Thermo-Electro-Mechanics at Individual Particles in Complex Colloidal Systems

Abstract

It has been well established that thermoelectric (TE) field can arise from different Soret coefficients of salt ions in the aqueous solution under constant temperature gradient. Despite their high relevance to cellular biology and particle manipulations, understanding and controlling of TE field in complex colloidal systems that involve micro/nanoparticles, salt ions and molecules have remained challenging. In such colloidal systems, the challenge arises from the thermal interactions with charged micro/nanoparticles that distort the TE field around the particles. Herein, we provide a framework for TE field in colloidal suspensions with various ions and surfactants at the single-nanoparticle level. In particular, we reveal the spatial variation of TE field around a dielectric particle under temperature gradient to determine the thermoelectric trapping force on the particle. Our theoretical results on the trapping force predicted from the TE force profile match well with the experimental opto-thermoelectric trapping stiffness of particles in the solutions where the temperature gradient was well-controlled by a laser beam. With their insight into TE field and force in complex systems, our framework and methodology can be extended to engineer the TE field for versatile opto-thermoelectric manipulations of arbitrarily shaped particles with non-uniform surface morphology and to advance the scientific research in cellular biology.

Graphical Abstract

INTRODUCTION

Thermophoresis, also known as Ludwig-Soret effect, is the response of molecules, ions, or micro/nanoparticles towards a thermal gradient. The thermophoresis-induced diffusion movement can be described by thermophoretic velocity u_th = −D_T∇T, where D_T is the thermal diffusion coefficient and ∇T is the externally imposed temperature gradient. In general, the sign of D_T is positive, indicating that molecules/colloids are thermophobic in nature¹. Thermophilicity (D_T < 0) has also been observed in a few cases experimentally². Because of their dependence on the inherent migration direction of the particles, applications based on thermophoresis are limited to colloidal particles with suitable compositions and migration direction. To tune the thermophoretic response of particles of expanded compositions, ionic compounds (salt or surfactant) are added to the solution to induce thermoelectric (TE) force for particle manipulations. Briefly, different Soret coefficients of individual charged ions in the solution lead to charge separation and result in a macroscopic TE field on the constituents of the system³. The direction of TE field can be reversed upon changing the constituent ions in the solution⁴, thereby controlling the migration direction of colloidal particles independent of the particle compositions. Applications exploiting this TE field have emerged in the fields of biology^5,6, microparticle manipulation^7,8 and microfluidic separation⁹. Moreover, electromagnetic waves have been used to generate opto-thermoelectric field through photon-phonon coupling to optically trap particles of various compositions and sizes^10–12.

Quantification and control of TE field is crucial to all the applications. Several theoretical studies were carried out to determine the TE field by analyzing thermal diffusion coefficient or Soret coefficient of micelles and charged colloids as a function of the nature of particle-solvent interface¹³, ionic strength¹⁴, colloid density^15,16, and temperature^17,18. Using such formulations, thermophoretic velocity of a particle was evaluated by assuming a constant temperature gradient across a particle. Navier-Stokes equations for the aqueous solution across constant temperature gradient have been solved to determine the volumetric force density on the solvent to evaluate TE force on the particle^19,20. However, in real-life applications, temperature gradient across colloidal particles is achieved through external stimuli such as laser heating, where constant temperature gradient is invalid. In addition, large colloidal particles alter the temperature gradient around the particles due to difference in thermal conductivities of solute and solvent, which also contradicts the constant temperature gradient assumption. Therefore, to precisely determine the TE force on particles in complex colloidal systems, one need a new methodology that considers the temperature gradient change caused by the existence of the particles.

Here, we present a new thermo-electro-mechanical framework for TE force at dielectric microparticles that takes into consideration the particle-induced temperature gradient distortion. Using a commercial finite-element analysis (FEA) solver, we apply an externally imposed macroscopic temperature gradient field to a particle-surfactant solution. The resultant TE field due to the surfactant is evaluated using small-gradient approximation²¹ and numerically integrated over the particle surface to obtain TE force. The significance of tangential and normal components of the TE field at the particle surface in controlling the particle motion is discussed. Subsequently, the theory is validated using experimentally measured trapping stiffness of polystyrene particles in cetyltetramethylammonium chloride (CTAC) solution under laser-heating-generated thermoelectric field or opto-thermoelectric field²². We observe a good match between the theoretical trapping stiffnesses from TE force profile of microparticles and the experimentally measured ones.

