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<h2 class="hd hd-2 unit-title">Tutorial Overview</h2>
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<h2>TutorIal Overview</h2>
<p style="font-size: 16px;">In these tutorial sections, you will find overviews of many of the important topics covered in the week's lecture. Here, you will find:</p>
<ul>
<li>Summaries of important concepts and equations</li>
<li>Definitions of important terms and variables</li>
<li>Practice problems with worked solutions</li>
</ul>
<p style="font-size: 16px;"></p>
<p style="font-size: 16px;">These tutorial sections are optional and are not graded, but are recommended if you would like extra practice with the material. The questions that you find here are generally simpler than those that you're encounter in the Homework and Quiz, but these problems should help to give you the fundamental skills that you will need to solve the graded problems </p>
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<h2 class="hd hd-2 unit-title">Part 1.1: The Structure of Materials</h2>
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<h3>The Structure of Materials</h3>
<p></p>
<p></p>
<p>The Structure of Materials refers to the quantitative description of materials structural arrangements at different length scales.</p>
<p></p>
<p><span style="text-decoration: underline;">This course deals primarily with solid materials:</span></p>
<ul>
<ul>
<li>Crystalline Materials</li>
<ul>
<ul>
<li type="circle">Atoms pack into periodic 3D arrays</li>
<li type="circle">Typical of metals, many ceramics, some polymers</li>
</ul>
</ul>
<li>Amorphous (non-crystalline) materials</li>
<ul>
<ul>
<li type="circle">Atoms have no periodic packing</li>
<li type="circle">Occurs in the case of complex structures, rapid cooling</li>
</ul>
</ul>
</ul>
</ul>
<p></p>
<p></p>
<p><span style="text-decoration: underline;">Important differences in the transformation from the liquid phase to the solid phase:</span></p>
<ul>
<ul>
<li>Crystalline materials</li>
<ul>
<ul>
<li type="circle">Crystallize at the melting temperature \(T_M\)</li>
<li type="circle">Have an abrupt change in specific volume at \(T_M\)</li>
</ul>
</ul>
<li>Glasses</li>
<ul>
<ul>
<li type="circle">Do not crystallize</li>
<li type="circle">At the glass transition temperature \(T_g\), there is a change in slope of the specific volume vs. temperature curve</li>
<li type="circle">At \(T_g\), the material solidifies and there is a large increase in viscosity</li>
</ul>
</ul>
</ul>
</ul>
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Tutorial Problem 1.1A
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><img height="518" width="600" src="/assets/courseware/v1/b66cf5cbf20cb335bb3104eb805af2c6/asset-v1:MITx+3.012S.1x+1T2019+type@asset+block/3012_NatureCurve.jpg" alt="Volume vs. Temperature curve for three different cooling conditions for a liquid. Solid one: As the liquid cools, the volume drops. When the liquid cools to the melting point, the volume drops dramatically. In the crystalline state, the volume continues to decrease linearly as we cool the crystal, but the slope is lower than when the material was liquid. For Material 2: This is Glass B. The volume of the liquid drops linearly as temperature decreases until we reach the glass transition temperature. At T_g, the slope of the volume vs. temperature curve decreases. At any given temperature, the volume of Glass B is higher than the Crystal. Material 3: Similar to Material 2, but the glass transition temperature is higher. This is glass A. At any given temperature, the volume of glass A is higher than both glass B and the crystalline phase."/>
<p>Image credit: Reprinted with permission from Nature, P.G. Debenedetti and F.H. Stillinger, Nature, 410, 259 (2001).</p>
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Tutorial Problem 1.1B
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Tutorial Problem 1.C
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Hint: Your answer should only be 2-3 words, and remember, you can always click on the "Show Answer" button if you think the auto-grader isn't accepting a correct answer.</label>
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<h2 class="hd hd-2 unit-title">Part 1.2: Structure of Non-Crystalline Solids</h2>
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<h3>Structure of Non-Crystalline Solids</h3>
<ul>
<ul>
<li>Disorder is an essential feature of noncrystalline solids</li>
<ul>
<ul>
