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Chapter 3
Bridging the Microscopic and Macroscopic in Thermodynamics with Molecular Dynamics Simulations: Lab Exercises for Undergraduate Physical Chemistry Matthew C. Zwier* Department of Chemistry, Drake University, Des Moines, Iowa 50311, United States *E-mail:
[email protected] A series of computational laboratory activities has been developed which bridges the gap between the macroscopic pictures of thermodynamics and kinetics and the microscopic picture of atoms, molecules, and their interactions. These activities, based on molecular dynamics (MD) simulations, are designed for an undergraduate thermodynamics and/or statistical mechanics course and require only freely available computer software and modest computing resources. Three activities, covering the behavior of ideal and real gases, the Boltzmann distribution and the meaning of thermodynamic β, and thermodynamic processes and phase changes are discussed in detail. In addition to providing visual reinforcement and accessible mathematical connections between molecular interactions and classical thermodynamics, these activities provide students with opportunities for curve fitting, numerical integration, and large-scale data analysis.
Introduction Physical chemistry, and in particular thermodynamics, is widely regarded among students as difficult to learn, particularly due to the abstract nature of © 2018 American Chemical Society Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
thermodynamic equations (1). The ability to visualize molecules and their interactions is known well among chemical educators as a strong predictor of (or critical prerequisite for) success in chemistry and related fields (2–4). Since chemical thermodynamics necessarily arises from the interactions of atoms and molecules with each other and their surroundings, it is perhaps natural to seek to use molecular visualizations to make thermodynamics somewhat less abstract. Because thermodynamics is concerned more with energy transfer than energy itself, a static picture of atoms and molecules is insufficient to explain thermodynamics; rather, the dynamics of atoms and molecules and how those dynamics are involved in the exchange of energy would seem to be necessary. The perceived pedagogical need for the visualization of dynamics of microscopic particles has been strikingly demonstrated by publications describing conceptually elegant machines for illustrating molecular motion. One paper published in 1972 details the design of a vibrating glass plate, suitable for displaying the dynamical behavior of ball bearings (representing particles, as in the kinetic molecular theory of gases) on a typical overhead projector (5). Another from 1995 describes the use of furnace blowers to bounce ping pong balls around in a Plexiglas box in another demonstration of the kinetic molecular theory (6). The subsequent explosive growth of computing technology — from personal computers to smartphones and tablets — has encouraged the development of computer-based molecular visualizations. These computer-based visualizations have proven pedagogical advantages (7), particularly with respect to students’ development of molecular (submicroscopic, particulate) reasoning about chemical and physical processes (8), though much work remains in characterizing how best to construct and use molecular visualizations (9). Molecular dynamics (MD) simulations are routinely used in chemical research to elucidate both qualitative and quantitative connections between submicroscopic dynamics and observed (bio)chemical behavior (10–12). The unique strength of MD simulations is their ability to model the behavior of individual atoms at atomic length (≥ Ångstrom) and time (fs – µs) time scales according to a realistic physical model. Though most typical MD simulations do not describe quantum phenomena or chemical change, they reproduce physical change and conformational dynamics with remarkable fidelity. (As might be expected, this fidelity typically improves as the system size and available thermal energy increases.) MD simulations use an interatomic potential energy function to determine the force on every atom simulated, and then this force is used to integrate Newton’s equations of motion — that is, to calculate the acceleration experienced by each atom at every point in time, and therefore the overall motion of each atom as a function of time. The potential energy functions typically used in MD emphasize both computational and conceptual simplicity, typically including terms for bond stretching, bond angle bending, bond rotation, electrostatic interactions, and van der Waals interactions (both attractive and repulsive). This potential function is parameterized to reproduce experimental data including observed bond lengths and angles, vibrational frequencies, or (in the case of biological molecules) prevalence of various types of secondary structures (e.g. helices or sheets in proteins). This means that the physics underlying MD simulations connects intuitively to the types of motions and interactions discussed in chemistry 34 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
