged particles penetrate the electric double layer at the interface.12 One could think that the only role of the thermal gradient is that of pushing the particles against the double layer of the bottom interface and that the observed interactions are ultimately electrostatic in nature. However like-charge attraction disappears on a metallic surface.15,16 We then checked (Figure 3d) that the attraction observed in our experiment persists on a metallic surface obtained through a gold coating that is 10 nm thick over the glass surface. The position of the laser beam inducing the thermal gradient is indicated by the brighter particle at the bottom, trapped by the beam and displaced from the bottom wall. Close-packed structures form both on a dielectric charged surface (i. e., the cover glass (visible as the clearer band)) and on a metallic surface (darker band). We cannot attribute the observed attraction to an imaging artifact17 because the formation and stability of small clusters clearly show the presence of a strong attractive force. Another candidate for the origin of the attraction could be the distortion of the isothermal surfaces in the fluid as a result of the different thermal conductivity of the particles and the spatial asymmetry induced by the presence of the wall. Because in our case thermophoresis drives particles along the negative thermal gradient, this effect may induce an attraction between particles touching a colder wall, when the thermal conductivity is higher than the solvent’s, or a repulsion in the inverse case. However, we always observe attraction in both cases of higher (silica) and lower (polystyrene) thermal conductivities. (15) Bowen, W.; Sharif, A. Nature 1998, 393, 663. (16) Carbajal-Tinoco; Gonzalez-Mozuelos, P. J. Chem. Phys. 2002, 117, 2344. (17) Baumgartl, J.; Bechinger, C. Europhys. Lett. 2005, 71, 487.
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Figure 4. Streamlines of the flow generated by a point force pointing away from a plane wall with sticky boundary conditions. The resulting flow can be represented as the superposition of the Stokeslet flow produced by the force F plus an image singularity sister consisting of a Stokeslet, a Stokes dipole and a source dipole.18 Such flow is characterized by a finite slip velocity on the particle surface 3 US ¼ - UT sin θ 2
Figure 3. Reversible self-assembly of colloidal particles in a thermal gradient. (a) Particles confined to the bottom wall by thermophoresis assemble into close-packed structures. (b) Reversible aggregation (first row, temperature gradient on) and disintegration (second row, temperature gradient off) of close-packed clusters. (c) Highly concentrated samples assemble in a stable 2D crystal. (d) Close-packed structures form both on a dielectric charged surface (i.e., the cover glass (visible as the clearer band) and on a metallic surface (darker band) obtained through a gold coating that is 10 nm thick over the glass surface. We suggest, in accordance with refs 7 and 8, that the underlying mechanism is hydrodynamic in nature. The basic idea is that even if the particles stop when they reach the bottom wall, Marangoni-like forces on particle surfaces drive a slip flow (Figure 4) moving tangentially from the planar wall toward the bulk liquid. To compensate for this tangential flow, an influx of liquid appears, which carries the other particles along. The resulting aggregation is then driven by pairwise attractive interactions and is not to be confused with the accumulation of thermophilic particles in hot regions,19 where particles are independently transported along temperature gradients by thermophoresis. Far from the boundary walls, a thermophoretic particle drifts along the temperature gradient at speed ð1Þ UT ¼ -DT rT where DT is the coefficient of thermal diffusion.5 The surrounding fluid will obey the Stokes equation with the boundary condition imposed by (i) the vanishing velocity field at infinity, (ii) no net force exerted on the particle, and (iii) the fluid velocity relative to the particle’s surface having a vanishing normal component. The resulting flow, in the laboratory frame, has the form of a source doublet:18 uðrÞ ¼
1 a3 ^ ^ ð3 r r -IÞ3UT 2 r3
To keep such a particle stationary in the temperature gradient, an external body force F has to be applied. Solving the Stokes equation with (i) vanishing flow at infinity, (ii) the application of a net force F, and (iii) the normal component of fluid flow at the particle surface vanishing in the laboratory frame because the particle is stationary, we get uðrÞ ¼
1 1 ^^ 1 a2 ^ ^ ð r r þ IÞ3F ð3 r r -IÞ3F 8πη r 8πη r3
ð4Þ
The flow is thus the superposition of a Stokeslet (first term) and a source doublet. The value of F can be determined by imposing the condition that fluid slips over the particle surface with the same speed as in free phoretic motion to obtain F ¼ -6πηaUT
ð5Þ
Following Wurger,20 we could impose the condition that the fluid stress over the particle surface has to remain the same to obtain F = -4πηaUT. At large enough particle separations, the ambient flow produced by one particle at the location of the other can be obtained as the flow propagating from a point force. In the absence of the wall, such flow would be described by the socalled Stokeslet,21 which would not have any attractive component on another particle at the same height. When the no-slip condition imposed by the wall is taken into account, the resulting flow will have a nonvanishing component ux pulling the two particles together. Such a flow can be evaluated using the Lorenz reflection method.22 In the Squires treatment of the problem,7 only the first reflection of the Stokeslet term was taken into account, which corresponds to the superposition of three image singularities: a Stokeslet, a Stokes doublet, and a source doublet. When evaluated at the center of a nearby particle touching the wall, the first two singularities cancel out and we are left with an image source doublet only. The reflection of the force-free (source doublet) component in eq 4 is therefore of the same order of the
ð2Þ
(18) Blake, J.; Chwang, A. J. Eng. Math. 1974, 8, 23. (19) Duhr, S.; Braun, D. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 19678.