<li type="circle">However, noncrystalline solids are not completely disordered</li>
<li type="circle">Noncrystalline solids exhibit <em>short-range order</em></li>
</ul>
</ul>
<li><em>Descriptors</em> are a conceptual scheme that provides quantitative characterization of some part of structure</li>
<li>Glasses and liquids have similar underlying structures</li>
<li>Because the structure of these materials vary in space (and sometimes in time), we use descriptors that represent statistical averages:</li>
<ul>
<ul>
<li type="circle">Free volume</li>
<li type="circle">Pair distribution function</li>
</ul>
</ul>
</ul>
</ul>
<p></p>
<p>Let's take a look at the first of these, <strong>free volume</strong>, in greater detail:</p>
<ul>
<ul>
<li>The free volume \(V_F(T) = V(T) - V_0(T)\)</li>
<ul>
<ul>
<li type="circle">Here, \(V\) is the total sample specific volume and \(V_0\) is the occupied specific volume</li>
<li type="circle">\(V_0\) can be approximated as \(V_{cryst}\), the specific volume of all the structural units and spaces between them in the crystalline state.</li>
<li type="circle">In other words, the free volume is approximately the specific volume of the glass (or liquid) phase minus the specific volume of the crystalline phase at the same temperature</li>
</ul>
</ul>
</ul>
</ul>
<p><img width="650" src="/assets/courseware/v1/ba940511c729876ae41391f43c09749c/asset-v1:MITx+3.012S.1x+1T2019+type@asset+block/3012_Free_Volume.jpg" alt="Image plotting the specific volume of a glass phase material and the specific volume of a crystalline phase material as a function of temperatuere. In all cases, the specific volume of the glass is greater than the specific volume of the crystal. The free volume of the glass is the difference between these two curves. " /></p>
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<p>Assume that you increase the rate at which the glass is cooled from the liquid. What happens to the free volume of the resulting glass:</p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>Now assume you decrease the rate at which the glass is cooled from the melt. What happens to the free volume of the resulting glass?</p>
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<h2 class="hd hd-2 unit-title">Part 1.3: Pair Distribution Function</h2>
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<h3>The Pair Distribution Function</h3>
<ul>
<ul>
<li>The pair distribution function tells us the probability of finding an atom a given distance \(r\) from a typical atom</li>
<li>We define the pair distribution function as \(g(r) = \frac{dn(r,dr)}{dV(r,dr)}\frac{1}{<p>}\), where</li>
<ul>
<ul>
<li type="circle">\(dn\): the number of atoms in the spherical shell</li>
<li type="circle">\(dV\): the volume of the spherical shell</li>
<li type="circle">\(r\): the distance of the shell from an arbitrary atom selected as the origin</li>
<li type="circle">\(<p>\): the average particle density</li>
</ul>
</ul>
<li>The pair distribution function is a unitless number</li>
</ul>
</ul>
<p><span style="text-decoration: underline;">Interpreting the pair distribution function:</span></p>
<ul>
<ul>
<li>In a crystal, the pair distribution function exhibits a series of discrete peaks</li>
<ul>
<ul>
<li type="circle">The magnitude and location of the peaks will depend on the exact crystal structure</li>
<li type="circle">An example:</li>
</ul>
</ul>
</ul>
</ul>
<p style="padding-left: 60px;"><img width="500" src="/assets/courseware/v1/b3a4fc96849e22eb91314971f0b9a46b/asset-v1:MITx+3.012S.1x+1T2019+type@asset+block/3012_Crystal_PDF.jpg" alt="Pdf that consists of a series of discrete peaks" /></p>
<ul>
<ul>
<li>In a gas:</li>
<ul>
<ul>
<li type="circle">The atoms are uncorrelated</li>
<li type="circle">Below the hard sphere diameter, the probability of finding an atom is zero</li>
<li type="circle">Above the hard sphere diameter, the probability of finding an atom is equal to the average gas density</li>
<li type="circle">An example:</li>
</ul>
</ul>
</ul>
</ul>
<p style="padding-left: 90px;"><img height="313" width="500" src="/assets/courseware/v1/145cf5bff0b8e6aa472df90e12088a69/asset-v1:MITx+3.012S.1x+1T2019+type@asset+block/3012_PDF_gas.jpg" alt="PDF with a value of zero at low r that jumps up to a constant finite value above the hard sphere diameter" /></p>
<ul>
<ul>
<li>In a liquid or a glass:</li>
<ul>
<li type="circle">There are broad peaks and valleys at short distances. This reflects the short-range order in these types of materials</li>
