Due to the definedness of the presented process, it is possible to derive a mathematical description for the stretching of a hcp structure. Starting with a conventional hcp particle array, the base vectors~e1and~e2of the hexagonal unit cell (Figure 4.10) are given as
~e1= 1 From this point, the actual orientation of the hcp structure at the water/air interface with respect to the immersion direction is specified by the rotation angleδ as shown in Fig-ure 4.10.
Figure 4.10: Definition of the initial hcp domain orientation angleδ, stretching angle α and stretching vector~S.
Thereby, the base vectors~aand~bfor the correctly rotated hcp structure prior to transfer can be calculated from~e1and~e2using a rotation matrix
~a=
The subsequent stretching of the vectors~a and~b is defined by the stretching vector~S (Figure 4.10). In this process, on one hand~Sdetermines the degree of stretching as the stretching factorS=|~S|states the magnitude of stretching. On the other hand,~Spoints in the direction of stretching, which can be related to the angleαbetween the stretching vector~S and the y-axis. Starting from~a and~b, the stretching process can be described mathematically as a sequence of matrix multiplications. In the first step, the hcp structure is rotated by an angle−α(M−α, equation 4.5), so that~Spoints now along the y-axis. In the following step, the uniaxial stretching is accomplished parallel to the y-axis with a factor
S(MS, equation 4.6). Finally, in order to return to the original orientation, the stretched structure is back rotated by an angle α(Mα, equation 4.5). In mathematical terms this leads us to the following individual matrices.
Mα = A multiplication of these matrices according to the above described stretching process gives the complete stretching matrix (M = Mα◦MS◦M−α) and the stretched vectors~a0 and~b0 of the final particle array.
M=
This calculation shows that the relation between the hcp structure (~e1,~e2) and the stretched, non-close-packed structure (~a0,~b0) is completely determined by the parametersδ,αandS.
For this reason, it is possible to theoretically predict particle arrangements after stretching for arbitrary hcp orientations and stretching vectors~S. A significant characteristic is that the obtained structures cover all of the Bravais lattice symmetries possible in two dimen-sional space. An overview over the accessible array symmetries is shown in Figure 4.11.
For the calculation of the diagram, the two orientation anglesδ and αare summarized to an angleβ = α−δ. βdescribes the effective stretching direction with respect to the orientation of the hcp structure at the water/air interface. Due to its 6-fold rotational symmetry the considered angular range can be reduced to 0◦ ≤ |β| ≤ 30◦. Equiva-lently, the same structures evolve for all angles|β±n·60◦|, withnbeing an integer, for a constant stretching factorS.
In two-dimensional space, there exist five Bravais lattices: square, hexagonal, centered rectangular, rectangular, and oblique. These non-close-packed structures can be realized for discrete values ofSandβas indicated in the phase diagram. An overview of realized Bravais lattice symmetries by a combination of different values ofS and βis shown in Figure 4.12.
Figure 4.11: Phase diagram for accessible Bravais lattice structures in two dimensions as a function of the stretching factorSand stretching angleβ.
Figure 4.12: Non-close-packed particle arrays with the five possible Bravais lattices in the two-dimensional space: (a) square, (b) hexagonal, (c) centered rectangular, (d) rectangu-lar and (e) oblique. Scale bars are 1µm.
A square array of particles (Figure 4.12 a) is obtained by stretching with a factor ofS =
√
3 ≈ 1.73 and a stretching direction along one of the connection vectors between the particles in the initial hcp structure (β = 30◦). An ideal non-close-packed hexagonal array (Figure 4.12 b) can be realized by stretching with a factor of S = 3 in the same direction (β= 30◦), whereas a centered rectangular array (Figure 4.12 c) is the result for S6=1.73 andS6=3 atβ=30◦. Rectangular arrays (Figure 4.12 d) are the result of discrete
combinations of stretching factors and directions that fulfill the following equation
For all other parameter combinations withβ6=30◦oblique lattices (Figure 4.12 e) are ob-tained. A special case of oblique symmetry are close-packed particle lines, corresponding to highly extended oblique lattices. They can be fabricated by choosingβ=0◦ and arbi-traryS. In brief, non-close-packed particle arrays with symmetries of any of the five two-dimensional Bravais lattices (square, hexagon, rectangular, centered rectangular, oblique) can be prepared by a uniaxial stretching of hcp structures.