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ð3Þ
(20) Wurger, A. Phys. Rev. Lett. 2007, 98, 138301. (21) Pozrikidis, C. Boundary Integral and Singularity Methods for Linearized Viscous Flows; Cambridge University Press: New York, 1992. (22) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Kluwer Academic Publishers, Dordrecht, The Netherlands, 1983.
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reflected Stokeslet and cannot be negleted. The streamlines of such first order reflected flow are reported in Figure 4. Such a flow will have, on the center of a second particle touching the wall at a distance s, a component parallel to the wall given by7 ux ¼ -
3F a3 s 15F a5 s þ 5=2 πη ðs2 þ 4a2 Þ πη ðs2 þ 4a2 Þ7=2
ð6Þ
The first term in eq 6 is an attractive component whose magnitude is twice that reported in ref 7 (the reflected forcefree flow adds another source doublet to the Stokeslet image singularities). The second term is a repulsive component that accounts for the fact that at short distances streamlines become tangent to the particle surface and the attractive component of the flow is reduced. In the limit of large interparticle distances s, the flow component parallel to the wall decays as -3Fa3/πηs4. Such an ambient flow on the second particle will be undistinguishable from an attractive force of intensity f ¼ 6πηaλux ¼ -f0
a4 s ðs2 þ 4a2 Þ5=2
þ 5f0
a5 s ðs2 þ 4a2 Þ7=2
ð7Þ
where λ = λ(a/h) is an a-dimensional correction factor on the Stokes drag resulting from the presence of the wall.22 Using eqs 1 and 5 and introducing the Soret coefficient for the diluted suspension ST = DT6πηa/kBT, the force scale F0 can be written as f0 ¼ 18λST kB TrT
ð8Þ
where rT is the vertical component of the temperature gradient. Micrometer-sized colloids may have a Soret coefficient on the order of 10 K-1,1 which with temperature gradients on the order of 10 K/μm will result in forces F0 in the piconewton range. Interacting particles will then be subjected to Langevin dynamics under the action of such interaction forces in addition to the usual drag and stochastic forces. Such an effective force field may be derived from a potential function; therefore, particle statistics will be governed by Boltzmann statistics with an effective interaction potential of VðsÞ ¼ -V0
a3 3ðs2 þ 4h2 Þ3=2
þ V0
a5 ðs2 þ 4h2 Þ5=2
ð9Þ
with an energy scale of V0 = f0a. The condition for stable aggregation (V(2a)/kBT ≈ STrTa . 1) can be roughly stated as the following: the temperature difference across a particle has to be large enough compared to the inverse Soret coefficient. To validate the above expression, we used optical micromanipulation to measure the force law with interparticle distance directly. Holographic optical tweezers allow us to isolate a single pair of particles and vary their relative distance while gauging their interaction. In particular, a high-intensity optical trap (P = 42 mW) is focused next to the interface with the top wall in order to produce the thermal gradient, and two traps of lower intensity (P = 3 mW) are used to hold two particles on the bottom wall. We checked that the intensity of the last traps alone is low enough not to induce further attraction between the particles. To avoid fluid instabilities induced by the high 4250
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Figure 5. Intensity of the attractive force as a function of distance. Open symbols are experimental data, and the solid line is a fit to eq 7 with f0 = 7.8 pN. The inset shows the interaction energy in units of kBT as obtained from eq 9 using the fitted value for f0. thermal gradient,23 we have chosen a high-viscosity suspension (a 56% w/w glycerol-water mixture) and a thin cell gap (13 μm thick). The thermally induced interaction tends to push the particles toward each other and out of the optical traps until the restoring trap force balances the attractive force. By measuring the difference Δs between the trap centers distance and interparticle distance, the value of the attractive force can be deduced as f = kΔs, where k is the equivalent elastic strength of the two-trap system24 and may be determined experimentally from the mean square displacement of particle distance as ÆΔd2æ = kBT/k. Under the reported experimental conditions, we found k = 0.1 pN/μm. However, an experimental determination of the prefactor f0 in eq 8 is very hard because of the strong dependence of λ on the exact value of h and difficulties in measuring the temperature gradient. On the other side, the dependence on interparticle distance contains only the particle radius and can be accurately checked. Figure 5 reports the experimental determination of force as a function of interparticle distance together with a fit to eq 7 with f0 = 7.8 pN as the only fit parameter. The corresponding energy curve for the same f0 value is reported as an inset in Figure 5, which apart from an overall scaling factor of order 1 looks very similar to the data reported in ref 8. In conclusion, we have shown that a thermal gradient pushing thermophoretic particles against an interface can induce a strong, long-ranged attraction between like-charged particles, leading to the prompt formation of stable, ordered structures. We provide a static measurement of this interaction and quantitatively test our prediction for a hydrodynamic pseudoforce driven by Marangoni-like forces on particle surface. The long-range nature of the interaction may open new routes to the fabrication of ordered structures. This mechanism may have already enhanced the self-assembly observed under convective flow and is uniquely attributed to a confining effect.25-27 Acknowledgment. We thank M. Ortolani and the IPNCNR laboratories for kindly providing us with the goldcoated cover glass. (23) Boon, J.; Allain, C.; Lallemand, P. Phys. Rev. Lett. 1979, 43, 199. (24) Di Leonardo, R.; Saglimbeni, F.; Ruocco, G. Phys. Rev. Lett. 2008, 100, 106103. (25) Cheng, Z.; Russel, W.; Chaikin, P. Nature 1999, 401, 893. (26) Toyotama, A.; Yamanaka, J.; Yonese, M.; Sawada, T.; Uchida, F. J. Am. Chem. Soc. 2007, 129, 3044 . (27) Duhr, S.; Braun, D. Appl. Phys. Lett. 2005, 86, 131921.
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