8.2 Theory
8.2.3 Shape dependency of the diffusion coefficients
Dx= Dobs2sin2 δ1 2 − Dobs1sin2 δ2 2 sin2 δ1 2 − sin2δ2 2 . (8.23)
How Dz and Dx depend on the shape and the degree of alignment will be
derived in the next subsection.
8.2.3 Shape dependency of the diffusion coefficients
1. No magnetic alignment
For a sphere, the diffusion coefficient is related to its radius, R, via the Stokes- Einstein equation [3, 4, 18]:
Dsphere=
kT
6πηR, (8.24)
with k the Boltzmann constant, T the temperature and η the viscosity of the solvent. Since a sphere is perfectly isotropic, the diffusion coefficient is isotropic as well. This is not the case for cylindrically symmetric particles. For such par- ticles the diffusion coefficient parallel to the symmetry axis is different from that perpendicular to the symmetry axis, as is shown in Figure 8.3. For all cylindri- cally symmetric structures Dk can be expressed in terms of D⊥ by introducing
the scaling constant a:
8.2 Theory
Figure 8.3: Cartoon showing the diffusion coefficients for a sphere, disc and a rod. The diffusion coefficient of a sphere is isotropic. A disc and a rod are both cylindrically symmetric structures. The symmetry axes (axes of rotation)
are indicated by the dashed arrows. The diffusion coefficient parallel and
perpendicular to the symmetry axes are defined as Dk and D⊥ respectively.
For a disc Dk= 23D⊥ while for the rod Dk= 2D⊥ [19].
where both a and D⊥are determined by the exact shape. Without any magnetic
alignment, all structures tumble around randomly and the diffusion coefficients in all three directions average out to:
Dav=
2 · D⊥+ Dk
3 =
2 + a
3 D⊥. (8.26) For spheres, a = 1 and D⊥ = Dsphere and hence Dav = Dsphere, as given by
equation 8.24. Rothenbuhler et al. showed theoretically that a = 2 for rods with high aspect ratio (infinite long rods) and a = 2/3 for discs with very low aspect ration (infinite flat discs) [19]. This is shown schematically in Figure 8.3. From this it follows that Dav= 4D3⊥ for rods and Dav= 8D9⊥ for discs. As
mentioned before, D⊥ is shape dependent. For rods D⊥ is given by [19, 20]:
D⊥ = kT 4πηLln L b , (8.27)
with L/b the aspect ratio. For discs, D⊥ is given by [19]:
D⊥= 3kT 16ηV1/3 πL 4b 1/3 , (8.28)
8 Towards dynamic light scattering in high magnetic fields
Figure 8.4: (a) The diffusion coefficients of a cylindrically symmetric object. (b) A cylindrically symmetric object in the lab frame. (x,y,z). The magnetic field is applied in the z-direction.
2. Magnetic alignment
In the previous subsection, we defined the diffusion rates for cylindrically sym- metric objects in terms of Dk and D⊥. Without magnetic alignment, the struc-
tures tumble around randomly and the diffusion coefficients average out in all three spatial dimensions in the lab-frame (x,y,z). In this subsection, we will derive the diffusion coefficients in case of (partial) magnetic alignment. We will choose the magnetic field direction to be along the z-axis. The object will have a certain orientation with respect to the applied magnetic field by angles θ and φ as is shown in Figure 8.4. The diffusion coefficients in the lab-frame (x,y,z) depend on the orientation of the cylindrically symmetric structure (θ, φ) by the following transformation:
Dx(θ, φ) = Dksin2(θ) cos2(φ) + D⊥cos2(θ) cos2(φ) + D⊥sin2(φ) ,
Dy(θ, φ) = Dksin2(θ) sin2(φ) + D⊥cos2(θ) sin2(φ) + D⊥cos2(φ) , (8.29)
8.2 Theory
Figure 8.5: Top: plots of a disc and a rod with equal surface area (A =
4πR2 with R = 250 nm), using the ∆χ obtained in chapter 6. Middle: The
calculated order parameters for both discs and rods as function of the applied magnetic field. Both structures are almost fully aligned at 30 T. Bottom: The normalized diffusion coefficients along the z and xy direction as function of the applied magnetic field. The average diffusion coefficient is given for comparison. The magnetic field is applied along the z-direction. For both discs and rods, the largest change is predicted to occur along the z-direction.
8 Towards dynamic light scattering in high magnetic fields
Equation 8.25 allows equations 8.29 to be written in terms of D⊥ only:
Dx(θ, φ) = a · D⊥sin2(θ) cos2(φ) + D⊥cos2(θ) cos2(φ) + D⊥sin2(φ) ,
Dy(θ, φ) = a · D⊥sin2(θ) sin2(φ) + D⊥cos2(θ) sin2(φ) + D⊥cos2(φ) , (8.30)
Dz(θ) = a · D⊥cos2(θ) + D⊥sin2(θ).
