tential interacting forces
Different from the former two approaches, the thermophoretic force can be obtained by summing all of the central interacting forces (i.e. −∂U (n)/∂n) along gradient direction. This is due to the force balance condition, namely how much external force we should apply on colloid in temperature gradient to fix the position.
FT = Z R+δ 0 n2dn Z π 0 sin θdθ Z 2π 0 ρ(n, θ)∂U (n) ∂n cos θdφ =2πP∞ kB Z π 0 sin θ cos θdθ Z R+δ 0 n2 1 T + n cos θ∇T exp − U (n) kB(T + n cos θ∇T ) ∂U (n) ∂n dn, (A.3) Similarly, the spherical coordinate is applied. The prefactor cos θ in the first line of Eq. (A.3) indicates the force is projected along the gradient direction. Due to the asymmetric density distribution, the unbalanced central forces come into the thermophoretic calculation. This is different from the two pre- vious approaches.
143
Appendix B
Analytical flow field for different
boundary conditions
This appendix services as the detailed analytical calculation for finite size effect correction in Chapter 3. Based on Eq. (3.13), the solutions to analyt- ically construct the thermophoretic flow field with different boundary con- ditions has been presented. Recalling the boundary conditions result in an additional friction force on the measurement of phoretic force, this methods provide a way to calculate the effective friction FγT. In this way, the finite size
effects factor λ in Eq.(3.21) for different boundary conditions is quantified by analytical calculation.
B.1
Thermophoretic flow field of fixed colloid con-
fined between walls
The first boundary condition we considered is colloid confined between walls in the center.
The flow field of fixed colloid is a combination of a Stokeslet and a source dipole. Colloids pump the fluid forward (opposite from FT) but bounces
back when it hits the wall and act on the central colloid. Meanwhile, the cooperation with other images perpendicular to the gradient leads to con- vection flow pattern in Ref.[60] and Fig.3.12(a).
Fig. B.1(b) indicates the schematic diagram of analytical calculation of flow pattern via including effects from walls (applying constant back flow as- sumption) and periodic boundary conditions along other two(x and y) di- rections.
z
x
OCo
ld
Hot
(a)y
x
O (b)Figure B.1: (a) Model consideration for measurement of thermophoretic force: colloid is fixed in the center of simulation box, temperature gradient is applied in z-direction with walls (were not visualized). (b) Schematic representation of calculation route for flow field considering periodic boundary conditions (x − y directions are considered).
To characterize FγT to thermophoresis, the calculation procedure is consid-
ered as follows: the first part comes from the thermophoretic flow at the walls bouncing back (constant back flow assumption) without changing the mag- nitudes, in which contributions from all images and center box are counted; the second part is the thermophoretic flow excited by other images whose directions are the same as the thermophoretic flow which decrease the mea- sured net force. Specifically,FγT(r) is the sum of the hydrodynamic interac- tions from the back flow of the colloid itself as well as its periodic images, and the thermophoretic flow of periodic images (which is opposite from the back flow direction). Hence the effective friction is represented by,
FγT(r) = 4πηR(vcenterb (r) + vimgb (r) − vimgc (r)). (B.1)
Fig.B.2(a) indicates the flow interactions between images and walls is long ranged, which doesn’t vanish crossing simulation boxes. This long range hy- drodynamic interaction due to Stokes term in eqn.(3.23) which is well known. In contrast, the second source dipole term decays 1
R3 which dominates near-
field flow and is thereby short ranged. Fig.3.14(b) renders the correspond- ing averaged z component of effective friction velocity calculated analyti- cally. Basing on the calculation outlined in Fig. B.2(b), we examined the value of effective flow velocity converges as the number of layers of images goes to 10. In this way, we make sure 10 layers of images (total number is
B.1. Thermophoretic flow field of fixed colloid confined between walls 145 -40 -20 0 20 40 60 80 0.0000 0.0002 0.0004 0.0006 0.0008 0.0010 v z x flow at walls (data is collected at z=50.0) (a) 10 0 10 1 10 2 10 3 0.00090 0.00095 0.00100 0.00105 0.00110 0.00115 0.00120 0.00125 v f ,z Number of layers Effective stokes friction: F eff =4 R(v self back +v img back -v img th )=4 Rv f (b)
Figure B.2: (a)Analytical calculation of z component of flow along x axis at walls without considering walls’ effect. Blue solid lines distinguish the center box boundaries along x or y direction, the gray dash lines mark the two nearest images, spheres are the representations of colloids.(b)
Convergence of average flow velocity acting on fixed colloid is calculated by averaging all of the fluid velocities in the colloid domain; the effective friction is therefore obtained by Stokes law.
(2 × 10 + 1)2 = 441) is sufficient to consider the effect of all the images. Then
we apply the previously mentioned back flow assumption to obtain the en- hanced part of thermophoretic force.
Using the same protocol as in simulations in reference [58], we vary the cubic box size into different values and measure the thermal diffusion factors by considering the effective friction contribution. Comparing the results with simulation data, we found a nice agreement between these two by applying the empirical linear fitting. It is important to note that the strongly depends on the error of the simulation results, thus the changing of potential type, radius of colloids leads to different λ. Therefore the numerical errors may lead to deviations from theoretical results (always are larger).
The comparison between simulation and analytical calculation reveals con- sistency of constant considerations of back flow assumption and periodic boundary conditions. And the characterization parameter λ ≈ 1.0 manifests the correctness of Stokes flow approximation.