Axisymmetric thermoviscous and thermal expansion flows for microfluidics

The controlled, precise manipulation of microparticles and biological cells in a liquid environment underpins applications across a wide range of disciplines and industries, from drug delivery and disease diagnostics to the construction of miniaturised devices and the study of single cells and molecules [1‐3]. Microparticle manipulation has been accomplished using a variety of physical mechanisms to achieve trapping, particle movement or assembly of structures [3]. Optical tweezers trap particles with optical forces associated with spatial gradients in light intensity, created by focused light [4‐6], while magnetic tweezers apply force to magnetic beads via a magnetic field [7], and acoustic tweezers use the acoustic radiation force induced by standing waves [8]. For optothermal micromanipulation, thermal gradients caused by laser heating result in thermophoresis, the migration of objects along a temperature gradient, providing control over microparticles [9]; active optothermophoretic manipulation involves particles that can establish their own temperature gradients [9].

With varied possibilities for driving mechanisms, fluid flow is instrumental in many techniques for micromanipulation. One of its advantages is that it need not rely on the optical or magnetic properties of the particles being manipulated; instead, recent studies have employed a vast range of flow field topologies in order to trap or manoeuvre particles hydrodynamically. For instance, time-averaged flow created by fluid oscillation, known as steady streaming, has enabled contact-free trapping of single cells in microeddies [8, 10]. Both driven by laser-induced temperature gradients, thermal convection and thermophoresis have been combined to manipulate microparticles [11], while Marangoni flows can be controlled optothermally [12] or via photoresponsive surfactants [13], to trap and guide the movement of particles at a gas-liquid interface, or to transport living cells [14]. Electroosmotic flow, which is actuated by electrodes that generate electric fields and hence flow of electrolyte solutions, has been used together with feedback control to independently steer and trap multiple particles at once [15]. In a microfluidic device with multiple channels, pressure-driven flow can be controlled via the channel flow rates; using this flow control, recent experiments have demonstrated precise manipulation of particles along user-defined paths, exploiting different flow modes [16‐20]. However, increasing the number of particles increases the number of degrees of freedom, necessitating a larger number of channels of the microfluidic device to control the particles independently [18].

Recent microfluidic experiments have investigated the use of thermoviscous flows for high-precision positioning of particles [21], trapping [22] and assembly [23]. In these studies, thermoviscous flow is driven by spatio-temporally varying heating of fluid, induced by a laser scanning along specified paths at kilohertz frequency. Although this technique, too, involves laser-induced heating, the physical mechanism for flow distinguishes it from the methods previously described. Here, the repeated scanning of the laser along a scan path results in fluid flow through the interplay between thermal expansion and thermal viscosity changes. This enables localised net transport of particles in the bulk fluid, without requiring direct laser exposure or physical channels [23, 24].

Highly versatile, these thermoviscous flows have also been applied inside living cells, and are known as focused-light-induced cytoplasmic streaming (FLUCS) in this context [25‐27]. This name contrasts FLUCS with naturally occurring cytoplasmic streaming: actively driven flows of the water-based, complex fluid (cytoplasm) inside cells, found in a wide variety of living organisms [28, 29]. For example, with thermoviscous flow perturbations, experiments have investigated how intracellular flows drive cell polarisation [25], and measured the rheology of cytoplasm [25] and the nucleus [26].

Previous theoretical work [24, 30, 31] has focused on the setup relevant to many of these microfluidic experiments [21‐25, 27, 30, 31], namely, with viscous fluid confined between parallel rigid plates. In Ref. [30], we considered a temperature profile with circular symmetry induced by the laser, which translates along a finite scan path, always in the same direction. We modelled the density and shear viscosity of the fluid as depending linearly on temperature change, a valid approximation for experimentally relevant temperature increases. The confinement provided by the parallel plates reduced the problem to two spatial dimensions, with a parabolic flow profile along the third dimension in the lubrication limit. We solved for the instantaneous flow field, and hence the average velocity of tracer particles throughout space due to repeated scanning. This revealed strong net transport near the scan path, together with inverse-square-law spatial decay, in agreement with experimental data [21]. Perhaps counter-intuitively, this net transport of tracers depends not on the magnitude, but instead on the rate of change of shear viscosity with temperature; hence, net thermoviscous flows are effective in fluids with vastly different viscosities [25, 30, 31]. Extending the method from one scan path to many, feedback algorithms have enabled the simultaneous manipulation of multiple particles, via net flow fields with complex topology [21, 23, 27], as shown in Fig. 1A and B.

