Article pubs.acs.org/Macromolecules
Photothermal Deformation of a Transient Polymer Network F. Schwaiger and W. Köhler* Physikalisches Institut, Universität Bayreuth, D-95440 Bayreuth, Germany ABSTRACT: Gold nanoparticles (GNPs) are embedded in a highly entangled solution of an ultrahigh molar mass polymer (polystyrene Mw = 16 800 kg/mol in toluene). When an isolated GNP is heated by a focused laser beam, the solvent is attracted to and the polymer is pushed away from the particle surface by thermophoretic forces, thereby elastically deforming the transient polymer network. Because of the long disentanglement time, the randomly distributed GNPs are trapped in the meshes of the transient polymer network and allow for a visual observation of the network deformation field using an optical microscope. The observed displacement field of the meshes is of long-ranged nature and only limited by the finite size of the cuvette, despite of the local heating of a single submicrometer-sized GNP.
1. INTRODUCTION Colloidal gold nanoparticles (GNPs) are well suited for thermo-optical experiments due to their high absorption cross section in the visible spectrum.1−3 They are chemically stable, and their absorption does not bleach even after prolonged irradiation in a focused laser beam.4 Because of their small size, they can serve as almost pointlike heat sources and allow for very localized heat release on length scales far below the optical diffraction limit.5 Because of their tunable size on the nanometer scale, GNPs are well matched to length scales relevant in soft matter problems, which makes them ideal probes for studying dynamic processes in these materials. Grabowski et al.6 have investigated isothermal diffusion dynamics of GNPs in poly(butyl methacrylate) melts. By variation of the chain length, they managed to tune the entanglement mesh size, which has a dramatic influence on the nanoviscosity experienced by a colloidal particle. As predicted by Brochard Wyart and de Gennes,7 they found a significantly (up to a factor of 250) lower friction experienced by their GNPs than expected from the macroscopic shear viscosity. Optically heated GNPs can change their dynamic properties by modifying the carrier fluid in their immediate vicinity. Rings et al. have studied “hot diffusion” of laser-heated GNPs in water8 by taking the temperature dependence of the viscosity into account. Laser-heated GNPs dispersed in polystyrene(PS)/toluene solutions have been shown to modify their environment by creating a localized pocket of increased solvent concentration.9 The polymer, which in our case (but not always10) has a positive Soret coefficient, is pushed away from the hot particle surface by the thermophoretic forces in the spherical temperature field that surrounds the colloid. If nonlinear saturation by excessive heating is avoided, the thus formed concentration cage for the GNP is confined within ∼1 μm (90% value). © 2013 American Chemical Society
The experiments reported in ref 9 have been performed with a relatively short polymer chain length (Mw = 17.7 kg/mol). Besides the cage formation around the colloid addressed by the laser, which has been visualized by phase contrast microscopy, no other change could be observed in the sample. In particular, none of the randomly distributed and not heated GNPs showed any kind of correlated motion. For the experiments reported in this work, also PS has been used, but now with a much longer chain length (Mw = 16 800 kg/mol). In contrast to the shorter chains, the GNPs not hit by the laser beam now show a correlated radial displacement away from the heated particle. This motion directly reflects the deformation pattern of the transient polymer network, which is visualized by the GNPs that are trapped in the meshes formed by the entangled polymer chains. In the following we will investigate this reversible deformation of the polymer network in detail. We will see that the displacement is of long range, despite the local perturbation by the single laser-heated colloid.