courses typically taken prior to physical chemistry, such as intermolecular interactions as described in general chemistry courses or torsional motion and steric effects as described in organic chemistry courses. The output of an MD simulation is a time-resolved sequence of atomic positions based on these motions and interactions (a physically-realistic “movie”), along with detailed reports of energy-related state quantities (such as kinetic energy, potential energy, temperature, and pressure) for every point in time. Thus, students can immediately engage with MD simulations in a meaningful way by building upon prior knowledge of molecular structure and motion, but MD simulations also provide the data necessary to construct a bridge from individual atomic motions to the state of large collections of particles, which is the typical domain of thermodynamics. The entire suite of tools necessary for using MD simulations in physical chemistry education is now available to the physical chemistry teacher. A number of research-grade MD software packages are available free of cost over the Internet, including NAMD (13), GROMACS (14), and AMBER (15). High-performance visualization software is necessary for viewing the “movies” of MD simulations, and several packages (notably VMD (16)) are likewise available. All of these software packages run on multiple platforms, including MacOS, Windows, and Linux, and their documentation has advanced to the point that specialized knowledge of either MD simulations or scientific computing is no longer absolutely required to install these packages and run simulations. Further, the continuous advances in computer hardware power over the last decade have made MD simulations of a few hundred atoms trivial and tens of thousands of atoms accessible even on inexpensive or outdated computers. Finally, powerful, efficient, user-friendly, and well-documented scientific data analysis systems such as the Jupyter interface (17) to the Python (18) or R (19) computing ecosystems is likewise freely available, allowing for ready analysis of the large amounts of data typically produced by an MD simulation. This represents an opportunity to expand the use of MD simulations in chemical education, particularly in order to introduce and reinforce concepts simultaneously from an intuitive, visual perspective and from the theoretical, mathematical, and data-driven domain more traditionally associated with physical chemistry. The following describes a series of MD-based laboratory exercises used in the undergraduate physical chemistry course at Drake University, a small liberal arts institution. Students in this course have typically two semesters of calculus and two semesters of physics prior to enrollment. An overview of the activities is given in Table 1, and detailed descriptions follow. The activities described here have been developed using the GROMACS MD engine (14), VMD for visualization (16), and Excel or Python (at the instructor’s or student’s discretion) for analysis. In the author’s experience, students find these labs helpful in connecting the macroscopic, submicroscopic, and mathematical domains of chemistry (20) in a particularly effective and engaging way (see “Student Response” below). The files necessary to perform these simulations, demonstration analysis scripts, student handouts, and instructions for installing the necessary software are freely available at https://github.com/mczwier/md_in_pchem. 35 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
Table 1. Overview of MD activities
Common Features of the Simulation Exercises All of the simulation exercises described below are constructed around the same molecular system: 100 atoms of argon. This allows a fairly clean separation between understanding the technical aspects of running the MD simulations themselves (largely covered in the first exercise) from their subsequent use in explicating thermodynamics concepts. Each of the exercises described below is designed to be completed in a single four-hour laboratory period. Subsequent exercises build somewhat on previous exercises, but each could stand alone with minimal adaptation (which would largely be confined to removing instructions to reflect on new data in the light of conclusions reached in prior exercises). At the beginning of the sequence, a minimal introduction to the theory of molecular dynamics simulations is provided, limited to a discussion of the relationships among potential energy, force, and acceleration. Detailed, step-by-step instructions for running the simulations and extracting data for analysis are provided to the students so that the students may focus on the results of the simulations rather than how they are run. The exercises themselves are designed so that students cycle through running simulations, examining preliminary data, and interpreting preliminary results. In-depth analysis of simulation data and building connections to thermodynamics occurs toward the end of each exercise. Reflecting the importance of mathematical modeling and numerical analysis in physical chemistry, this latter analysis is integral to the structure of each exercise, and is intended to be completed in the same lab period in which simulations are performed. Students submit their work to the instructor as concise narrative reports, emphasizing the presentation and interpretation of data and connections to thermodynamics over the somewhat more traditional scientific manuscript-like format sometimes used in upper-level laboratory courses. 36 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
Software and Hardware Considerations As noted above, all of the software necessary for running and analyzing MD simulations is freely available both to individuals and to institutions. The instructions for students to gain access to the MD software will necessarily vary from institution to institution. The primary difficulty in using MD simulation in education is in making the software available to students. In the case of the author, the simulations are performed using remote desktop software connecting to a research computing cluster, but the necessary MD, visualization, and analysis software could easily be installed on individual laboratory or student-owned computers. Instructions for installing the software required for these activities on personal computers are available at https://github.com/mczwier/md_in_pchem.
