2d-colloidal crystal
P. Keim, U. Herz, G. Maret, H.H. von Gr¨unberg
Using positional data from video-microscopy and apply-ing the equipartition theorem for harmonic Hamiltonians, we determine the wave-vector-dependent normal mode spring constants of a two-dimensional colloidal crystal and compare the measured band-structure to predictions of the harmonic lattice theory. We find good agreement for both the transversal and the longitudinal mode. For q→0, the measured spring constants are consistent with the elastic moduli of the crystal. Next to the solid→ hex-atic phase transition, the moduli have to be renormalized because of the appearance of lattice defects according to Kosterlitz-Thouless-Halperin-Nelson-Young theory [1-3]. An essential element of this theory is the universal prediction that Young’s modulus must approach 16π at the melting temperature. This is indeed observed in our experiment.
The experimental setup is described as followed: Spher-ical colloids (diameter d = 4.5 µm) are confined by gravity to a water/air interface formed by a water drop suspended by surface tension in a top sealed cylindrical hole in a glass plate. The flatness of the interface can be controlled within±half a micron. The field of view has a size of 835×620µm containing typically up to 3·103 particles, whereas the whole sample has a size of≈ 50mm2 with about105 particles. The number of particles in the field of view is controlled by the curvature of the droplet via an active regulation with an accuracy of 1% . The variation of the inclination of the sample is in the range ofα ≈ 1µRad providing best equilibrium conditions for long time stability. The particles are super-paramagnetic due to Fe2O3 doping. A magnetic field B~ applied perpendicular to the air/water interface induces in each particle a magnetic momentM~ = χ ~B which leads to a repulsive dipole-dipole pair-interaction energy ofβv(r) = Γ/(√πρr)3with the dimensionless in-teraction strength given byΓ =β(µ0/4π)(χB)2(πρ)3/2 (β = 1/kT inverse temperature, χ susceptibility, ρ= 2/√
3a22D density,alattice constant of a hexagonal lattice). The interaction can be externally controlled by means of the magnetic fieldB. Γ is the only parameter controlling the phase-behavior of the system. ForΓ>60 the sample is a hexagonal crystal (see Fig. 1).
The property we here consider is the crystal’s elastic re-sponse to thermal excitations, specifically, the phonon elastic dispersion relations. In this regard, colloidal crys-tals are rather special in that their phonons are almost
Figure 1: Videoimage (420×310µm) of a typical colloidal crys-tal; the two-dimensional system consists of paramagnetic col-loids confined at the air-water interface of a hanging water drop.
always overdamped: the ratio between the wave-vector dependent frequency ω(~q) = p
λ(~q)/m, characteristic of the harmonic forces with spring constants λ(~q), and the friction factors Λ(~q)for the modes of lattice motion through the host liquid, is typically of the order10−3 to 10−4in colloidal systems. Therefore, the time autocorre-lation function of a phonon normal mode coordinate de-cays exponentially with a rate given byλ(~q)/Λ(~q).
For each of all N particles in a given configuration, we determine the displacement~u(R)~ of the particle from its equilibrium position R. Using the theory of harmonic~ crystals , we now derive an equation guiding us from the measured displacement vectors~u(R)~ to the eigenvalues of the dynamical matrix. LetDµ,ν(~q)(µ, ν ∈ {x, y}) be the dynamical matrix, connected through a Fourier trans-formation to the matrixDµ,ν(R, ~~ R′)which is essentially the matrix of the second derivatives of the pair-potential v(r)∼Γ/r3. It is obvious thatDµ,ν(~q)depends linearly on the interaction strength parameterΓ; therefore we write Dµ,ν(~q) = (kTΓ/a2) ˜Dµ,ν(~q)and obtain the dimension-less dynamical matrixD˜µ,ν(~q)which is independent ofΓ.