2. EXPERIMENT The experimental setup consisted of an inverted light microscope (Olympus IX71) equipped with a phase contrast condensor (Olympus IX-LWPO) and objective (Olympus LCACH 40XPHP). Images were recorded with a CCD camera (PCO pixelfly) mounted on a trinocular tubus. A diode-pumped solid-state laser (Coherent Verdi V-5, λ = 532 nm) was coupled into the sample through the same objective as used for observation. A notch filter (AHF F40-531) protected the camera from reflected and scattered laser light. Polystyrene with a molar mass of Mw = 16 800 kg/mol and a polydispersity of Mw/Mn = 1.33 was purchased from PSS Polymer Standards Service GmbH and toluene p.a. (≥99.9%) from Merck. The aqueous gold colloid solution was obtained from British Biocell Received: December 31, 2012 Revised: January 30, 2013 Published: February 12, 2013 1673
dx.doi.org/10.1021/ma302665p | Macromolecules 2013, 46, 1673−1677
Macromolecules
Article
International. The colloid diameter of 2R = 282 ± 14 nm was measured by TEM. A small amount of PS was weighted into a screw cap vial and diluted with toluene until a polymer mass fraction of c = 0.03 was reached. This stock solution was additionally sealed with Parafilm and very slowly agitated by a magnetic stirrer (RPM < 20 min−1) for at least 3 days. To prepare the sample for a measurement, a tiny amount of the stock solution was dried on the coverslip of a demountable cuvette (Hellma, QS, layer thickness 100 μm). A drop of colloidal solution was allowed to dry, leaving the colloids on the surface of the polymer film. The cuvette was filled afterward with polymer solution and sealed. The sample was then allowed to equilibrate for 1 day. The final polymer concentration was c = 0.03 ± 0.003.
3. RESULTS AND DISCUSSION Experimental Observation. Figure 1 shows a micrograph with the randomly distributed colloids. The laser is focused
Figure 2. Displacement of a GNP along a radial coordinate (shown in Figure 1). The slope of the baseline toward r → 0 is caused by the intensity modulation due to the concentration cage.
Figure 1. Radial displacement field (distance exaggerated, not all arrows plotted, image cropped) of the colloids around the laser-heated GNP in the center. The circular artifacts around the colloids are caused by the phase contrast technique employed to visualize the temperature and concentration cage around the heated GNP.
Figure 3. Colloid displacement ΔR as a function of radial distance r from the heated GNP. Transients after 1 and 2 s and stationary state after 11 s. The solid red line with horizontal asymptote is a prediction based on eqs 5 and 7 for an infinite system. The dashed line is the constant asymptotic shift for the approximate expression eq 8. The approximating fit accounts for a finite system size.
onto the central GNP. The temperature field around the particle and the polymer-depleted cage show up as a bright halo in the phase contrast image. As soon as the laser is switched on, the other GNPs undergo a radial outward motion away from the heated GNP. The arrows indicate the directions of the particle displacements, but note that their length is not on scale. This particle shift is fully reversible, and the GNPs return to their initial position once the laser beam is switched off. The stationary state is rapidly reached within less than 20 frames, corresponding to 10 s. When the same experiment is performed with a low molar mass polymer (Mw = 100 kg/mol instead of 16 800 kg/mol), a comparable cage formation is observed, but no particle displacement. This absence of a motion of the GNPs in case of shorter polymer chains under otherwise identical conditions is a clear proof that there is no noticeable direct thermophoresis of the GNPs themselves. Image analysis allows for a determination of the particle position to better than 50 nm by fitting a 2d-Gaussian function to the diffraction limited colloid images, and the displacement is obtained from the difference between the on and the off image. Figure 2 shows, in a simplified 1d-representation, the gray value plotted along a radial coordinate, shown in Figure 1 as a black line, through a colloid with the heated GNP defining the origin. The colloid can be fitted by a Gaussian function, and the shift between the on and the off state is clearly visible. The displacements of all colloids have been determined by an automated 2d tracking procedure and are plotted in Figures 3 and 4 as a function of the radial distance from the heated GNP together with a model function discussed below. Surprisingly,
Figure 4. Stationary colloid displacement ΔR as a function of radial distance r from the heated GNP for two different colloid temperatures.
