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During the initial stages of the growth of the nanowire of Problem 3.109, a slight perturbation of the liquid catalyst droplet can cause it to be suspended on the top of the nanowire in an off-center position. The resulting nonuniform deposition of solid at the solid-liquid interface can be manipulated to form engineered shapes such as a nanospring, that is characterized by a spring radius \(r\), spring pitch \(s\), overall chord length \(L_{c}\) (length running along the spring), and end-to-end length \(L\), as shown in the sketch. Consider a silicon carbide nanospring of diameter \(D=15 \mathrm{~nm}, r=30 \mathrm{~nm}, s=\) \(25 \mathrm{~nm}\), and \(L_{c}=425 \mathrm{~nm}\). From experiments, it is known that the average spring pitch \(\bar{s}\) varies with average temperature \(\bar{T}\) by the relation \(d \bar{s} / d \bar{T}=0.1 \mathrm{~nm} / \mathrm{K}\). Using this information, a student suggests that a nanoactuator can be constructed by connecting one end of the nanospring to a small heater and raising the temperature of that end of the nano spring above its initial value. Calculate the actuation distance \(\Delta L\) for conditions where \(h=10^{6} \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}, T_{\infty}=T_{i}=25^{\circ} \mathrm{C}\), with a base temperature of \(T_{b}=50^{\circ} \mathrm{C}\). If the base temperature can be controlled to within \(1^{\circ} \mathrm{C}\), calculate the accuracy to which the actuation distance can be controlled. Hint: Assume the spring radius does not change when the spring is heated. The overall spring length may be approximated by the formula,

Step by step solution

01

Calculate the number of turns and the spring length L

We are given the overall chord length \(L_c\) and the spring pitch \(s\). To find the number of turns \(N\), we can use the following formula: \[N = \frac{L_c}{s}\] We plug in the given values of \(L_c\) and \(s\): \[N = \frac{425 \ \mathrm{nm}}{25 \ \mathrm{nm}} = 17\] Now that we have found the number of turns, we can find the length of the spring \(L\). Intuitively, the length of the spring (when viewed from the side) is equal to the product of the spring pitch \(s\) and the number of turns \(N\): \[L = N \times s\] Plugging in the given values of \(N\) and \(s\): \[L = 17 \times 25 \ \mathrm{nm} = 425 \ \mathrm{nm}\]

02

Relate the change in spring pitch 螖s to the actuation distance 螖L

The actuation distance, \(\Delta L\), is the change in the end-to-end length of the spring as a result of the change in the average spring pitch, \(\Delta \bar{s}\). According to the given relationship between the spring pitch and temperature, we have: \[\frac{d \bar{s}}{d \bar{T}} = 0.1 \ \mathrm{nm/K}\] Therefore, for the given change in the base temperature, \(\Delta T_b = 1^\circ \mathrm{C}\), we can obtain the change in the average spring pitch, \(螖\bar{s}\): \[螖\bar{s} = \Delta T_b \times \frac{d \bar{s}}{d \bar{T}} = 1^\circ \mathrm{C} \times 0.1 \ \mathrm{nm/K} = 0.1 \ \mathrm{nm}\] Now, we can relate the actuation distance, \(螖L\), to the change in spring pitch, \(螖\bar{s}\), using the number of turns, \(N\): \[螖L = N \times 螖\bar{s}\] \[螖L = 17 \times 0.1 \ \mathrm{nm} = 1.7 \ \mathrm{nm}\]

03

Calculate the accuracy of actuation distance

As we have calculated the actuation distance \(\Delta L\) for the given change in base temperature \(\Delta T_b = 1^\circ \mathrm{C}\), the accuracy to which the actuation distance can be controlled is equal to \(\Delta L\) itself: Accuracy of actuation distance = \(螖L = 1.7 \ \mathrm{nm}\) This means that any change in the base temperature could be controlled within \(1^\circ \mathrm{C}\), which makes the actuation distance accurate up to \(1.7 \ \mathrm{nm}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Thermal Expansion

Thermal expansion refers to the tendency of a material to change its dimensions in response to a change in temperature. When materials are heated, their molecules move faster and tend to occupy more space, causing the material to expand. This property is crucial in understanding how temperature variations affect physical structures and functionality.

For silicon carbide nanosprings, thermal expansion manifests as a change in the spring pitch. The relationship is described by a linear coefficient indicating how much the pitch increases with a temperature rise. With a given coefficient of 0.1 nm/K, a temperature increase will elongate the spring by increasing its pitch. This principle is key in designing nanoactuators as it can precisely control the expansion to achieve desired mechanical movements.

When dealing with nanoscale elements, even a minor temperature change can lead to significant expansion. This precision is what makes thermal expansion an exciting area of study in nanoengineering applications like nanosprings.

Nanowire Structures

Nanowire structures are astonishingly small, wire-like materials with diameters measured in nanometers. They are vital components in developing modern technology, including in the fields of electronics and nanomechanics. Nanowires are fabricated from various materials, such as silicon carbide, which possess unique electrical, thermal, and mechanical properties due to their nanoscale size.

In our context, nanowires serve as the foundational structures from which nanosprings are constructed. By manipulating the conditions during their growth, such as the position of a catalyst droplet, engineers can influence the final shape and properties of the nanowire. This careful control can yield specific, advanced geometries like nanosprings which are utilized for their mechanical flexibility and reactivity to thermal stimuli.

Unlike bulk materials, the surface-to-volume ratio in nanowires is significantly larger, thereby accentuating their interaction with environmental factors, such as heat. This makes meticulous design and control a necessity in applications that harness nanowire structures.

Nanoengineering

Nanoengineering is the practice of designing, fabricating, and utilizing nanoscale materials and devices. It combines principles from multiple disciplines like chemistry, physics, and nanotechnology to create innovations at the molecular level. This field has significant implications across various industries, including medicine, electronics, and materials science.

In the case of nanosprings, nanoengineering is applied to create structures with specific mechanical and thermal properties. Working at such a small scale requires precise manipulation of material properties and environmental factors to achieve the desired outcomes. Techniques in nanoengineering allow researchers to form complex shapes like nanosprings that utilize thermal expansion for actuation, showcasing the integration of multiple scientific principles.

The goal is to produce devices with specific functionalities and respond predictably to external stimuli such as temperature changes. The application of nanoengineering enables the creation of nanoactuators, which use thermal expansion to produce controlled movement.

Heat Transfer Coefficient

The heat transfer coefficient is a measure that describes how well heat is conducted through a material or between two fluid surfaces. It quantifies the efficiency of heat transfer in thermal systems, and higher values indicate better heat transfer capabilities.

In nanoscale applications, like with nanosprings, this coefficient is crucial for understanding how quickly and efficiently heat can be absorbed or released. It's particularly relevant when designing systems that involve temperature control, such as nanoactuators. The value provided in the problem, 10鈦� W/(m虏路K), is notably high, suggesting efficient heat transfer across the surface of the nanospring.

A comprehensive understanding of the heat transfer coefficient is essential for predicting system behavior as temperature changes. It plays a crucial role in determining how quickly the nanospring will respond to changes in temperature, thus influencing its actuation precision and reliability in practical applications.