Self Assembly of Magnetically Interacting Cubes by a Turbulent Fluid Flow

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Written by Kevin Tian, AP 225, Fall 2011

Title: Self assembly of magnetically interacting cubes by a turbulent Fluid Flow

Authors: Madhav Mani, Filip Ilievski, Michael Brenner, George Whitesides

Journal: Physical Review E (2011). Volume: 83, Issue: 1, Pages: 3-6

Keywords: self-assembly, electromagnetism, fluid dynamics, statistical mechanics, turbulent flow, PDMS

Paper Summary

A common method for the assembly of macroscopic objects has been vibration. Previous works have performed the shaking of a Polydimethylsiloxane (PDMS) sheet embedded with small magnets, resulting in the spontaneous folding of this sheet into a closed structure. Many other such examples of combining vibration and magnetism to perform self-assemmbly, however most do not have a theoretical basis underlying their experiments. There appears to be a lack of quantitative theory to describe the self-assembly process, and how to adjust parameters in order to obtain high yield of a desired pattern or structure.

This paper's aim is to propose a simple model system for which only very basic physics is required, yet for which we can adequately describe and predict the behaviors of the system. The system of choice is one for which the yield can be accurately predicted through theory; this consists of a number of PDMS cubes embedded with magnets in a closed container placed in turbulent fluid flow.

Through the use of simple statistical mechanics and fluid mechanics, the authors of the paper provide some foundation for use of magnetism and turbulent flow in self-assembly.

Theory

Background

The essential description of the system can be arrived at via first principles description of the mechanics of particles in turbulent fluid flow, combined with the known form of magnetic binding energy and statistical mechanics. To begin with we observe the translational equation of motion for the center of mass for the 'i'th monomer:

<math>(1)\qquad m_p {{d^2 x_i}\over{d t^2}} = c_D {{d x_i}\over{d t}} - \nabla V(X_i) + \xi (t) </math>

for mass of the particle <math>m_p\,\!</math>, drag coefficient <math>c_D\,\!</math>, an interactive magnetic potential <math>V(x_i) = \sum_{i \ne j} U(\vert x_i - x_j \vert) </math>, summing up the magnetic interation <math>U(\eta)\,\!</math>. The agitation on the particle due to the turbulent flow is given by <math> \xi (t)\,\!</math>.


We note that both <math>c_D, \xi (t)\,\!</math> depend on particle size relative to the turbulent eddies in the fluid flow. Typical eddy size is equivalent to the length scale at which the local Reynolds number is unity, known as the Kolmogorov Microscale, <math>l_{\eta}</math>, which in the case of this experiment is found to be <math>l_{\eta} \approx 10^{-3} cm</math>. Comparing to cube size, <math>d_p \approx 1cm</math>, we note that the Kolmogorov Microscale is significantly smaller than particle size. This implies viscous stresses dominate the drag on the particle, allowing us to estimate the drag coefficient as:

<math>(2)\qquad c_D = 24 \pi \mu d_p ( 1 + 0.13115 Re_{r}^{n} )</math>

...where <math>Re_p = (d_P / l_n)^{4/3}</math> and <math>n= 0.82 - 0.05 log_{10} (Re_p)</math>.


In order to further simplify the problem we must understand the turbulent force, <math>\xi (t)</math>.

  • Typical deceleration time scale for a particle is <math>( m_p / c_D ) \approx </math> 25 times slower than turnover time scale of turbulent eddies, which heavily implies the fluid force is effectively time-uncorrelated.
  • In combination with the Central Limit Theorem, the authors claim a fluid force that is a Gaussian that has zero mean and is temporally uncorrelated.
    • <math><\xi (t)>=0\,\!</math>
    • <math><\xi (t') \xi (t)> = 2q \delta (t'-t)\,\!</math>, where 'q' is the strength of the fluctuation (aka Noise).
    • This yields the result that the particle diffusion constant is <math>D = m^2 q / 2 c^{2}_{D} </math>
  • We estimate q

With all these assumptions we can thus realize that (1) is simply a classical Langevin equation, which has a well known solution. We thus make the analogy of this system with a system of particles in contact with a thermal bath, and finding the probability distribution of chain lengths of linear monomers of length N, <math>P_N</math>. The equivalence comes with allowing for an effective thermal energy to be defined as <math> k_b T_{eff} = c_D D \,\!</math>.

We must then compute the partition function as defined in statistical mechanics in order to derive the probability distribution. After decomposing the partition function into several independent parts (translational, rotational and vibrational represented by <math>q_t,~q_r,~q_v</math> respectively) we obtain:

<math>(3) \qquad Q_N = \left [q_{t} q_{r} q_{v} e^{-V^*/k_B T_{eff}} \right]^N </math>, for a magnetic binding energy of two magnets <math>V^*</math>.

Evaluating the partition function requires some though, however by making the following approximations, an explicit form can be computed.