Its eigenvalues are denoted byλs(~q)a2/(kT Γ). Here the polarization subscriptsstands for the longitudinal (s=l) and transversal (s=t) mode. The harmonic potential en-ergy of the crystal can be written in the following form
U =1 2
X
~ q,µ,ν
u∗µ(~q)Dµ,ν(~q)uν(~q) (C.2) withuν(~q)being theν’th component of the Fourier trans-form of the displacement vectors ~u(R). The equiparti-~ tion theorem for classical harmonic Hamiltonians states that on average every mode has an energy ofkT /2. Thus hu∗µ(~q)Dµ,ν(~q)uν(~q)i/2 =kT /2and this leads us to
hu∗µ(~q)uν(~q)i=kT D−1µ,ν(~q) (C.3) where in our case the average has to be taken over all mea-sured configurations. Introducing withps(~q)an
abbrevia-C2. DISPERSION RELATION AND ELASTIC MODULI NEAR MELTING OF A 2D-COLLOIDAL CRYSTAL33
Figure 2: Dispersionrelation of harmonic lattice spring constants of a 2D colloidal crystal. Symbols for constants experimentally determined with eq. C.3 and C.4 from the relative displacement of the particles from their equilibrium position for a soft (Γ = 75, empty circles), hard (Γ = 175, empty squares) and very hard (Γ = 250, filled triangles) crystal; solid lines for the theoretical band-structure calculated from standard harmonic crystal theory.
The inset shows the first Brillouin zone of the hexagonal lattice and labels for high-symmetry pointsΓ, K, M. The upper curve corresponds to the longitudinal, the lower one to the transversal mode. the measured set of displacement vectors for Γ = 250, 175 and 75, and compares it to the theoretical band-structure (solid lines) of a harmonic crystal having a two-dimensional hexagonal lattice (a = 12.98µm). The lat-ter is based on the second derivatives of the known pair-potential and results from diagonalizingD˜µ,ν(~q). We find good agreement for both the transversal and longitudinal mode. No fit parameter has been used [4].
Atq→0the elastic moduli of the crystal can be read off from the elastic dispersion curves:limq→0λt(~q) =µv0q2 andlimq→0λl(~q) = (K+µ)v0q2whereKandµare the bulk and shear elastic moduli of continuum theory (with the cell volumev0 = 1/ρ) [3]. Moreover, in the elastic limitλl(~q)andλt(~q)are particularly simple to obtain from the measured displacement vectors. After decomposing the displacement field~u(~q) into parts~u||(~q) and~u⊥(~q), parallel and perpendicular to~q, one finds that
v0(2µR+λR)
3a2/2is the area per colloid in a triangular lattice [5].
For the pair-potential∼Γ/r3the elastic constants can be calculated in the limitΓ→ ∞(T = 0) using simple ther-modynamical relations involving essentially lattice sums of the pair-potential. One findsλ+µ= 3.46 Γandµ= 0.346 Γ. For convenience, we divide in the following all moduli byΓ. Fig. 3 shows(λ+2µ)v0/ΓkTandµv0/ΓkT, obtained from thisT = 0calculation, as thick solid ar-rows, and compares it to the expressions q2h|u||(~q)|2i−1
and q2h|u⊥(~q)|2i−1
, as obtained from the measured tra-jectories for three different values ofΓ.
While for all our measurements aboveΓm = 60the re-sulting bands lie on top of the dashed thick lines in Fig. 3, one finds a systematic shift to smaller values forΓ<Γm. Fig. 3 shows, as an example, one out of the four bands of the measurement in the fluid phase (Γ = 52). It lies an order of magnitude below the crystalline bands.
In order to infer functionsµR(Γ)andλR(Γ)from these bands, we need to take the limitq → 0. Since at lowq we have to expect finite size effects, and at highq, near the edges of the first Brillouin zone, effects resulting from the band dispersion of the discrete lattice, we choose an intermediateqregime (0.8< qa <2.5), indicated by the thick solid bar in Fig. 3, to extrapolate the bands toq= 0, applying a linear regression scheme. Fig. 4 shows the re-sulting moduli, for all values ofΓ studied. Black sym-bols refer to systems in the crystalline state (Γ > Γm), grey data points to those in the fluid/hexatic phase. The thick dashed lines in Fig. 4 represent theT = 0 calcula-tion which holds down toΓvalues close toΓ = 75. The thick solid line shows the theoretical curve forµR(Γ)and µR(Γ)+2λR(Γ), which we computed following the renor-malization procedure outlined in Ref. [3]. Theory and ex-periment agree well, considering that no fit parameter has been used. For Γ > Γm, all our results are converged, meaning that the computed moduli do not depend on the
Extrapolation quantity is plotted for two different directions inq-space (Γ→ M andΓ→Kin the first Brillouin zone). Thick solid arrows for aT = 0prediction of the elastic moduli, dashed solid lines for the predictions of harmonic crystal theory. ForΓ = 52, deep in the fluid phase, just one band is shown.
