U bT V T[1

where bT is the particle thermophoretic velocity, and V T is the transversal gradient of the temperature in the channel. The particle thermophoresis is commonly related to the osmotic pressure gradient produced in the surface layer due to the temperature gradient. This excess osmotic pressure is established in the particle surface layer due to accumulation of the solvent or dissolved molecules or ions in the particle surface layer. This accumulation is related to the particle surface potential This surface potential may have the electrostatic nature, when the particle surface carries electric charge, or represent some kind of dipole-dipole interaction, when ion adsorption or surface group dissociation is impossible. The latter situation should be characteristic for organic solvents, where dispersion interaction between the particle surface and solvent molecules should play the main role. The theory of thermophoresis is developed mainly for particles larger in size than the characteristic thickness of the surface layer and having moderate surface potential of several kT (k is the Boltzmann constant), which interacts with dissolved ions or molecules. These ions or molecules should be present at a concentration low enough to avoid the excluded volume effects in their accumulation in the particle surface layer. In this situation, the particle ther-

mophoretic mobility may be written as dy (l + »f )[e-»"T(kT+i)-i]

where c0 is the solute concentration in the carrier liquid, y is the carrier liquid viscosity, R is the particle diameter, n is the particle-to-liquid thermal conductivity ratio, and y is the transverse coordinate in the surface layer. The immediate physical factor for the particle thermophoresis is the 'slip' liquid flow in the particle surface layer due to the osmotic pressure gradient, which is related to the temperature gradient in the particle surface layer established along the macroscopic temperature gradient in the liquid. Thus, the main physical events in thermophoresis take place near the particle surface, though the temperature gradient near the particle surface playing the role of the driving force for the particle is defined by the particle and liquid thermal conductivity, which are bulk properties. However, one can expect that these bulk properties will be the same for small particles and larger samples of the material, and the particle thermal conductivity can be obtained from literature data on thermal conductivity of the material or independent experimental determination. It means that particle thermophoresis is mainly related to particle surface properties. It becomes absolutely true for metal particles with very high thermal conductivity, when the parameter n in eqn [1] is very large. For metal particles, the particle thermophoretic mobility is a function of the particle surface properties only. For smaller particles with higher surface potentials, eqn [1] is not true due to intensive solute transport in the particle surface layer.

This surface transport should be compensated by the solute diffusion outside the surface layer, and which, in turn, leads to the solute concentration gradient and related electric field establishment (in electrolytes) around the particle (so-called concentration polarization). However, for a particle with moderate size and thermal conductivity having a surface potential about two to four kT, we can state that the particle thermophoretic mobility is defined by the particle surface properties and does not depend on its size. For emulsion droplets, the thermophoretic mobility in the absence of concentration polarization is determined as:

y dy

where ^ is the viscosity of the liquid inside the droplet. For homopolymer chains, it is shown by ThFFF experiments, that chain thermophoretic mobility does not depend on chain length and branching, and one can expect that eqn [2] will define it accurately, where R will be the monomer radius. The theory of particle thermophoresis may be true, if some solutes present at low concentration, for example, salt ions, are accumulated around the monomers. However, in true polymer solutions, where no dissolved extrinsic solutes are present, excluded volume effects cannot be neglected, and eqns [2] and [3] cannot be used for the description of thermophoretic behaviour.

For calculation, the Boltzmann exponent indexes in eqns [2] and [3] may be simplified using the approximation:

where s is the depth of the surface potential well in kT units, and h is the characteristic width of this well. Typical orders of values for different kinds of surface potentials are present in Table 1, where A is the Hamaker constant, d is the solute radius, q is the solute electric charge, ( is the particle zeta-potential, and X is the Debye length.

For the characteristic relaxation time, the Boltz-mann transverse particle distribution is established in the system by forcing injected particles toward the wall of the channel and by their diffusion motion. In ThFFF, particles of the same size with higher ther-

mophoretic mobility will be accumulated more closely to the wall, while particles of lower thermophoretic mobility will form a more diffuse layer that extends further into the flow of the carrier liquid. For particles with about the same thermophoretic mobility, the thickness of this layer may also be different, if particles have different diffusion coefficients, D. Particles with higher diffusion coefficient (i.e., with smaller size) will be accumulated in a more extensive layer due to more intensive thermal movement. The ther-mophoretic mobility related to the surface potential is an important parameter interrelated to the particle surface charge density (where it represents the electrostatic potential) and characterizing the particle surface properties and the possible exchange of substances between the particle and the surrounding liquid. Also, the separation of particles of the same material but with different sizes may be important in the characterization (molecular mass distribution) of commercial latex and polymer particles.

ThFFF is carried out in a thin channel of rectangular cross-section with a width to thickness ratio (aspect ratio) of about 100 (thickness about 10-100 microns). It allows the separation system to approximate to the laminar flow between infinite parallel plates, which is characterized by a parabolic velocity profile, where the fluid velocity at the channel walls is zero, and a maximum in the centre of the channel. Thus, if a group of particles are accumulated, maintaining an average distance from the wall different from another group of particles, their velocities along the channel will be different. As a consequence, they will leave the channel at distinct times, related to the particle thermophoretic mobility and size, which defines the particle diffusion coefficient. There are no other direct methods, where temperature gradients are used for particle, droplet or mac-romolecule separation. FFF systems are elution methods and allow the collection of fractions during a separation.

Table 1

Surface potential Analytical expressions for the ® (y), s and h The ranges of values for parameters s and h

Table 1

Surface potential Analytical expressions for the ® (y), s and h The ranges of values for parameters s and h

®0)

£

h

£

h

Van der Waals forces

— A(d/r)6

A/kT

d/3

(low-molecular surfactant)

Coulomb electrostatic forces

— q{ e-y)

qUkT

A

0-

-10

10-7-10~4 cm** (aqueous electrolytes)

Adsorption forces

None

None

None

0-

-10

+10~7 cm

Structure forces

None

None

None

0-

-10

+10~5 cm

*The maximum values of Hamaker constant are reached for metals

**The maximum value of the Debye length is calculated for the deionized water.

*The maximum values of Hamaker constant are reached for metals

**The maximum value of the Debye length is calculated for the deionized water.

Solar Panel Basics

Solar Panel Basics

Global warming is a huge problem which will significantly affect every country in the world. Many people all over the world are trying to do whatever they can to help combat the effects of global warming. One of the ways that people can fight global warming is to reduce their dependence on non-renewable energy sources like oil and petroleum based products.

Get My Free Ebook


Post a comment