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1. Objectives

Vibrational modes of solid materials are important parameters that can be used to determine their thermodynamics properties. The modes can also provide insight into various behaviors of solids such as phase transitions and many of their applications in areas such as catalysis, photonic, materials fabrication, etc. Microscopic dynamical measurements are an ideal method to determine the vibrational modes of solid materials. Unfortunately, such microscopic dynamical measurements within the bulk of a three-dimensional (3D) atomic or molecular crystal are impossible due to difficulty in tracking atoms or molecules. The goal of this project is to employ thermally responsive colloidal spheres and real-time video light microscopy to experimentally determine the vibrational modes in colloidal crystals in situ. The diameter, and therefore the volume fraction which controls phase transition, of the microgel spheres can be precisely adjusted by tuning the temperature, and will allow us to extend our approach to the investigation the melting transition in colloidal crystals in the future.

2. Results

2.1. Particle Synthesis.

To create the thermally sensitive particles required for our experiments, we synthesized core-shell particles where the core is a fluorescently labeled silica particle and the corona is made out of N-isopropylacrylamide polymer (PNIPAm). PNIPAm have a lower critical solution temperature (LCST); they are hydrophilic below the LCST and hydrophobic above it. As a result, the thickness of the PNIPAm shell and therefore the volume fraction decreases with an increase in temperature. A core-shell particle with a fluorescently labeled core and unlabelled corona made locating the particle center easier and increased the spatial resolution. We synthesized the fluorescent silica core using a modified Stober method with slight modification to incorporate the fluorescence dyes. The PNIPAm shell on fluorescent silica core was synthesized using atom transfer radical polymerization.

2.2. Data Collection and Analysis

The temperature sensitive colloids were loaded into hermetically sealed glass chamber of dimensions 18 × 6 × 0.1 mm. The sample had a volume fraction, f = 0.59 and  contained face-centered-cubic crystal (FCC) domains of few mm sizes. The volume fraction was kept constant by keeping the temperature of the sample fixed (± 0.1°C) using stage and objective warmers. We imaged the crystal structure and the particle motion using a multiphoton confocal microscope (Leica SP5) with 100x oil objective (N.A. 1.4) and with oil condenser (N.A. 1.25). We found our crystal to have face-centered-cubic (FCC) symmetry. We collected 21500 frames with each frame containing 2700 particles and measured particle displacements in each frame.

Experimental equilibrium points of each particle were calculated by averaging particle positions for 21500 frames. The averaged mean squared displacement (MSD) of the particles plateaued at long time indicating that the particles were caged. The displacements along the x and y axis of the lab frame, u(x) and u(y) respectively, were calculated by taking the difference between instantaneous displacements of particles from their equilibrium positions.  We then calculated the covariance matrix from the displacements of each particle and averaged for 21500 frames, diagonalized the covariance matrix to find the eigenvalues and eigenvectors. The eigenvectors are the vibrational modes of the crystal. The inverse of the eigenvalues are energies of the measured vibrational modes.

2.2. Results

The lowest energy vibrational modes for our FCC colloidal crystal are shown in Figure 1.  In general, rotational fields (vortices) are related to the transverse waves in a crystal. The energy cost to create rotational modes is less than the outward and inward (divergence) type of motion, which are related to longitudinal modes. We found that the lowest energy modes in our FCC colloidal crystals are mostly transverse plane waves. In mode 1, we observe a single vortex spanning the entire field of view. Mode 2 consists of vortices of length that is shorter than that in mode 1. As a result, mode 2 has part of a second vortex within the field of view. In mode 3 and mode 4, we observe two vortices in the field of view. At higher energy modes, the longitudinal modes start to appear (not shown).

3. Conclusions

We used temperature sensitive colloidal particles and real-time video microscopy to experimentally determine vibrational modes in colloidal crystals. The lower energy modes are transverse whereas higher energy modes have longitudinal contribution. The methodology of experimentally determining vibrational modes will allow us to investigate how modes change in crystals during melting.

Figure 1: The vibrational modes in a FCC colloidal crystal.

4. Educational Outreach

The emerging concepts in self-assembly, diffusion, and the relationship between structure and dynamics from this project are incorporated in an existing course on soft materials. Furthermore, the highly visual and interactive nature of soft materials have been successful at attracting and inspiring undergraduates to high-level science by making complex ideas more tangible. The PI has supervised 4 undergraduate students through NSF Research Experience for Undergraduates (REU) program. Three of the participating students are now in graduate school.