The First Activity: Microscopic Dynamics of Ideal and Real Gases The first exercise introduces MD simulations and provides both concrete visualization and mathematical analysis of the differences in behavior between ideal and real gases. Students first perform constant-volume, constant-energy MD simulations at temperatures of 100 K, 200 K, and 300 K on a system of 100 ideal gas particles each with the mass of argon. Ideal gas behavior is obtained by disabling the van der Waals interactions that would otherwise result in physically realistic behavior of the argon particles. The simulations are 5 ns long and atomic positions are stored every 0.1 ps. Students view these trajectories using the VMD visualization program (Figure 1), and are asked to comment on the directions and speeds of particle travel. This provides immediate visual reinforcement that at equilibrium, particles travel isotropically (in any direction with equal probability), and that particles do not travel with equal speeds (a common misconception that develops when students are first presented with the relationship between temperature and average kinetic energy). Students are further asked if particles appear to be interacting in any way; they are not, but students frequently answer incorrectly until presented with the contrasting behavior of real gases. Finally, students observe that particles travel faster, on average, at higher temperature. Students then examine the numerical output of these ideal gas MD simulations, which include average energies and temperature (Figure 2). For each of the three temperatures, students observe that the average potential energy is identically zero, which follows from the ideal gas behavior of the particles. Students then verify that the average kinetic energy áEKñ = (3/2)RT as is expected from the kinetic molecular theory. Finally, given the volume V of the simulation and using the average pressure and temperature áPñ and áTñ respectively, students verify the ideal gas law áPñV = nRáTñ. (Here, averages are over the instantaneous values of pressure and temperature reported by the MD software. These values are calculated from the instantaneous velocities and forces acting upon atoms in the MD simulation, and the values of pressure and temperature reported by the MD simulation are related to the thermodynamic temperature and pressure 37 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
only in the average.) Upon these verifications, the instructor may wish to note that this suggests (though does not prove) that the simulations just performed are physically reasonable.
Figure 1. Visualization of an ideal gas of 100 argon-like atoms from an MD simulation. (see color insert)
Figure 2. Example output from a GROMACS simulation of 100 ideal argon-like gas particles. The total kinetic energy is commensurate with the temperature, and potential energy is entirely absent.
38 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
Students then go on to perform simulations of real argon particles at temperatures of 100 K, 200 K, and 300 K. Upon viewing the simulation trajectories in VMD, students immediately note that argon atoms scatter off each other routinely and also frequently dimerize. For some students, this is the first firm indication that noble gas particles can interact at all, addressing a misconception lingering from general chemistry conflating chemical and physical interactions of particles. Dimers can also be observed to vibrate in internuclear distance, introducing the concept of noncovalent, non-ionic chemical bonding. The instructor may wish to point out that dimerization is readily visible, but trimerization is rare or absent; this would be a useful scaffolding point in a discussion of the relative importance of terms in a virial expansion for the pressure of a real gas, or of the nearly universal absence of termolecular elementary reaction steps in kinetics. Students finally verify that áEKñ = (3/2)RT and áPñV ≈ nRT, tending from approximate equality toward true equality as temperature increases; this reinforces the independence of the kinetic molecular theory from the nature of the gas (ideal vs. real) under consideration and that gases do indeed behave more ideally at higher temperatures. The instructor may wish to point out that one part of this tendency toward ideal behavior as temperature increases is because the gas particles are moving too fast for intermolecular attractions to deflect their paths substantially in near-scattering events, which are clearly visible in the simulation trajectories (“movies”). Finally, students undertake a detailed analysis of the interparticle distance to connect the above largely qualitative observations to the detailed quantitative information provided by MD simulations (namely that every atomic position is available in sub-Ångstrom spatial and subnanosecond temporal resolution). The analysis tools packaged with MD engines can calculate the radial distribution function g(r), typically with only one command and producing g(r) in a format readily imported into analysis tools. The radial distribution function is defined as the probability pobserved(r) of observing a given interparticle distance r relative to observing that interparticle distance in an ideal gas (21):
Students plot g(r) for all six of their simulations in a plotting tool of their choice (typically Microsoft Excel). They then observe that g(r) = 1 for the ideal gas simulations and further that g(r) → 1 as interparticle separation increases in the real gas simulations, corresponding to ideal gas behavior at low densities. There is a maximum in g(r) at an internuclear separation of about 3.8 Å, corresponding to the equilibrium non-covalent bond length in the argon dimer. The height of this maximum in g(r) decreases as temperature increases, indicating a decreased probability of dimerization at higher temperatures. The radial distribution function g(r) is related to the interparticle potential energy (pair potential) V(r) according to (21)
39 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
At large interparticle separations, it is expected that V(r) = −C/r6, and the constant C is related to the parameter a in the van der Waals equation of state according to
In Equation (3), σ is the interparticle distance where V(r) = 0 (that is, where V(r) crosses the r axis as the potential goes from being repulsive to attractive) and NA is Avogadro’s number. Similarly, the constant b in the van der Waals equation is similarly related to the pair potential according to (21)
Students construct V(r) from g(r) in an analysis tool of their choice, then perform a linear least-squares fit of V(r) vs. 1/r6 to extract the constant C in Equation (3), ideally with an error estimate (see Figure 3). Students obtain σ directly from their plot of V(r). From these values, van der Waals constants for argon can be calculated from the simulation data. Propagation of error is used in conjunction with Equations (3) and (4) to calculate error bounds on a and b in terms of uncertainties in C and σ. The values for a and b compare favorably with reference values (see Table 2). The difference between MD and experimental results is likely a combination of two factors: (i) the nonbonding interaction potential used in MD (the Lennard-Jones potential) is highly approximate in the repulsive region of the pair potential; and (ii) Equations 3 and 4 are based on a hard-sphere model (as is the van der Waals equation itself), and the MD simulation does not treat atoms as hard spheres. Thus, the difference between MD and experimental values for the van der Waals constant can be used to drive a discussion about the merits and perils of approximation in physical modeling.