the particle motion is a very nonlocal process, where even GNPs that are almost 100 μm away from the center undergo a notable shift of their position. This long-ranged nature of the displacement field is observed, although ∼90% of the amplitude of the temperature and concentration change are limited to a 1674
dx.doi.org/10.1021/ma302665p | Macromolecules 2013, 46, 1673−1677
Macromolecules
Article
radius of ∼1 μm around the heated GNP.9 Although no satisfactory time resolution was possible with the available experimental equipment, the transients after 1 and 2 s in Figure 3 nicely show the distance-dependent buildup of the network deformation by the thermodiffusive process. The stationary displacement is readily reached after 2 s for the nearest particles, where the distant GNPs are still at their initial position. Network Deformation. The concept of a long-lived transient polymer network is essential for the understanding of these experimental observations. Short polymer chains are either not entangled or disentangle much faster than the characteristic diffusion times of the polymer solution over the mesoscopic length scales of the experiment. There is no longterm memory in the polymer network, and the temperaturedriven interdiffusion of polymer and solvent is very similar to the case of a binary mixture of small molecules. The relatively large colloids do not move because there is no net flow of the binary carrier liquid associated with this process. Very long chains, on the other hand, are highly entangled even in semidilute solution. The disentanglement time τd scales with a high power of the chain length, τd ∼ M3+ε,11,12 and can become rather long for large M. When τd significantly exceeds the diffusion time, as in our experiments, the polymer network is deformed like a cross-linked gel, thereby preserving its topology. The GNPs that are trapped in the meshes of this entanglement network have to move together with the polymer against the net diffusion flow of the solvent. This situation is illustrated in Figure 5, which shows the transient polymer
a heated particle dispersed in a polymer solution without longlived entanglements has been discussed in detail in ref 9, and only a few relevant steps will be repeated here. A GNP of radius R, embedded in an homogeneous infinite medium and illuminated by a laser beam, is surrounded by a stationary temperature field T (r ) = T0 +
Pabs 1 R = T0 + (T (R ) − T0) 4πκ r r
(1)
Pabs is the absorbed laser power and κ the thermal conductivity of the medium. T0 is the temperature at infinite distance and T(R) = Pabs/(4πκR) + T0 the surface temperature of the colloid. If the medium is a binary liquid mixture, like a polymer solution, of density ρ, the temperature gradient induces a thermodiffusion current density jT = −ρc(1 − c)DT ∇T of the polymer. c is the polymer weight fraction. In the stationary state the thermodiffusion current is balanced by the isothermal Fickian diffusion current jc = −ρD∇c. D and DT are the diffusion and the thermodiffusion coefficient, respectively.13 For small temperature and concentration changes, the Soret coefficient ST = DT/D can be taken as constant, and the spherically symmetric concentration field is readily obtained as −1 ⎡ 1 − c0 ⎛ R ⎞⎟⎤ ⎜ c(r ) = ⎢1 + exp ST (T (R ) − T0) ⎥ ⎝ c0 r ⎠⎦ ⎣
(2)
by integrating the condition for vanishing total diffusion current ∇c = −ST c(1 − c)∇T
(3)
c0 is the equilibrium polymer mass fraction. In case of stronger heating and corresponding significant concentration changes in the vicinity of the GNP, the concentration and temperature dependence of ST must explicitly be taken into account. An empirical parametrization for the system PS/toluene has been given in ref 9 ST (M , c , T ) =
γ a ⎛ T0 ⎞ ⎜ ⎟ 1 + bc β ⎝ T ⎠
(4)
with T0 = 298 K and γ = 2.4. The coefficients a, b, and β are functions of the molar mass M of the polymer (see ref 9). Using eq 4 and introducing the dimensionless temperature rise ϑ = (T(R) − T0)/T0, the condition of stationarity (eq 3) can be rewritten after separation of variables as ⎛ R ⎞−γ 1 + bc β R dc = ⎜ϑ + 1⎟ T0 ϑ 2 dr ⎝ r ⎠ ac(1 − c) r
Figure 5. Deformation of the transient polymer network with respect to the solvent continuum (gray background) in the presence of a temperature gradient. The arrows show the local displacement field. The meshes where the GNPs are trapped persist on the time scale of the polymer/solvent diffusion.
