First Effective Mathematical Model Allowing the Dramatic Improvement in the Efficiency of Biorotary Nanomotors

Through the Application of Maxwell's Demon

Charlotte Kavaler

What is a nanomotor?

Biorotary NanomotorsA biorotary nanomotor is a rotating, submicroscopic motor either naturally occurring or modeled off of naturally occurring motors.

In the design used here, the motor acts like a water wheel, turned by electrons travelling from high concentration in one of two electron source reservoirs to low concentration in an electron drain. These reservoirs are powered by a voltage. The electrons bond to quantum dots along to wheel of the motor, and as the travel to the drain they turn the motor. This creates torque, and the movement of the electrons creates a current.

Applications of Nanomotors

• Nanomotors have endless potential within medicine. They can be used to created targeted and powered treatment methods; that is, medication that actively seeks out the afflicted regions. They will be used to treat cancer, heart disease, and countless other ailments, as they can target treatment to individual cells [3].

• However, nanomotors are far from achieving these goal. Current nanomotor designs do not produce nearly enough output to achieve these goals [4].

Maxwell's Demonor

The Information-to-Energy Conversion

Szilard's 'Intelligence'In 1929, Leo Szilard proposed a system where there is a particle on a staircase (representing an electric potential). Because of quantum fluctuations in energy, that particle will randomly move up or down a step every now and then. He suggested that an intelligent body could place a block behind the particle each time it climbed a step. Over time, this would cause a substantial increase in energy at an insignificant cost (the energy it takes to place the block is negligable).

The Information to Energy ConversionInitially, Szilard's work was controversial, as many scientists argued it violated the Law of Conservation of Energy. However, decades later, it was realized that energy was not created, but transformed from information about the particle's location. Because of this, Maxwell's demon is often called the information-to-energy conversion. In this work, I will utilize an information-to-energy conversion the improve the efficiency of nanomotors.

Literature Review

Research on the NanomotorThe discovery of the biorotary nanomotor (1997, Noji, Yasuda,

Yoshida, and Kinosita, "Direct Observation of the rotation of F1-ATPase")

The creation of the model that provides the basis for my work (2009, Junge, "Torque Generation and Elastic Power Transmission in the rotary F0F1-ATPase"; 2009, Smirnova, Mourokh, Savel'eva, and Noria, "Bio-mimicking rotary nanomotors")

The first nanomotors within a cell (2014, Penn State)

Theoretical Research on Maxwell's Demon Szilard's Initial Hypothesis of the power of an intelligence (1929,

Szilard, "On the Decrease of Entropy in a Thermodynamic System by the Intervention of Intelligent Beings")

The resolution of the problem by viewing the effect of memory on the entropy of the system (Landauer and Bennett)

But what does this mean in terms of energy?

The Physical Demonstration of the DemonThe information to energy conversion was for the first time experimentally shown in 2010 by a group of Japanese scientists who placed a particle on a glass surface and created a periodic electric potential. They were able to create what registered as a net gain on energy on the system, demonstrating Szilard's theory.

Research HypothesisIn order to create the next generation of nanomotor, I decided

to apply the concept of the information to energy conversion to the motor. As I explained earlier, the conversion can be thought of as using knowledge of a particle's location to derive energy. In this model, the position of the electrons travelling on the bonding sites through the motor is monitored, so it can be used to improve the efficiency of the system. The source-drain voltage of the motor can be varied based on whether or not particles need to be added to or removed from the bonding sites at various times.

I hypothesize that by doing such, the motor will be able to have an increased overall output of current, rotational speed, and torque, by decreasing the wasted energy used for the source-drain voltage.

Methodology

The proton distribution, ρ𝜇, for , υ ∈ {1,2,...,8}, can be used to produce the following differential 𝜇equation:

where a ;σ μν is the matrix created by the annihilation operator of the system, ħ is the modified Plank’s constant, εi

is the energy of state i, and fα( )ω is the Fermi-Dirac Distribution for the electrons in the various reservoirs.

After solving for ρ in the differential equation, it is used to find the population of a site nX using the

following:

Electron-State

First, I defined a function depending on the electron state of the motor;

that is, a function was created that described which of the bonding sites

was occupied at some time. For each of the bonding sites, A,B, and C, a

voltage, Vμ=500 mv if the site is empty, or -500 mv if the site is

occupied, where μ∈{A,B,C}. The source drain voltage of the overall

motor was redefined using the angle of the motor. If φ is the angle of

the motor (see Figure 1), then the voltage of the motor depends on

which one of the sites is closest to the sources and the drain.

Fermi-Dirac Distribution

This in turn affects other aspects of the system. The Fermi-Dirac (F-D)

Distribution in a system is defined as 1e(εi-μ)/kT+1, where εi is the

energy of state i with just one particle, T is the temperature, k is the

Boltzmann constant, and μ is the chemical potential, where μ is

dependent on V. Here, there are three reservoirs—the two sources and

the drain—in which Fermi-Dirac distribution applies, and each one of

them was rewritten in order to take into account the varying of V over

time.

Proton-Distribution

The F-D distribution is used to determine the proton distribution, ρμ, of

the system, as discussed above, for each of the sites, so these likewise

became dependent on the electron state and the angle of the motor.

The population for each site relies on the proton distribution, and can be

defined using the equation in the above section. So, finally, the

population of each site at some time t was rewritten using each of the

previous rewritten equations to make it depend on the angle and

electron state.

Results

Results SummaryAs all of these figures show, the model of nanomotor proposed in this paper

dramatically improves upon previous designs. It increases the output of the motor in

torque, in number and speed of rotation, and, most substantially, in current. By making the

source-drain voltage dependent on the rest of the motor, I was able to increase the

efficiency of the overall system by a large margin, and therefore create a design for a

motor that has potential for utility, unlike prior designs that have been far too inefficient to

provide practical value.

Areas for Continued ResearchBut as dramatically improved as this new model is, it does have a few

important limitations. First of all, this motor is based on the biological rotary motors seen in nature. Similar structures exist within the mitochondria of the human cell. And so this motor is subject to many of the limitations that those natural structures are subject to. Most significantly, these nanomotors will break down at high temperatures. While this can be a problem even within the human body, if the scientific community aims to integrate nanomotors into so-called 'smart materials,' the motors will need to be able to withstand a larger range of environments than they are currently able to.

In addition to that, this work is entirely theoretical. It is important that the design put forward within this paper is tested experimentally. The next step in developing this research would be to evaluate this design in a real-world environment.

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