METHODS

Theory

Under steady conditions, the motion of an uncharged particle due to thermal gradient is balanced by the consequential concentration gradient achieved, and is mathematically represented as

$$J_{uncharged}=0=-D\nabla c-c{D}_T\nabla T$$ (1)

where c is the concentration of the solute and D is the diffusion coefficient of the solute. Soret coefficient (S_T) is defined as the ratio D_T/D, which related the concentration to temperature at steady state. In an electrolyte solution, molecules (i.e., salt and surfactant) dissociate into ions and each of the species moves differently under the thermal gradient due to dissimilar S_T, which results in an electric field. The ion flux under steady state is, therefore, a balance between concentration and thermal gradients along with the subsequent thermoelectric field, and is represented as¹⁸

$$J_{charged,i}=0=-Di\nabla c_i-c_i{D}_{T,i}\nabla T+\frac{Z_{i}e{c}_i{E}_{T}Di}{k_{B}T}$$ (2)

where i is the represents index of ion, Z is electron charge, is the valency of ions, e is electron charge, E_T is the thermoelectric field generated and k_B is the Boltzmann’s constant. The direction of TE field would be parallel or anti-parallel to the temperature gradient depending on the charge of the species with the larger S_T⁴ and is given as¹⁷

$$E_T=\frac{k_{B}T\nabla T}{e}\left(\frac{\sum X_i{c}_i{S}_{Ti}}{\sum X_i^2{c}_i}\right)$$ (3)

where X_i = ± 1 for positive and negative ions respectively. The net thermoelectric force F_th on a charged colloidal microparticle is then evaluated as qE_T where q is the charge of the particle. However, equation (3) is valid only under the assumption that ∇T is uniform throughout the microparticle volume and that the net bulk charge density is zero. Figure 1(a) represents a system with an imposed uniform temperature gradient across the particle. Due to discontinuity in thermal conductivity at the particle-solvent interface, the temperature gradient deviates from the ideal constant temperature gradient in near-vicinity of the polystyrene (PS) particle as shown in Figure 1(b). Therefore, introduction of colloidal particles in a constant temperature gradient system inherently introduces a distortion in the temperature gradient, which necessitates the consideration of spatial varying temperature gradient and TE field to accurately determine TE force on the particles.

Figure 1.

(a) Schematic of a constant-temperature-gradient system where the electrolyte ions diffuse at different rates. (b) Introduction of particle in such a system distorts the uniform temperature gradient (∇T₀) around the particle surface. (c) Using small-gradient approximations, the salinity of the system for a temperature gradient of 20K/μm is evaluated for 10mM electrolyte solution. (d) The ratio magnitude of E_T,‖ and E₀ (the bulk thermoelectric field due to a constant thermoelectric effect) shows that the effective tangential component is higher than that estimated at the surface. (e) Schematic for ion distributions around the particle surface. The chloride ions form a screening layer called Stern layer and the free ions in the diffuse layer would be polarized while maintaining a near zero-charge density. Beyond the screening length (Debye length), E_T,⊥ depends only on the temperature gradient (solid line) and tends to zero in the case of uniform temperature system (dot-dashed line).

While TE field is being discussed, an important aspect to analyze is whether the temperature gradient is strong enough to overcome the electrostatic attractions, resulting in a macroscopic separation of ions. To understand such a separation, we need know the salinity of the electrolyte solution. Salinity can be defined as the effective concentration of a charged electrolyte assuming that the bulk charge density due to ion dissociation is zero. In the current scenario, colloidal PS particles are dispersed in CTAC solution. The cation has a positive hydrophilic end and a hydrophobic polymer chain, enabling it to form micelles, which act as a macro-ion with a larger S_T due to the decreased diffusion coefficient [Supplementary Note I]. Owing to positive S_T, both micelles and chloride ions move towards the colder side, effectively lowering the ‘salinity’ in the hot region. Figure 1(c) represents the salinity obtained as [Supplementary note II]

$$\frac{\nabla n_0}{n_0}+\left(\frac{Z_2{S}_{T_1}-Z_1{S}_{T_2}}{Z_2-Z_1}\right)\nabla T=0;n_0=\frac{Z_1{Z}_2}{Z_1-Z_2}\left(c_1+c_2\right)$$ (4)

where salinity (n₀) is expressed as combination of concentrations of ions such that net charge density is zero.