(5)
The concentration field c(r) is obtained by numerical integration of eq 5 as described in ref 9. It must be kept in mind that eq 4 may only give an approximate description. It is mostly based on experiments on shorter polymer chains, and entanglement has not explicitly been considered in the TDFRS experiments used to measure the Soret coefficient.14 Up to now there has been no systematic investigation on how the measured Soret coefficient is affected by the entanglement and disentanglement of long polymer chains and how it relates to the slow mode frequently observed in diffusion of entangled polymer solutions.15 Network Displacement Field. The motions of the colloids reflect the displacement of the polymer network, which can be calculated from the concentration field c(r) using conservation of mass. Since the network topology is left unchanged, any polymer segmentand any GNPfound in the stationary
network with statistically distributed GNPs in the equilibrium state and after application of a temperature gradient. The time scale is still shorter than the disentanglement time. The polymer with its positive Soret coefficient is pushed toward the cold side, whereas the solvent moves to the warm side, thereby diluting the polymer solution near the hot surface (the laserheated GNP). The displacement of the GNPs is visualized in the microscope and can be used to map the deformation pattern of the polymer network. Concentration Field. For a quantitative description, it is necessary to calculate the deformation field of the transient polymer network. As a first step, the polymer concentration field in the heated state is required. The cage formation around 1675
dx.doi.org/10.1021/ma302665p | Macromolecules 2013, 46, 1673−1677
Macromolecules
Article
Finite Size Effects. Real systems, such as our sample, are never truly infinite, and the displacement of the polymer network must necessarily vanish at some finite distance. Our samples are limited in the z-direction parallel to the optical axis by the windows of the optical cell. Within the horizontal plane it has much larger dimensions (centimeter), but also here the temperature field does not decay like 1/r due to a screening effect of the parallel plates (Suprasil, κ = 1.38 W/(K m)) of ∼7 times higher thermal conductivity than the liquid (toluene, κ = 0.2 W/(K m)). Formally, this screening problem is equivalent to the electrostatic problem of a point-charge between two metal plates, where the far field in the symmetry plane is screened by mirror charges and decays exponentially with a characteristic length of the order of the plate spacing.16,17 Since the thermal conductivity of the windows is still finite, the screening will be less perfect. Instead of solving the full problem, an approximate and tractable treatment of the finite size effects can obtained by considering two concentric spheres with the polymer solution in between. The inner spere of radius R represents the heated gold colloid, whereas the outer sphere has a radius R′ of the order of the cuvette thickness and is kept at the temperature T(R′) = T0 + (T(R) − T0)R/R′. This leaves the temperature field within the sample volume unchanged from the one in the infinite case. The only difference is that the outer sphere represents, like the cell windows, an impermeable boundary, where the diffusion current vanishes. The lines in Figures 3 and 4 that approximate the measured displacements have been obtained from this simplified model with R′ = 100 μm and a temperature of the heated GNP of T(R) = 340 and 310 K, respectively, as discussed above.
heated state at a distance R2 from the heated GNP originates from an isothermal distance R1 < R2 (Figure 6). Thus, in an
Figure 6. Heated GNP (left) and displaced GNP (right).
isotropic system, any polymer material initially contained within a sphere of radius R1 will be found within a larger sphere of radius R2 in the heated state:
∫R
R1
4πr 2ρ(r )c0 dr =
∫R
R2
4πr 2ρ(r )c(r ) dr
(6)
c(r) is obtained from eq 2 or 5. The density ρ(r) ≈ const does not change very much with composition and temperature. Equation 6 allows to calculate for every final distance R2 the initial distance R1 ⎛ 3 R1 = ⎜R3 + c0 ⎝
∫R
R2
⎞1/3 r c(r ) dr ⎟ ⎠ 2
(7)
The GNP displacement is readily determined as ΔR = R2 − R1. An analytic solution can be obtained in the weak perturbation limit ST(T(R) − T0) ≪ 1, for which the exponential function in eq 2 can be expanded up to the linear term, resulting in R ΔR = R 2 − R1 = (1 − c0)ST (T (R ) − T0) (8) 2 Note that this approximation also holds in the far-field (r ≫ R) for arbitrary values of ST(T(R) − T0), where the argument of the exponential function in eq 2 becomes small because of R/r → 0. A surprising result of eq 8 is that the displacement of the colloids and, thus, of the meshes of the transient polymer network is constant and independent of the distance from the heated GNP. This result is the explanation for the experimentally observed nonlocal shift of the colloids as far as 103 colloid radii away from the center. The displacements calculated by numerically solving eqs 5 and 7 for an infinite system and the approximate solution according to eq 8 are plotted as solid lines in Figure 3. For realistic situations, a significant deviation from the asymptotic value would only be expected very close (a few colloid radii) to the heated GNP. For short distances, there is an agreement with the experimental data assuming a surface temperature of T(R) = 340 K. For larger distances, however, the experimental ΔR values do not reach a constant plateau. We attribute this slow decay to the finite system size as discussed below. The assumed surface temperature is a realistic estimation that is obtained by matching the theoretical curves with the experimental data for for short distances r. The true surface temperature cannot be measured in the present setup. An estimation based on the assumption of a geometric absorption cross section of the colloid and a beam focus of 1 μm yields a temperature rise of ∼150 K. A slight lateral motion of the heated GNP was an indication that the laser focus was not fully centered around the GNP, thus reducing the absorbed power.