The Second Activity: The Meaning of Thermodynamic β The constant β in the Boltzmann probability distribution
is usually identified as β = 1/kBT (or β = 1/RT if considering molar energy) by comparison between equations of statistical mechanics and classical macroscopic thermodynamics. Particularly if emphasizing energy and entropy on the submicroscopic level of atoms and molecules in a physical chemistry course, it seems necessary to introduce the definition of β prior to having developed the physical and mathematical machinery necessary to fully justify it in terms of classical thermodynamic logic. As an alternative, MD simulations can be used to provide a data-driven justification of this relationship. Having established from the previous exercise that MD simulations are a reasonable physical model for argon, the probability distribution for velocity of (real-gas) argon particles from the MD simulation can be transformed into a Boltzmann distribution while simultaneously justifying that β = 1/RT. 40 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
Figure 3. Fit of the long-distance tail of the pair potential V(r) in an MD simulation of 100 argon atoms at 100 K. From this fit, students obtain estimated values for the van der Waals a and b parameters for argon gas.
Table 2. Van der Waals constants for argon from an MD simulation Van der Waals Parameter a (L2 bar mol−2) b (L mol−1) a
Typical MD Result
Experimental Valuea
1.73 ± 0.02
1.3483
0.0500 ± 0.0009
0.031830
From Reference (21)
Importantly a constant-energy MD simulation is equivalent to a perfectly isolated system, so no contact with a (virtual) heat bath (as in a constant-temperature MD simulation) is required for maintaining the system temperature. Such heat baths (more technically, thermostat algorithms) use the Boltzmann distribution to adjust the velocities of particles in the simulation in order to adjust the system temperature. In the constant energy system considered in this laboratory exercise, the Boltzmann distribution is not built into the simulation in any way, but rather arises from the exchange of energy among argon atoms according to a realistic physical potential. In this way, a constant energy MD simulation is a readily accessible “experiment” that allows the observation of the Boltzmann distribution, rather than the assumption (or enforcement) of behavior governed by the Boltzmann distribution. For mathematical simplicity, the velocity distribution in only one dimension (for example, the x-direction) is considered. The x-velocity distribution is readily available from completed MD simulations using tools packaged with the MD engine. The analysis necessary for this exercise can be performed in 41 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
a typical spreadsheet application by suitable data transformations followed by linear least-squares analysis, or by nonlinear least-squares fitting as provided in a numerical analysis system like Python (with Numpy and Scipy), Matlab (or its free workalike Octave), or R. This exercise is typically performed in an inquiry-based manner. With the instructor’s guidance, the x-velocity distribution is extracted from a constant-energy simulation of (non-ideal) argon gas at 100 K, such as that performed in the previously described exercise. Students plot the distribution and observe that the resulting curve appears Gaussian, that is, a curve of the form exp(−x2). Based on this observation, students are guided toward selecting an initial functional form for the x-velocity probability distribution f(vx)dvx of
[For clarity of discussion, the differential volume element dvx, after explaining its meaning, is then dropped from the discussion until students arrive at a final form for f(vx)]. Students obtain values and error estimates for the curve height A and width parameter b through a least-squares fit, determine their units by dimensional analysis, and finally plot the resulting curve along with their observed x-velocity distribution data, obtaining a remarkably good fit. At this point, students are reminded of the form (5) of the Boltzmann distribution, and asked to recall the relationship between kinetic energy and velocity —EK = (1/2)mv2; taken together, in the students are guided to explain why this suggests that the presence of exponent of Equation (6) be rewritten to account for particle mass:
Either per-particle or molar mass may be used for m, corresponding to students arriving at β = 1/kBT or β = 1/RT at the end of the exercise, respectively. (Molar mass has typically been used in this exercise, and mass in kilograms per mole is assumed in the following discussion.) A value and associated error estimate is obtained for the constant c, and students conclude from dimensional analysis that c must have units of inverse molar energy. Students are then asked to investigate how the constant c in Equation (7) varies with simulation temperature. Recapitulating the analysis described above on MD simulations performed at different temperatures, students discover that doubling the simulation temperature from 100 K to 200 K roughly halves the value of c, and tripling the simulation temperature from 100 K to 300 K causes c to decrease to roughly one third of its value at 100 K. Students are asked to account for this by altering the form of Equation (7), eventually arriving at