40 60 80 100 120 140 160 180 inverse temperature (Γ)
0.1 1
elastic constants
3.806 (at T=0)
0.346 (at T=0) (λR+ 2 µR) a2 / Γk T
µR a2 / Γk T Nelson/Halperin
Γ < Γm
Figure 4: Elastic moduli of a 2D colloidal crystal as a function of the inverse temperature, obtained from extrapolating the bands in Fig. 3 down toq= 0. The melting temperature is atΓm= 60.
Thick dashed lines for aT = 0prediction of the elastic moduli, λ+ 2µ = 3.806Γand µ = 0.346Γ; thick solid line for the theoretical elastic constants, renormalized as described in [3].
length of the trajectory. This is demonstrated by means of theΓ = 100measurement for which the moduli were computed taking only the first third of all configurations (open square symbol in Fig. 4). ForΓ < Γm the mod-uli depend on evaluation time (open square symbol of the Γ = 49measurement). Physically, one could interpret this in terms of frequency-dependent moduli. Especially a shear modulus for non-zeroω ∼ 1/texpdoes seem to exist in the fluid phase.
A very strong prediction of the KTHNY theory has never been verified experimentally. It states that the renor-malized Young’s Modulus KR(T), being related just to the renormalized Lam´e coefficientsµRandλR, must ap-proach the value16πat the melting temperature [3],
KR(T) = 4µR(1−µR/(2µR+λR))→16π, (C.7) if T →Tm−.
This is obviously an universal property of 2D systems at the melting transition. Fig. 5, indeed, confirms the the-oretical prediction expressed by eq. C.7. KR(Γ)is eval-uated using the elastic moduli from Fig. 4. Using the theoretical values from theT = 0calculation, we obtain K(Γ) = 1.258 Γ, shown in Fig. 5 as dashed line. The thick solid line shows the theoretical curve forKR(Γ)which we computed with Lam´e coefficients that were renormalized following the procedure explained above. The main result of this work is that the experimental data points closely follow the theoretical curve and indeed, they cross16πat Γ = Γm in excellent agreement with the predictions of Nelson and Halperin. The length of the remaining error bars correlate with the total measurement time, and the q-range chosen in the extrapolation step [4].
To conclude, we have measured particle trajectories of a two-dimensional colloidal model system and computed
40 60 80 100 120 140 160 180
inverse temperature (Γ) 50
100 150 200 250
Young’s modulus (KR(Γ)) T=0: 1.258 Γ
KR(Γ) Nelson,Halperin
16 π Γm
Figure 5: Young’s modulus, eq. C.7, as a function of the in-verse temperature, evaluated with the experimentally determined Lam´e coefficients of Fig. 4 (symbols). The solid curve isKR(Γ) renormalized according to [3], while the dashed curve is based on the T=0 prediction.
elastic dispersion curves which at low~qgive access to the elastic constants. We thus measuredµR,λRand Young’s modulus KR as a function of the inverse temperatureΓ.
All three quantities compare well with corresponding pre-dictions of the KTNHY theory. Young’s modulus, in par-ticular, tends to16πwhen the crystal melts, as predicted in [3].
[1] J. Kosterlitz, D. Thouless, J. Phys. C. 6, 1181 (1973).
[2] A. Young, Phys. Rev. B. 19, 1855 (1979).
[3] D. Nelson, B. Halperin, Phys. Rev. B. 19, 2457 (1979).
[4] P. Keim, G. Maret, U. Herz, H.H. von Gr¨unberg, Phys. Rev.
Lett. 92, 215504 (2004).
[5] H.H. von Gr¨unberg, P. Keim, K. Zahn, G. Maret, Phys. Rev.
Lett. 93, 255703 (2004).
C3. MELTING OF ANISOTROPIC CRYSTALS IN 2D 35