For a better estimate of TE field around the particle, individual concentrations of ions must be analyzed as deviations from the resultant salinity, as salinity acts a reference for zero ion dissociation. This can be obtained from Gauss law, where the net charge density at any position can be given as a function of TE field, ρ/ϵ = ∇.E_T, where ρ is the net charge density in a differential volume given as N_Ae(Z₁c₁ + Z₂c₂), NA is the Avagadro number and ϵ is the permittivity of water as a function of temperature⁸. Using Gauss law along flux equations for micelles and counterions simultaneously, TE field can be determined around the particle. However, three-dimensional analysis of the system results in increased complications while solving coupled differential equations. To ease this numerical evaluation, small-gradient approximation is utilized, which leads to ∇n₀/n₀ ~ ∇c_i/c_i ~ ∇T/T when δT≪T²¹. This is a valid approximation as the maximum temperature difference considered in this work is 20K, indicating that effective concentrations of ions can be approximated to salinity. Treating the micelle as a macro-ion, including the effects of accumulation number and effective charge on the micelle, we write the net thermoelectric field that is simplified as²¹

$$E_T=\frac{k_{B}T\nabla T}{e}\frac{\left[\frac{Z_1{S}_{T_1}}{N_{agg}}+S_{T_2}\right]}{1-\frac{Z_1{S}_{T_1}T}{N_{agg}}}=SC\nabla T$$ (5)

where $S_{T_1}$, $S_{T_2}$ are the Soret coefficients of the micelle¹⁵ and chloride ion (~7.18×10⁻⁴ K⁻¹)²³, N_agg is the number of ions forming a micelle aggregate (~89 for CTAC²⁴), Z₁ is the effective charge on the micelle and SC is the Seebeck Coefficient. equation (5) is only valid for solution containing monovalent cation and macro-micelle.

Although E_T is along the temperature gradient, the tangential and normal components of E_T at the particle surface has different effects on particle due to the electric charge on the particle. The tangential component E_{T, ‖} does not vanish at the surface and interacts with the particle’s charge. Figure 1(d) shows that the magnitude of E_{T, ‖} has high deviations from TE field estimated using a constant temperature gradient (|E₀| = |SC∇T₀|). The tangential component is numerically integrated around the surface to determine the effective TE force on the particle. On the other hand, the normal component of the thermoelectric field (E_T, _⊥), is attenuated due to double layer formation. The counterions in the solution are attracted to the charge of the particle and form an immovable layer known as ‘Stern layer’, and a diffuse layer, which screens the remaining effect of the particle charge²⁵. Figure 1(e) represents the Stern layer and diffuse region around the particle surface. The E _⊥ component is deviated by a magnitude of SC∇T _⊥ in the bulk region in the presence of an external temperature gradient. The deviation minimizes while approaching the particle surface (Figure 1(e) – left). The net E _⊥ field at the particle surface is solely due to the charge on the particle, which does not influence motion of the particle²⁶.

The effective TE force on the particle is subsequently evaluated by estimating the surface charge of the particle. A constant charge density can be assumed as charge re-distribution does not occur under the influence of electric field on a dielectric particle²⁶. The PS particles used are inherently negatively charged, however, due to CTAC adsorption on the particle, the net charge on the particle surface is positive, resulting in a positive zeta potential^22,27. The relation between the effective surface charge and the measured zeta potential is given as²⁸

$$\sigma =\frac{ϵ\kappa k_{B}T}{e}\left[\text{exp}\left(\frac{e\zeta }{2k_{B}T}\right)-\text{exp}\left(-\frac{e\zeta }{2k_{B}T}\right)-\frac{4}{\text{κa}}\frac{\left(\text{exp}\left(\frac{e\zeta }{2k_{B}T}\right)-1\right)}{\left(\text{exp}\left(\frac{e\zeta }{2k_{B}T}\right)+1\right)}\right]$$ (6)

