Measuring the Soret coefficient of nanoparticles in a dilute suspension

  • Chao Zhao
  • Jinxin Fu
  • Alparslan Oztekin
  • Xuanhong Cheng
Research Paper


Thermophoresis is an efficient process for the manipulation of molecules and nanoparticles due to the strong force it generates on the nanoscale. Thermophoresis is characterized by the Soret coefficient. Conventionally, the Soret coefficient of nanosized species is obtained by fitting the concentration profile under a temperature gradient at the steady state to a continuous phase model. However, when the number density of the target is ultralow and the dispersed species cannot be treated as a continuous phase, the bulk concentration fluctuates spatially, preventing extraction of temperature-gradient-induced concentration profile. The present work demonstrates a strategy to tackle this problem by superimposing snapshots of nanoparticle distribution. The resulting image is suitable for the extraction of the Soret coefficient through the conventional data fitting method. The strategy is first tested through a discrete phase model that illustrates the spatial fluctuation of the nanoparticle concentration in a dilute suspension in response to the temperature gradient. By superimposing snapshots of the stochastic distribution, a thermophoretic depletion profile with low standard error is constructed, indicative of the Soret coefficient. Next, confocal analysis of the nanoparticle distribution in response to a temperature gradient is performed using polystyrene nanobeads down to 1e–5 % (v/v). The experimental results also reveal that superimposing enhances the accuracy of extracted Soret coefficient. The critical particle number density in the superimposed image for predicting the Soret coefficient is hypothesized to depend on the spatial resolution of the image. This study also demonstrates that the discrete phase model is an effective tool to study particle migration under thermophoresis in the liquid phase.


Discrete phase model Nanoparticles Thermophoresis Soret coefficient 

List of symbols


Spatial coordinates (µm)


The thermal diffusion coefficient (µm2 s−1 K−1)


The Soret coefficient (K−1)


The diffusion coefficient (µm2 s−1)


Concentration of the nanoparticles (kg m−3)


Absolute temperature of the fluid (K)


Power density (mW m−3)


The laser power (mW)


Absorption coefficient (m−1)


Reflection coefficient


Laser pulse x standard deviation (µm)


Laser pulse y standard deviation (µm)


Density of the fluid phase (kg m−3)


Velocity of the fluid phase (m s−1)

\(\Delta t\)

Time step (s)


Dynamic viscosity of the fluid (Pa s)


Kinematic viscosity of the fluid (m2 s−1)


Static pressure (Pa)


Thermal expansion coefficient of the fluid (K−1)


Gravitational acceleration (m s−2)


Heat capacity of the fluid (J kg−1 K−1)


Thermal conductivity of the fluid (W m−1 K−1)


Thermophoretic force per unit particle mass (m s−2)


Density of the nanoparticles (kg m−3)


Mass flow rate of the particles (kg s−1)


Volume of fluid element (m−3)


Diameter of the nanoparticles (µm)


Velocity of the nanoparticles (m s−1)

\(\bar{\bar{\tau }}\)

Stress tensor (Pa)

\(F_{D} (\varvec{u} - \varvec{u}_{\varvec{p}} )\)

Drag force per unit particle mass (m s−2)


Reynolds number


Relative Reynolds number


Drag coefficient


Brownian force per unit particle mass (m s−2)


Lift force per unit particle mass (m s−2)


Boltzmann constant (m2 kg s−2 K−1)


Deformation rate tensor (s−1)



We are grateful for the helpful discussions on confocal microscopy with Prof. H. Daniel Ou-Yang and Ming-Tzo Wei. Funding for the research is provided by the National Institute of Health under Grant No NIAID-1R21AI081638 and Pennsylvania Department of Health CURE Formula Funds.

Supplementary material

11051_2014_2625_MOESM1_ESM.tif (539 kb)
Fig. S1 Dependence of fluorescent intensity of BCECF as a function of temperature. The microscope stage was heated uniformly and the temperature in the detection chamber was measured using a thermocouple. The intensity decreases in response to increasing temperature with a slope of −1.6 % K−1. (TIFF 538 kb)
11051_2014_2625_MOESM2_ESM.tif (122 kb)
Fig. S2 Measurement of the temperature profile in the detection chamber from laser heating. a) Fluorescent intensity measurement by BCECF. b) Radially averaged fluorescence intensity is converted to the temperature profile, which indicates a linear temperature gradient of 0.5 K µm−1 spanning radially from the focused laser for ~20 μm (c) The simulated temperature gradient matches the experimental observation. This temperature gradient is used in both the continuous and discrete phase models. (TIFF 121 kb)


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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • Chao Zhao
    • 1
  • Jinxin Fu
    • 2
  • Alparslan Oztekin
    • 3
  • Xuanhong Cheng
    • 1
  1. 1.Department of Materials Science and Engineering and Bioengineering ProgramLehigh UniversityBethlehemUSA
  2. 2.Department of PhysicsLehigh UniversityBethlehemUSA
  3. 3.Department of Mechanical Engineering and MechanicsLehigh UniversityBethlehemUSA

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