<li type="circle">At larger distances, the pair distribution function reaches a constant value. This reflects the fact that liquids and glasses have no long-range order</li>
<li type="circle">An example:</li>
</ul>
</ul>
</ul>
<p style="padding-left: 60px;"><img height="334" width="500" src="/assets/courseware/v1/4322b8fdba4c55c2dd81d75889f6254b/asset-v1:MITx+3.012S.1x+1T2019+type@asset+block/3012_Glass_PDF.jpg" alt="Pair distribution function for a glass. The function rises rapidly at r=2R_0 and has several peaks (the first two of which are marked A and B). The function finally decays to a steady value after a distance xi." /></p>
<ul>
<ul>
<ul>
<ul>
<li type="circle">\(2R_0\): below the hard-sphere diameter, the probability of finding another atom is zero</li>
<li type="circle">\(A\): This point represents the average distance to the first shell of nearest neighbors</li>
<li type="circle">\(B\): This point represents the average distance to the second shell of nearest neighbors</li>
<li type="circle">\(\xi\): This is the correlation distance of the glass. Beyond this distance, the locations of the atoms are no longer correlated, and the pair distribution function reaches a value of one.</li>
</ul>
</ul>
</ul>
</ul>
</div>
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<h2 class="hd hd-2 unit-title">Part 1.4: Polymer Structure</h2>
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<h3>Polymer Structure</h3>
<ul>
<ul>
<li>Polymers are long molecules composed of repeating units called monomers</li>
<li>Most polymers are organic molecules</li>
<li>The random walk model can be used to describe the structure of amorphous polymers</li>
<ul>
<ul>
<li type="circle">A random walk describes a path of successive steps, where the direction of each step is independent of the previous step</li>
</ul>
</ul>
<li>The chain end-to-end distance \(\vec{R}_n\) gives the distance between the beginning and the end of a polymer chain</li>
<li>The root-mean-square (rms) end-to-end distance \(R_o\) is a descriptor we can use to characterize the average spatial extent of a polymer</li>
<ul>
<ul>
<li type="circle">\(R_0 = \sqrt{<R_n^2>}\)</li>
</ul>
</ul>
<li>We can demonstrate that \(R_0 = \ell n^\frac{1}{2}\), where:</li>
<ul>
<ul>
<li type="circle">\(\ell = |\vec{r}_i|\) is the distance between each monomer unit</li>
<li type="circle">\(n\) is the number of monomers</li>
<li type="circle">\(\frac{1}{2}\) is a scaling factor</li>
</ul>
</ul>
<li>The scaling factor of \(\frac{1}{2}\) assumes that the position of each monomer is completely independent of the position of its surrounding monomers</li>
<li>The scaling factor will increase if the monomer positions are not completely independent of one another (i.e. the self-avoiding random walk)</li>
</ul>
</ul>
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Tutorial Question 1.3A
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>A polymer consists of \(n=3000\) monomer units, and the distance between each monomer unit is \(0.4 \; nm\). Assuming that the polymer can be described as an idealized random walk, what is the RMS end-to-end distance \(R_0\) of the polymer?</p>
<p>\(R_0\) (in \(nm\)):</p>
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<h2 class="hd hd-2 unit-title">Part 1.5: Polymer Tacticity</h2>
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<h3>Polymer Tacticity</h3>
<ul>
<ul>
<li>Polymers tend to form noncrystalline solids</li>
<li>However, depending upon the tacticity of the polymer, some degree of crystallization is possible</li>
<li>Tacticity describes the spatial arrangement of side groups \(R\) along the polymer chain</li>
<li>Isotactic-- all \(R\) groups are on the same side:</li>
</ul>
</ul>
<p style="padding-left: 60px;"><img width="400" src="/assets/courseware/v1/a8d0815d3458138fdad418281499573e/asset-v1:MITx+3.012S.1x+1T2019+type@asset+block/3012_Tutorial_Isotactic.jpg" alt="Polymer with all of the R groups on one side" /></p>
<ul>
<ul>
<li>Syndiotactic-- \(R\) groups alternate sides:</li>
</ul>
</ul>
<p style="padding-left: 60px;"><img height="187" width="400" src="/assets/courseware/v1/acf63f3613eed8bf28e6e91cc57b777d/asset-v1:MITx+3.012S.1x+1T2019+type@asset+block/3012_Tutorial_Syndiotactic.jpg" alt="Polymer with R groups alternating sides" /></p>
<ul>
<ul>
<li>Atactic --\(R\) groups are randomly positioned:</li>
</ul>
</ul>
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<ul>
<ul>
<li>Isotactic polymers are the most likely to exhibit crystalline behavior</li>
</ul>
</ul>
<p></p>
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