Students use curve fitting (Figure 4) to obtain a value for the constant g at all three temperatures (100 K, 200 K, and 300 K), along with error estimates, and observe that values for g are equal within error at all three temperatures. Students further note that g bears units of mol·K/J. Upon being asked if they have seen that 42 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
combination of units before, students observe that the units of g are the inverse of the units of the R = 8.314 J/mol·K universal gas constant . Calculating R = 1/g and associated error, students obtain the expected value for the universal gas constant.
Figure 4. Fit of the x-velocities from an MD simulation of argon to a Gaussian function, demonstrating that exchange of energy due to a realistic physical potential results in the Boltzmann distribution for energy. Students thus arrive at the conclusion that the x-velocity distributions for 5-ns MD simulations of 100 argon atoms are given by 43 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
Translational energy in the x direction indeed obeys the Boltzmann distribution, and β = 1/RT. A straightforward discussion of the isotropic distribution of particle travel direction at equilibrium leads immediately to the conclusion that translational energy as a whole obeys the Boltzmann distribution. Though it is not shown in this exercise that other types of energy commonly discussed in physical chemistry (including vibrational, rotational, and electronic degrees of freedom) also follow the Boltzmann distribution, students have nonetheless observed that the Boltzmann distribution arises naturally in an intuitively accessible context.
The Third Activity: Thermodynamic Processes and Phase Changes The final exercise described here uses MD simulations to provide a concrete intuitive understanding of thermodynamic processes, phase changes, and the differences between physical states. The now-familiar system of 100 argon atoms is compressed under 10 bar of pressure and cooled from 140 K to 85 K over three nanoseconds. This causes condensation, and the liquid state is then simulated for two more nanoseconds. The resulting simulation data is used for qualitative description of thermodynamic processes and a phase change, quantitative examination of state variables throughout the compression and condensation process, and finally to calculate the enthalpy of vaporization of argon. After performing the simulation, students view the simulation trajectory in VMD and record qualitative observations about the compression process, the condensation process, and the atoms in the liquid phase. Students have audibly gasped when condensation occurs; it is one thing to define a phase transition as a discontinuity in a thermodynamic state variable, but quite another to see a thousand-fold decrease in volume occur nearly instantaneously. Students then use VMD to highlight a single argon atom in the liquid state and note the difference between the largely inertial motion of the gas phase and the largely diffusive motion of the liquid phase. Students go on to extract and plot the total energy, potential energy, volume, density, and temperature of the simulation as functions of time. From these plots, students identify stages of the thermodynamic process simulated here: compression (a rapid decrease in volume and potential energy), cooling (a slower decrease in volume and potential energy), condensation (an extremely rapid decrease in volume and potential energy), and liquid dynamics. The conditions simulated are very close to the liquid/solid phase boundary for argon, and occasionally a solid forms for roughly 100 ps. When this occurs, students invariably ask about transitory decreases in volume or potential energy in the liquid state, and are directed to view the simulation trajectory in VMD, wherein they observe the vibrating lattice characteristic of the solid state. Students use pressure and temperature from their trajectories to calculate the density of argon before and after the phase transition using the ideal gas law and the van der Waals or Redlich-Kwong equations of state. They compare these densities 44 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
to the density at corresponding points in their simulations, demonstrating the sometimes-overlooked fact that even a cubic equation of state will fail to predict the physical properties of a liquid. By construction, this problem requires using the Newton-Raphson method or fixed-point iteration to solve a cubic equation, reinforcing that for all the analytical rigor and elegance of thermodynamics, sometimes a brute-force numerical approach is the most effective way to solve a problem in physical chemistry. Finally, students use their plots of total energy and volume as a function of simulation time to calculate the enthalpy of vaporization of argon. Because the MD simulation reports internal energy and volume at every time point, by identifying the values these quantities have before and after the phase transition, students can calculate ΔHvap = ΔUvap + PΔVvap directly from their simulation data. This requires students to solve the subtle problem of identifying when the