4. SUMMARY AND CONCLUSIONS We have discussed the effect of localized heating of a GNP embedded in a polymer solution by a focused laser beam. Because of the Soret effect, the temperature field leads to a polymer depletion near the particle. This effect is purely diffusive without long-term memory in the case of small molecules or short polymer chains. In the case of long entangled polymer chains, there is an additional time scale, the disentanglement time, over which the topology of the network persists. GNPs of sufficient size are trapped within the meshes of this transient network, and the motion and deformation of the network can be visualized by these tracer particles using optical microscopy. Surprisingly, the displacement of the tracers and, hence, of the transient network is long-ranged and independent of distance in an infinite medium. Even particles far away from the local perturbation will move by approximately the same amount as particles in the immediate neighborhood of the heated GNP. In finite systems the displacement field is still far-reaching, but it necessarily decays to zero for distances of the order of the system size. In our experiments we have been able to observe significant displacements in a distance of 100 μm, corresponding to 103 colloid radii. After these initial experiments it will be of interest to investigate the dependence of the effect on experimental parameters like the molar mass and concentration of the polymer, which both determine the characteristic lifetime of the network topology, the disentanglement time. Particularly interesting scenarios are expected if the diffusion time becomes comparable to this time scale. In this case, we expect the effect to be no longer fully reversible. So far, we have not yet been 1676
dx.doi.org/10.1021/ma302665p | Macromolecules 2013, 46, 1673−1677
Macromolecules
Article
able to follow the buildup and decay of the elastic deformation in an experiment with a sufficient time resolution, which would require a high speed camera with a higher frame rate than the one employed in the present work.
■
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
■ ■
ACKNOWLEDGMENTS This work has been supported by the Deutsche Forschungsgemeinschaft (Research Unit FOR608/TP6, Ko 1541/6-1). REFERENCES
(1) Jones, C. D.; Lyon, L. A. J. Am. Chem. Soc. 2003, 125, 460. (2) van Dijk, M. A.; Tchebotareva, A. L.; Orrit, M.; Lippitz, M.; Berciaud, S.; Lasne, D.; Cognet, L.; Lounis, B. Phys. Chem. Chem. Phys. 2006, 8, 3486. (3) Spill, R.; Köhler, W.; Lindenblatt, G.; Schaertl, W. Phys. Rev. E 2000, 62, 8361. (4) Jain, P. K.; Lee, K. S.; El-Sayed, I. H.; El-Sayed, M. A. J. Phys. Chem. B 2006, 110, 7238. (5) Govorov, A. O.; Zhang, W.; Skeini, T.; Richardson, H.; Lee, J.; Kotov, N. A. Nanoscale Res. Lett. 2006, 1, 84. (6) Grabowski, C. A.; Adhikary, B.; Mukhopadhyay, A. Appl. Phys. Lett. 2009, 94, 021903. (7) Wyart, F. B.; de Gennes, P. G. Eur. Phys. J. E 2000, 1, 93. (8) Rings, D.; Schachoff, R.; Selmke, M.; Cichos, F.; Kroy, K. Phys. Rev. Lett. 2010, 105, 090604. (9) Schwaiger, F.; Zimmermann, W.; Köhler, W. J. Chem. Phys. 2011, 135, 224905. (10) Kita, R.; Kircher, G.; Wiegand, S. J. Chem. Phys. 2004, 121, 9140. (11) de Gennes, P. G. J. Chem. Phys. 1971, 55, 572. (12) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford University Press: New York, 1986. (13) de Groot, S. R.; Mazur, P. Non-equilibrium Thermodynamics; Dover: New York, 1984. (14) Rauch, J.; Köhler, W. J. Chem. Phys. 2003, 119, 11977. (15) Li, J.; Ngai, T.; Wu, C. Polym. J 2010, 42, 609. (16) Glasser, M. L. Am. J. Phys. 1970, 38, 415. (17) Pumplin, J. Am. J. Phys. 1969, 37, 737.
1677
dx.doi.org/10.1021/ma302665p | Macromolecules 2013, 46, 1673−1677