In general, micromanipulation can be conducted in two  [7, 9, 11, 16‐18] or three [4, 7‐9, 19, 20, 32] dimensions. So far, thermoviscous flows have been used for two-dimensional micromanipulation experimentally [21‐24], while existing theoretical studies consider essentially two-dimensional flow in the parallel-plate geometry [30, 31]. However, the possibility of three-dimensional thermoviscous and thermal expansion-driven net flows and micromanipulation, for example, with highly focused heating [24, 30], has yet to be explored. Unlocking more degrees of freedom could result in increased versatility of the technique for future experiments.

Geometry has a significant influence on the flows and transport induced by the focused light. For example, it can strongly modify the far-field behaviour, which, importantly, determines the impact of directing a particle towards its target on other particles, in microfluidic experiments [23]. Thus, to make progress towards three-dimensional micromanipulation, fundamental, quantitative understanding of three-dimensional flow is required. In this article, we study thermoviscous and thermal expansion-driven flows of three-dimensional, unconfined fluid.

As a consequence of the unconfined geometry, we must also include a qualitatively new physical ingredient, known as the bulk viscosity (or volume viscosity [33]) of the fluid, in our theory. This material property is relevant only for compressible flows. For the parallel-plate geometry, it could be safely neglected [30, 31]; however, without the parallel plates, bulk viscosity could play a role. From a continuum perspective, just like shear viscosity, bulk viscosity is a phenomenological coefficient that characterises a Newtonian fluid, and is independent of its state of motion [33]. While shear viscosity relates stress to linear deformation rate, bulk viscosity relates stress to volumetric deformation rate [33], i.e. dilatational-compressional motion of the fluid described by divergence of flow velocity [34]. At the molecular level, bulk viscosity reflects rotational and vibrational degrees of freedom in molecular motion, while shear viscosity is associated with translational motion [35, 36].

The bulk viscosity of a liquid may be determined experimentally via various techniques [34, 36‐38]. Motivated by recent microfluidic thermoviscous flow experiments [24, 25, 31], we may focus here on both glycerol-water mixtures and pure water. The authors of Ref. [37] demonstrated that the bulk viscosity of aqueous glycerol is comparable in magnitude with the shear viscosity, and that it depends on temperature. Furthermore, different groups have shown that the bulk viscosity of water decreases with temperature, at a rate comparable to that of shear viscosity, over the temperature range relevant to thermoviscous flow experiments [34, 37, 38]. Therefore, we cannot neglect the magnitude and rate of change with temperature of bulk viscosity in favour of those of shear viscosity, and here we retain both bulk viscosity and its temperature dependence in our theory.

In this paper, we solve systematically for the flow and net transport of particles that result from the scanning of a model spherical heat spot in unconfined viscous fluid. This article is organised as follows. We begin in Sect. 2 with the heat transport problem, solving numerically for the temperature field induced by a prescribed translating heat source in the experimentally relevant limit. Based on our results, we then choose a temperature profile that will serve as an input to our fluid flow model, representing the effect of the spatio-temporally varying heating. Next, in Sect. 3, we introduce our model for the compressible fluid flow, including the effects of thermal expansion, thermal shear viscosity changes and bulk viscosity with arbitrary temperature dependence. We derive an analytical expression for the flow induced during the translation of the heat spot up to quadratic order. Although our model includes the effect of bulk viscosity, we demonstrate that the flow is independent of it, instead depending on thermal expansion and thermal shear viscosity changes, just as for the parallel-plate geometry. We then compute in Sect. 4 the net transport of passive tracers induced by the flow, due to a full scan of the heat spot. In contrast with the instantaneous fluid flow, the leading-order net transport occurs at quadratic order in the thermal expansion coefficient and thermal shear viscosity coefficient. The direction of the net transport generated depends on the relative importance of these two effects, which can provide competing contributions: one from the interplay of thermal expansion and thermal shear viscosity changes (thermoviscous net transport, shown in the top panel of Fig. 1C), and the other purely driven by thermal expansion (Fig. 1C, bottom). Furthermore, the average velocity of tracers decays as a hydrodynamic source dipole in the far field, more strongly than the instantaneous flow during one scan of the heat spot. Finally, in Sect. 5, we compare the results of our axisymmetric model with our previous work for the parallel-plate setup, and propose future experiments to validate our theory, along with potential applications.

In this section, we solve numerically for the temperature field induced by a prescribed heat source that moves along a scan path, in three-dimensional, unbounded space, in the limit relevant experimentally to FLUCS [25, 27]. Motivated by our results, we then introduce the temperature profile that we will prescribe in our model of the flow induced by the scanning heat source (Sect. 3).