phase transition begins and ends in order to calculate change in internal energy and volume across the phase transition. Though the phase transition is remarkable in how quickly state variables change, at the particulate scale the phase transition is not truly discontinuous; in particular, the volume changes much more slowly than internal energy as the phase transition begins. Students are free to solve this problem however they see fit, but the most accurate results are obtained by identifying the phase transition time as the moment when the time derivative of internal energy — calculated by finite differences — attains its maximum magnitude, then performing linear extrapolation of energy and volume forwards in time from before the phase transition and backwards in time from after the phase transition to the calculated phase transition time (Figure 5). Choosing different time periods upon which to base the fits used to perform these extrapolations allows the estimation of error in the obtained values of ΔU and ΔV. An appropriate solution to the problem of accurately calculating ΔU and ΔV results in a calculated ΔH (6.41 ± 0.12 kJ/mol) in excellent agreement with experiment (6.53 kJ/mol (22)).
Student Response Student responses to these lab activities have been overwhelmingly positive. No experiment has been conducted to isolate the pedagogical effectiveness of these laboratories, but several students (the majority of whom have proceeded to graduate studies) have shared the impact these activities have had on their chemical education: • •
•
[The MD simulations] revolutionized my understanding of chemistry simply by allowing me to visualize it in 3D and time. While I love the visual element of these simulations, I sometimes feel like people only appreciate simulations for their visual elements, when really computation can provide a lot of quantitative theoretical insight into a system. The MD simulations that we used during physical chemistry were great because they bridged the gap between theory and practical application. It 45 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
•
is so easy to get caught up in the math, but the computational work we did helped me connect the numbers to what they were saying. For example, when we learning about ideal gases, we were studying argon-argon interactions. We were able to calculate the energy of the interactions and supplement that with simulations at different temperatures. By doing this, I could see that increasing the temperature would increase the energy. The atoms moved faster, which makes sense, but just seeing the math wouldn’t always make it as apparent. The simulations were exactly the supplement I needed for the lecture. They allowed me to see the interactions or actions of molecules we had talked about in class like the argon gas molecules as real versus ideal. It was also helpful to use data from the simulation and analyze it with mathematics like we did for the translation energy and probability lab. I could see the molecules moving on the screen, make observations and then see the plotted data. It made the chemistry concepts more tangible, so when we got to fitting a Gaussian curve and talking about the mathematics, I didn’t feel lost in it. It made the mathematics easier to learn, while still focused on chemistry. Overall, it was a tool I had never used before and was a good bridge to more mathematics than what I had seen in chemistry before. What I learned helped me in my other chemistry classes as well; being able to visualize what molecules actually look like when moving and how they interact with each other.
Figure 5. Identifying ΔU for the condensation of 100 argon atoms from extrapolations (dotted lines) of the total energy in an MD simulation. Here, ΔU = 6.145 ± 0.006 kJ/mol. The error estimate is calculated by evaluating ΔU at the beginning, middle, and end of the phase transition (3107.0 ps, 3132.5 ps, and 3158.0 ps, respectively, for this particular simulation). 46 Teague and Gardner; Engaging Students in Physical Chemistry ACS Symposium Series; American Chemical Society: Washington, DC, 2018.
Conclusions A series of molecular dynamics laboratory exercises for have been constructed for use in undergraduate physical chemistry education. Each of these activities (summarized in Table 1) is designed to take place in a single laboratory period using freely available software, and emphasizes connections among qualitative observations of the submicroscopic behavior of matter, quantitative measures of the behavior of the simulated submicroscopic system (e.g. radial distribution functions, velocity distributions, and internal energy), and macroscopic thermodynamic behavior. Students found these activities uniquely engaging, and have commented on their usefulness in helping them to develop mental models of the behavior of matter in all three domains of chemistry (submicroscopic, macroscopic, and symbolic). The files and detailed instructions necessary to set up and run these simulations are available at https://github.com/mczwier/md_in_pchem. MD simulations like those considered here provide an accessible and effective bridge between the microscopic and macroscopic in the teaching of thermodynamics.
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