  • <math>q_t \sim \left (N m k_B T_{eff} \right)^{3/2} \,\!</math>, obtained from considerations of thermal translational motion
  • <math>q_r \sim (I_{AN} k_B T_{eff})^{1/2} (I_{BN} k_B T_{eff}))\,\!</math>, obtained from the independent rotational partition function for a rod
    • <math>I_{AN},~I_{BN}</math> are defined as the moments of inertia for rotating rod of N aggregate segments along long and perpendicular axes respectively.
  • Taking these together we note that <math>Q_N \sim N^5 x^N \,\!</math>, where <math>x \equiv e^{-V^*/ k_B T_{eff}}</math>
    • [NOTE] Here the authors do something that seems not very intuitive to me. They for some reason ignore the vibrational portion of the partition function. My best guess for this is that the reason for this part does not depend on the number of linearly aggregated cubes, N, and thus does not quite influence the resulting probability distribution. However this is mere speculation. Verification is required.


Theoretical results

By combining all of the above, we arrive at the probability distribution:

<math>(4) \qquad {P \over P_{N_m}} = \left[{N \over N_m} exp \left( 1 - {N \over N_m}\right) \right]^5 </math>

We note that this function has a most probable configuration <math>N_m</math>:

<math>(5) \qquad N_m \sim - {5 k_B T_{eff} \over V*}</math>

This predicts that the 'shape' of the distribution depends only on the parameter <math>N_m</math>, and that re-scaling all distributions measured by the measured <math>P(N_m)</math> should collapse all results onto a single curve, described by equation 4.

Additionally there is predicted to be a linear relationship

Experimental Design

Figure 1. Various pictures depicting the set-up of the experiment. (a) shows a schematic sketch of the equipment, which consists of a closed container of size order 8cm enclosing N magnetic cubes, placed inside a rotating disk (which generates turbulent flow). (b) depicts the Polydimethylsiloxane blocks with embedded magnets and NiCu pieces. (c) depicts the results of agitation that cause the spontaneous assembly of the block-chains.

Monomers

The magnetic blocks used as the self-assembling unit consist of PDMS cubes (with side length 1cm) embedded with two items. (See Figure 1b)

  • A small disk-shaped NdFeB magnet (1/8" diameter and 1/32" thickness) in the center of one face
  • A square prism of NiCu (1/4" x 1/4" x 1/8")in the opposing face.

NiCu was selected due to it's low curie temperature of <math>T_{Curie} = 165^{\circ}</math>.


Environment

Vibration tuning was done by placing N of these cubes in a closed containers (diameter 8cm, height 10cm). The medium is water with 0.3M CsCl (to match cube density) and 10mM Triton-X 100 (to minimize bubble formation upon agitation).

This jar is then attached to a 60cm diameter disk which is rotated at a frequency between 9RPM and 80RPM.

Experiment

The experiment itself involves N=12 monomers rotated in the jar for 50 full rotations at varying RPM. Chain lengths are then observed (see figure 1c) and the manually disassembled before repeating the experiment. Two temperatures were used in this experiment, Room temperature and <math> T=80^{\circ}</math>.

Results

Figure 2. Histograms of the experimental chain length distributions for varying rotation RPM and temperatures (a) Room Temperature, (b) T=80°.

Initial distribution of results can be seen in Figure 2 for the two temperature used in the experiment. We note that the distribution appears rather broad, yet has a distinct chain length of which there is a maximum yield.

Figure 3. Normalized experimental results being compared with theoretical predictions. Statistical error bars are given. Solid line is theoretical prediction, with symbols representing experimental data.

Figure 3 shows the testing of the predictions given by Equation (3), which predict that if we normalize the measured probability then all experimental data would fall onto a single curve that is a function of <math>N/N_m</math>. AS one can see the agreement is fairly good if we account for the statistical error.

Figure 4. Position of the peak chain length, <math>N_m</math>, vs angular velocity of the jar, <math>\omega</math>. Lines given are from best linear fit though respective data, suggesting evidence for a linear relationship between the two quantities.

Figure 4 is testing the second prediction which notes a relationship between the most probable value of N, <math>N_m</math> and the magnetic interaction energy between monomers <math>V^*</math>. More specifically however, this energy is influenced by the rotation frequency, and thus there should be a linear dependence of <math>N_m</math> on angular frequency <math>\omega</math>. As one can see the relationship is fairly linear, and the change in the slope of the linear relation is exactly as one would expect for the real change in temperature.

Discussion & Conclusions

The purpose of the paper has indeed been realized, in that a turbulent flow has been effectively demonstrated to predictably control the mascroscopic self-assembly of a simple experimental system. Changeing both strength of the turbulence and the binding energy of the magnets has been shown to alter the probability distribution of chain lengths in a well-described fashion by simple use of statistical mechanics.

However as one should note, the equation for the probability distribution is valid only in the dilute limit (since hydrodynamic effects between cubes have been ignored due to their size relative to the magnetic interactions when the cubes are in close proximity). Although I personally have a few questions regarding the derivation of the describing theory, it appears to follow the measured experimental data fairly well. However regarding further refinements to the theory, it begs the question whether vibration contributions to the partition function are significant or not.

On the other hand, the idea of using fluid flow to control the assembly of mesoscopic objects holds great potential, especially if one can construct a theory to deterministically predict the behaviors of various systems, as the authors of the paper have done. How well this can be done for more complex structures remains to be seen, however the idea still has yet to be fully explored.