where ζ is the measured zeta potential, and κ is the inverse Debye length and is dependent on the net concentration of the freely moving ions in the solution, given as

$$\kappa =\sqrt{\frac{e^2{N}_A\left(2c_{cmc}+Q*\left(c_0-c_{cmc}\right)\right)}{ϵk_{B}T}}$$ (7)

where c_cmc is the critical micellar concentration beyond which micelles are formed (~0.13mM for CTAC) and N_A is the Avogadro number. The net TE force on the particle can be numerically integrated as

$$T{E}_{Force}={\int }_A\sigma E_{T,\parallel }dA=\sum _{\theta =0}^{\pi }\sum _{\varphi =0}^{2\pi }\sigma E_{T,\parallel }{a}^2\text{sin}\left(\theta \right)d\theta d\varphi$$ (8)

Experiments

Our methodology can also be applied to colloidal systems with spatially varying temperature gradient field as long as the small-gradient approximation is valid. For experimental verification of our theory, opto-thermoelectric trapping (OTET) is employed to study the effects of thermoelectric force on PS particles in solution with CTAC surfactant. OTET uses laser heating of quasi-continuous gold nanoparticle substrate²², where photon-phonon conversion at the substrate and diffusion of temperature at the focal spot lead to a thermal gradient. Directed towards the center of the focal beam, the thermal gradient leads to the opto-thermoelectric field that enables trapping of the PS particles at the heating spot. For detailed experimental procedure and sample preparation, please refer to Supplementary Note IV. Along with TE force, optical and osmotic forces on the particle are also considered [Supplementary Note V]. We limit this work to in-plane trapping stiffness evaluation using TE field, and additional forces like depletion, van der Waals and electrostatic repulsion act in the evaluation of z-directional trapping stiffness.

Figure (2a) describes the computational geometry of the colloidal system under a laser-beam-generated thermal gradient. The gold nanoparticle substrate is at the interface of water and glass, and the heat retained within the substrate is negligible. The absorbed power by water is modelled as a Gaussian heat influx at the interface of glass and water. The simulation volume is large enough to assume a constant temperature as boundary conditions for glass and water media. Inside the water medium, a PS particle is introduced to account for the distortion of temperature gradient around the particle. The effective temperature gradient and the parallel component of the thermoelectric field are evaluated on the PS surface, and numerically integrated to evaluate the TE force on the particle using equation (8). This simulation is repeated for several positions of the particle along the substrate and TE force is represented as a function of position in Figure 2(b). For a comprehensive account of FEA simulations, please refer to Supplementary Note VI.

Figure 2.

(a) Schematic for opto-thermoelectric trapping of a single PS particle. The arrow direction indicates thermal gradient towards the hotspot. Scale bar 1μm. (b) The TE force on PS surface is evaluated as a function of position of particle using equation (8). The TE force along with optical and osmotic forces constitutes a trapping well of depth ~400 k_BT for a low laser power of ~135 μW.

The net force F = F_TE + F_optical + F_osmotic is used to evaluate the in-plane trapping potential as U = −∫Fdx, although F_TE is dominant by two orders of magnitude over F_optical,F_osmotic. Figure 2(b) shows that, for a low laser power of ~135μW, the theoretically obtained in-plane trapping potential is roughly 400k_BT for a PS particle of 1 μm in diameter. For a similar laser power, the net trapping force is 2–3 orders higher in magnitude than that of optical tweezers²⁹. Due to thermal diffusion of heat generated at the interface, the observed particle trapping is a long-range phenomenon. The force on the particle follows Hooke’s law around the beam center, where trapping stiffness k_x is defined as −∂F/∂x at the beam center.

RESULTS AND DISCUSSION

Laser power

Figure 3(a) shows the variation of trapping stiffness of 1 μm PS particle with respect to the laser power at a constant CTAC concentration of 1mM. Increase in laser power increases the temperature within the same beam spot, which increases the TE force on the particle. This indicates that the trapping stiffness varies linearly with increasing laser power. An increasing trend is also observed in the experiments, although the trend is not linear at higher powers. This is due to increase in Rayleigh Barnard convection that causes destabilization of the particle within the trap. Also, the temperature of the particle within the trap increases, leading to higher Brownian motion. These destabilization effects were not included in the simulation, which lead to an over-estimation of the trapping stiffness at the higher laser powers, contradicting the expected linear trend in the system. Conventional optical tweezers do not have such destabilizing effect, thereby obtaining a near-linear trend²⁹.