In this section, we introduce a three-dimensional model for the thermoviscous and thermal expansion-driven flows induced during one scan of a translating heat spot, in unbounded fluid. Here and in what follows, the term “heat spot” refers to the model temperature perturbation detailed in Sect. 2.6. With the instantaneous flow during one scan, we will compute in Sect. 4 the net transport of particles due to a full scan, which, by analogy with the two-dimensional case [30], we anticipate will allow trapping and manipulation of micron-sized particles in future experiments.

We introduced in Sect. 3 our model for the thermoviscous and thermal expansion-driven flows induced by a translating spherical heat spot in three-dimensional, unbounded fluid, solving for the instantaneous flow during one scan of the heat spot, up to quadratic order in the thermal expansion coefficient \(\alpha \) and thermal shear viscosity coefficient \(\beta \). In two-dimensional experiments, the relevant physical quantity observed is the net displacement of tracer beads due to repeated scanning of the laser, instead of the instantaneous fluid flow during one scan; furthermore, the theoretical average velocity of tracers from hydrodynamic modelling has been used quantitatively to design scan patterns to manipulate particles [23, 27, 30, 41]. We therefore anticipate that future three-dimensional experiments, like their two-dimensional counterparts, will measure and exploit the time-averaged trajectories of tracers; consequently, predicting the net displacement of tracers due to our three-dimensional, unconfined fluid flow over the course of one scan is a key step towards applications in micromanipulation. We thus now examine the kinematics of tracer particles in this flow and compute the net displacement of tracers due to a full scan of the heat spot from \(z=-\ell \) to \(z=\ell \).

In this article, we considered thermoviscous and thermal expansion-driven fluid flow in three-dimensional, unbounded fluid: a new geometry compared with previous work on viscous fluid confined between parallel plates [30]. We first examined heat transport. By solving numerically for the temperature field induced by a scanning heat source in the limit relevant to experiments [25, 27], we motivated a simplified model of the temperature perturbation to act as an input to our flow model. We then derived analytically the fluid flow and net transport due to the scanning heat spot, to quadratic order in the thermal expansion coefficient \(\alpha \) and thermal shear viscosity coefficient \(\beta \). Our model included bulk viscosity, a key new physical ingredient present because of the compressible nature of the flow. Bulk viscosity did not play a part in the parallel-plate setup due to the geometry, as shown using a scaling argument [30, 31]. Here, for the three-dimensional, unconfined fluid, we treated the bulk viscosity as a general function of space and time, without specifying its temperature dependence. Through our analysis, we found that even though the bulk viscosity impacts the pressure field, it does not influence the fluid velocity. We obtained the same physics and qualitative results as the parallel-plate setup. Specifically, the leading-order instantaneous flow during one scan is driven by thermal expansion via the spatio-temporally varying temperature field, occurring at order \(\alpha \) and linear in the peak temperature change. In contrast with this, the leading-order net transport occurs at both order \(\alpha \beta \) and order \(\alpha ^2\), quadratic in the temperature change. The two typically competing contributions at these two orders originate from two fundamentally different physical mechanisms, with the order-\(\alpha \beta \) thermoviscous term generated by the interplay between thermal expansion and thermal shear viscosity changes, while the order-\(\alpha ^2\) effect arises purely from thermal expansion. The far-field average velocity of tracers is given by a hydrodynamic source dipole, but now in three dimensions.

We now comment on the validity of our spherical approximation for the heat spot. In existing microfluidic experiments, the scan path can have length on the order of \(50~{\upmu \text {m}}\), but there can be a relatively large length scale of heat absorption by the fluid, perpendicular to the scan path, of around \(300~{\upmu \text {m}}\). However, for a scan path long enough that heat spot is small in comparison, the geometry would be approximately axisymmetric; our model could therefore serve as a first approximation in this situation. Furthermore, methods of achieving thermoviscous transport in three dimensions that have been suggested include highly focused heating of the fluid [24].

Our work provides a first model of fully three-dimensional thermoviscous and thermal expansion-driven net flows, which we anticipate will be key to explaining experimental data (Moritz Kreysing, personal communication). The unconfined geometry of our model may help with understanding experiments where boundaries are far from the heat spot [25] or where the lubrication approximation (employed in theory for the parallel-plate setup [21, 30]) is no longer valid. Our theory will also allow us to analyse the effect of boundaries and confinement on three-dimensional thermoviscous and thermal expansion flows and transport in future modelling work. The quantitative theoretical results, in combination with scan-path selection via feedback algorithms [21‐23] or global optimisation [41], could contribute to the design of new experiments that apply net thermoviscous and thermal expansion flows to trap or manipulate particles in three dimensions.