It has to be mentioned that the presented phase diagram in Figure 4.12 is only an extract of the complete phase diagram. It shows the first stretching branchΣ, which summarizes accessible structures for the lowest values ofS. A consideration of higher stretching fac-torsS > 5 yields an expanded phase diagram with more stretching branches, Φ,Ψ,Ω.
These branches feature repeatedly the same Bravais lattice symmetries for higher values ofS. Thereby, the structures of each stretching branch are based on a different pair of base vectors (~a0
i,~b0
i; with i =Φ,Ψ,Ω). These vectors have in common that they are all described by linear combinations of~a0and~b0.
Figure 4.13 shows the expanded phase diagram divided into two parts. The upper dia-gram displays the angle between the base vectors as a function ofSandβ(Figure 4.13 a) and the lower diagram screens the relative length of the base vectors (Figure 4.13 b). As the unit cells of the Bravais lattices are specified according to the angle between the base vectors~a0
i and~b0
i of the regarded structure and the ratio of the vector lengths, the five symmetries can be identified for the additional branches using the combination of the expanded phase diagrams.
Each lattice symmetry can be described by two conditions, vector lengths and angle be-tween two unit cell vectors (Figure 4.14): square (~a0
i = ~b0
The mathematical description above represents a detailed tool for the prediction of pos-sible non-close-packed structures starting from hcp particle arrays. In the opposite case experimentally observed particle arrangements can be also assigned to a distinct stretch-ing process. Then, the stretchstretch-ing direction α and magnitude S are determined by the following equations.
α=cot−1(y
0−y(δ)
x0−x(δ)) (4.11)
Figure 4.13: Expanded phase diagram as a function of the stretching factorSand stretch-ing directionβ. (a) Phase diagram with the color bar showing the angle between the base vectors of the individual stretching branches (Σ,Φ,Ψ, Ω). (b) Phase diagram with the color bar showing the ratio between the lengths of the base vectors.
Figure 4.14: Definition of Bravais lattices in two dimensions.
S= x
0−x(δ)·cos2α+y(δ)·sinα·cosα x(δ)·sin2α+y(δ)·sinα·cosα
(4.12) where x0, y0 are the coordinates of the stretched vectors ~a0 and ~b0, and x(δ), y(δ) are the coordinates of the initial vectors~a and~b. For this purpose, the stretched vectors~a0
and~b0 with the coordinates are associated with particle positions in SEM images. As the initial vectors are unknown, the calculation ofαandS is dependent on a screening of the parameterδ, which defines the orientation of the original hcp structure,x(δ)and y(δ). An exemplary calculation of the reconstruction of a stretching process is shown in Figure 4.15 and Table 4.1.
Figure 4.15: Reconstruction of stretching parameters from SEM image. (a) SEM image of stretched particle array symmetries with highlighted vectors. Scale bar is 2µm. (b) Schematic illustration of initial hcp structure with δ = -2◦ and stretched structure as a result of a stretching process withα= 132◦anS= 1.73.
Table 4.1: Calculation of stretching parameters for SEM image in Figure 4.15 with~a0:
The averaged coordinates of the indicated vectors are extracted from the SEM image and inserted into the equations for the calculation ofαandS. The values of αandSare cal-culated forδ = [−180◦; 180◦]. An extract of this calculation is given in Table 4.1. It can be seen thatα andS usually differ for the individual calculations with~a0 and~b0. Only for one value ofδ there is a match ofαandS. In the present case, this is applicable for δ= -2◦. Consequently, the stretching parametersα= 132◦ andS = 1.73 can be assigned to the SEM image in Figure 4.15 a. This procedure can be applied to any SEM image of a stretched monolayer in order to ascertain the proceeded stretching. Furthermore, it was shown that the calculated stretching parameters are valid throughout the entire substrate (Figure 4.16). This means, that independent of the initial orientation of a domain, similar stretching parameters are applied during immersion. The domains A and B had signifi-cantly different orientations in the hcp structure,δ= 23 ◦andδ= -16 ◦. Nevertheless, the individually calculated values forαandSmatch very well.