Figure 3.

Experimental and theoretical values of trapping stiffnesses of polystyrene particles dispersed in CTAC solution as a function of (a) laser power (CTAC 1mM & PS 1μm) (b) CTAC concentration (power 135μW & PS 1μm) and (c) size of PS (power 135μW & CTAC 1mM)

Surfactant concentration:

A change in the concentration of the surfactant alters the zeta potential and the effective charge of the PS particle. As particle-solvent interfacial characteristics are not well established in the literature, zeta potential is experimentally evaluated for dispersions of PS particles in varying concentrations of CTAC. The trapping stiffness is measured as a function of concentration of CTAC at a constant laser power of 135μW. The trapping stiffness increases with the surfactant concentration in the lower regime before a saturation is observed. Figure 3(b) show an excellent match between theoretical estimates and experimental values. However, at relatively high concentrations (>12mM), the size of the micelle increases, increasing the vertical depletion forces on the PS particle and thus resulting in suppression of Brownian motion. At such high concentrations, experimental realization of dynamic manipulation of particles is not achievable at low laser powers. Therefore, the experimental value of trapping stiffness is higher than the theoretically expected one. To avoid complications involved with manipulating particles in high CTAC concentration, the concentration of CTAC at the onset of saturation (5mM in this work) is preferred for maximizing TE force from application point-of-view.

Size dependence:

The trapping stiffness as a function of size of particle is studied to further test the versatility of our theory. As particle size increases, trapping stiffness increases due to an increase in surface charge of the particle as shown in Figure 3(c). Also, increase in size of the particle leads to a decrease in the diffusion coefficient of the particle, leading to enhanced trapping stiffness. The matching values of the theoretical and measured trapping stiffnesses suggest that the effect of size has already been inscribed within the model during the numerical integration of E_T,‖ carried over the particle surface. However, correction factors must be included when we deal with smaller particles (<100 nm) to account for surface curvature, thus changing the σ − ζ relationship³⁰.

CONCLUSIONS

The modelling technique in this work is based on the decomposition of the TE field around the particle, which can also be extended to model dielectric particles’ motion under electrophoresis. For metallic particles, electrons are free to move within the particle under the influence of TE field and therefore, the effect of perpendicular and parallel components of TE field varies, compared to dielectric particles. However, this variation is not expected to be very high for nanoparticles in OTET as the dielectric layer of CTAC adsorbed on the particle surface makes the particle behave like a dielectric towards the thermoelectric field.

In summary, we have developed a thermo-electro-mechanical framework to calculate TE forces on dielectric particles under small temperature gradients by incorporating the temperature variation caused by the particle. Although the variations do not alter the bulk TE field much, they must be accounted in evaluating TE force on the particle, thereby suggesting a role of thermal conductivity of particle in TE field evaluation. A corrective term to include the thermal conductivity contrast was used to estimate the thermo-diffusion coefficient and thermophoretic drift of the particle in several works^1,3,19,31. Please refer to Supplementary Note VII for comparison of thermoelectric forces based on the current work and those using a point-particle assumption and correction factor. A good agreement between both the quantities is observed. However, in the previous works, the corrective term was limited to spherical particles and did not consider particles with sub-particle thermal conductivity variation, which can easily be implemented in the current work by altering the computational geometry of the particles. The methodology in this work can be further applied to light-absorbing particles, where the temperature gradient arises from the inherent non-uniform optical heating of the particle. We can further incorporate the particle surface characteristics into the model at sub-particle resolution, leading to a better TE force and torque estimate for a broader range of particles (e.g. Janus particles and core-shell particles). With recent advances in numerical solvers and visualization techniques, temperature distributions of arbitrarily shaped objects in complex environments can be retrieved to determine TE force accurately, which would lead to the optimum performances of end-applications such as thermoelectric trapping and manipulation and provide a new way towards exploring cellular biology.