Because partially aligned structures can still rotate around the magnetic field lines (along the φ direction), the diffusion in the x and y directions are therefore equal when averaged over time. Therefore, Dx(θ, φ) = Dy(θ, φ) = Dxy(θ),
with: Dxy(θ) = 1 2a · D⊥sin 2(θ) +1 2D⊥cos 2(θ) +1 2D⊥. (8.31) When a magnetic field is applied in the z-direction, the structures will start to align. The degree of alignment follows a Boltzmann distribution, as was already explained in chapter 3, section 3.3.2. We can therefore express the magnetic field dependency of the average diffusion coefficients in the z and xy direction by: Di(B) = Rπ θ=0Di(θ) · f (θ) · sin (θ) · dθ Rπ θ=0f (θ) · sin (θ) · dθ , (8.32)
with i being either xy or z and f (θ) the Boltzmann distribution function as given earlier in chapter 3 (equation 3.40). To gain more insight in how Dz and
Dxy depend on the applied magnetic field for various polymersome shapes, we
plotted equation 8.32 for both a polymersome disc and a polymersome rod. Figure 8.5 shows the degree of alignment as function of field (the order pa- rameter, see chapters 3 and 6) and the normalized diffusion coefficients Dz/D⊥
and Dxy/D⊥ as function of magnetic field for both a disc and a rod. For both
shapes, Dz/D⊥and Dxy/D⊥are identical to Dav/D⊥at zero field since there is
no orientational order. As the magnetic field increases, the degree of alignment increases and the diffusion coefficients in the z and the xy direction start to dif- fer. For both a polymersome disc and a polymersome rod, the largest changes occur in the z-direction, which is only 12.5% for discs and 50% for rods.
In a DLS measurement, one measures the scattered light at an angle be- tween 0◦ and 180◦. According to equation 8.19 the diffusion coefficient that is measured is a mix between Dzand Dxy and depends on the scattering angle. In
case of (partial) magnetic alignment, Dz and Dxy are not equal and therefore
the observed diffusion coefficient is strongly depending on the scattering angle, as was already shown by equation 8.19. Figure 8.6 shows how the normalized observed diffusion coefficient, Dobs/Dav, depends on both the magnetic field
8.2 Theory
Figure 8.6: The normalized observed diffusion coefficient (Dobs/Dav) as func-
tion of magnetic field and scattering angle for discs (left) and for rods (right) shown both as a color and 3D plot. For all shapes the observed diffusion coef- ficient at 0 T is independent from the scattering angle. This is not the case at non-zero magnetic fields. Since the degree of alignment increases with increas- ing field, the diffusion coefficients along the z-direction and xy directions will become increasingly different and therefore the observed diffusion coefficient will start to depend the scattering angle, according to equation 8.19. Only at a scattering angle of 70.5◦, indicated by the dashed lines, the observed diffu- sion coefficient will equal the average diffusion coefficient independent of the degree of alignment.
rods. Similar calculations for tubes and cross-linked tubes are given in the Ap- pendix, figure 8.14. At 0 T the observed diffusion coefficient of both shapes is clearly independent of the scattering angle. This is due to the fact that there is no orientational order and thus the diffusion coefficients in the xy plane and the z-direction are equal to the average diffusion coefficient for discs and rods respectively. The stronger the magnetic field, the larger the difference between Dz and Dxy, until saturation occurs. This will cause the observed diffusion
8 Towards dynamic light scattering in high magnetic fields
equation 8.19. For both the discs and the rods the observed diffusion coeffi- cient decreases with increasing magnetic field at scattering angles below 70.5◦ degrees. Above this angle, the observed diffusion coefficient increases with in- creasing magnetic field. At 70.5◦ the observed diffusion coefficient equals the average diffusion coefficients at all magnetic field strengths for both discs and rods. This is no coincidence, since all cylindrically symmetric structures show this behavior at this particular angle. To prove this we use equations 8.19, 8.26, 8.30 and 8.31 and calculate the scattering angle at which the observed diffusion coefficient equals the average diffusion coefficient:
Dav = Dobs 2 + a 3 D⊥ = Dz(θ) sin 2 δ 2 + Dxy(θ) cos2 δ 2 , 2+a 3 D⊥− Dxy(θ) Dz(θ) − Dxy(θ) = sin2 δ 2 1 3(a − 1) + 1 2(1 − a) sin2(θ) (a − 1) +32(1 − a) sin2(θ) = sin 2 δ 2 , 1 3 = sin 2 δ 2 δ = 2 · sin−1 1 √ 3 ≈ 70.53◦. (8.33)
As can be seen, the scattering angle δ at which Dobs = Davis not a function of
θ or a. Instead, it is constant for all cylindrically symmetric structures. Figure 8.6 also shows that the largest changes in the observed diffusion coefficient occur at the largest scattering angle. Therefore, it will be best to measure DLS at the largest scattering angle possible. Measuring at large angles has also an additional benefit, namely that disturbances caused by dust parti- cles is minimal at high scattering angles [18,21]. This is due to the fact that the scattering intensity drops with increasing scattering angle. The rate at which the intensity decreases with increasing scattering angle depends heavily on the size of the particles: the larger the particles, the faster the drop in intensity. Since dust particles are relatively large compared to the particles to be investi- gated, the contribution of contaminants will therefore be minimal at the highest possible scattering angle. Finally, the unwanted effect of multiple scattering is also at a minimum at 180◦ [21, 22]. So also for this reason, measuring at the largest possible scattering angle is recommended.