Chemistry on-line Conference, 1997. Paper 6.

Nature doesn't solve equations, so why should we?

Mathematically-lean simulations in chemistry


Hugh M. Cartwright

Physical and Theoretical Chemistry Laboratory, Oxford University

South Parks Road, Oxford, England OX1 3QZ

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1 Introduction

2 The place of simulation in the undergraduate course

3 Categories of simulation

4 The advantages of simulation

5 Advantages of mathematically-lean simulation

6 Comment

1. Introduction

2. The place of simulation in the undergraduate course

3. Categories of Simulation

Fig. 1. The spectrum calculated by a "Black Box" NMR instrument. This display is created by a Java applet [1]. Full size version of this figure.

Fig. 2. Data from an on-line optical rig, gathered remotely and displayed using Java applets [2]. Full size version of this figure.

3.2 Simulation of equations.

3.3 Visualisation of idealised behaviour

3.4 Simulation at the molecular level

4. Advantages of simulations.

Interactivity is also a powerful argument in favour of simulation. Users quickly become bored if they are just passive consumers of information. They learn better when they need to respond frequently to, and have control over computer software.

It should be appreciated, though, that interactivity carries with it some dangers. A conventional program which channels all users along a pre-defined track depends on that track being chosen with care, and on it being suitable for almost every user. Interactive software by contrast presents the user with many facilities and numerous choices. This flexibility may remove from the user any sense of direction, so that they become lost within the software package. The academic objectives of the work may become unclear, and the user may wander inefficiently through attractive but meaningless exercises. Interactive software must therefore be sufficiently structured that users do not lose sight of the educational target.

The use of html wrappers and embedded Java applets can provide this essential structure, though there are of course other equally effective ways of ensuring that users are not submerged in a complex, multi-faceted simulation.

4.2 Flexibility in topic ordering

    In any set of computer-mediated exercises the instructor must decide upon the most suitable order in which topics are to be presented. In mathematically-based or rule-based exercises a progression from mathematical simplicity to sophistication may be used.

    Exercises might progress from point masses to masses of finite volume and eventually to molecules. This may be a justifiable course, but there is no reason why an order determined by an increase in mathematical complexity should coincide with the order which is academically most desirable. It might instead be desirable to present a comprehensive simulation initially, and gradually uncover its mathematical basis as the underlying science is studied.

4.3 Development of scientific intuition

    Other, more controversial, arguments also exist in favour of simulation.

    Women are under-represented in most areas of science, particularly in the physical sciences. For many people who teach at school, college or university, this is a cause of concern, and much effort has been devoted to searching for ways to increase the number of women in science.

    The limited number of female science teachers and professors, and the rarity of female scientists on the television may have an effect, but there are probably other factors at work. Anecdotal evidence - drawn from observation of a group of students at Oxford - suggests that female students are often harder working than male students, but may have less "flair" for science at the university level.

    By contrast, their male compatriots sometimes seem to have a greater sense of scientific "intuition". What does this mean? Suppose that, during a walk in the country or on a beach, we come across a stream blocked by a child's dam. We can assess whether it is a "good" dam (in the sense of being strong enough to hold the water back) without considering any hydrostatic or hydrodynamic equations. Indeed, the problem of whether a particular sticks, stones and mud dam is "good" would present a testing computational problem.

    But how then do we assess the quality of the dam, if not by using equations? Presumably, through having built dams - or having seen them built - and in so doing, having gained practical experience of what a "good" dam looks like. We have accumulated a fund of knowledge which is ill-defined, incomplete and fuzzy, yet still useful scientifically. This is not "intuitive" knowledge in the normal sense, but a collection of experience which gives us a feel for how things behave.

    Those students who possess this sort of scientific intuition have built this up from childhood, perhaps through building dams, throwing stones and falling off trees, but also - crucially - through the challenge provided by science teachers whose approach is first to get their students to understand the physical principles of science, and only later to learn the rules, facts and equations.

    Scientific intuition may help when students have to evolve appropriate ways to tackle a novel problem, or rationalise a newly-discovered equation, trying to understand why it has some particular form. Inasmuch as simulations allow the user to experiment and interact with the material world, they will stimulate the development of this intuition. Simulations thus teach not only a single topic, but provide also broader scientific experience.

5. Advantages of mathematically-lean simulations

Fig. 8. A frictionless hinged beam falling under gravity. [6] Full size version of this figure.

5.3 Avoidance of mathematical complexity

    A strong mathematical background is an asset to an honours chemist. Nevertheless, many students find the language of mathematics difficult or obscure. Simulations provide a means by which the mathematics can be made less intimidating and the chemistry more palatable and understandable.

    Students can, for example, learn qualitatively about the relationships between the temperature of an assembly of molecules and the vapour pressure without first (or ever!) meeting the Clausius-Clapeyron equation. They can develop an understanding of heat capacity, or colligative properties, from simulations such as that shown in Fig 11.

The Clausius-Clapeyron equation expresses the dependence of vapour pressure of a volatile solid or liquid on temperature. It contains differentials (or logs in the integrated form). It is an important equation in thermodynamics, but though it is not particularly complex, it still presents students with difficulties, and they may not appreciate that vapour pressure rises approximately exponentially with temperature. A simulation of a collection of molecules, assuming a simple interaction potential, will allow students quickly to investigate the dependence of vapour pressure on temperature.

The experimental behaviour is not "built-into" the simulation - indeed thermodynamic properties are defined only for bulk systems, and the simulation is of individual molecules. Nevertheless, with a sufficiently large number of molecules, the user can measure vapour pressure and investigate P/T relationships.

In a two phase (gas/solid) system, one can illustrate the principles of Langmuir or BET behaviour using again simple interaction potentials. (Fig 12).

Fig. 12 Simulation of argon above a charcoal surface. The movement of the argon atoms is illustrated by the tracks. [5]. Full size version of this figure.

This leads us to a further point of crucial importance:- molecular simulations also simulate what we do not see.

This sounds like nonsense. What can it mean? Although the primary role of the simulation shown in Fig 12 might be to allow students to study adsorption on solids, further physical phenomena may be investigated using the same model. For example, the molecular tracks in Fig 12 hint that the molecules adsorbed on the surface are not actually stationary, but move around a little. This behaviour is not "what we are looking for" if we are using the simulation to study Langmuir adsorption; indeed, this movement across the surface could not be observed if the simulation was based solely on the Langmuir equation. In this sense then, the molecular level simulation may include within it behaviour we are not looking for (and may not even notice). The sudden discovery by a student of quite unexpected phenomena, such as movement of adsorbed molecules, is both a stimulant to learning, and often a source of delight to the student.

5.4 Insight into the physical origin of equations

5.5 Revelation of complex behaviour from simple equations
    It is clear that complex behaviour can often result from simple equations. The growth of fractals in solution, for example, can be simulated using a random walk model in which ions in solution are attracted to, and deposited on an electrode.

    With the simplest of equations, quite complex behaviour can be modelled. Fig 14 shows the symmetrical fractal obtained when a centro-symmetric field is provided by an electrode in a dilute solution.

Fig. 14. Fractal growth resulting from a small centro-symmetric field applied to an electrode in a dilute solution of copper ions [7]. Full size version of this figure.

Fig. 15. Fractal growth by deposition of ions released from a point source. Full size version of this figure.

Fig. 16. A fractal growth generated by an electrode which provides a field aligned along the x-y axes. Full size version of this figure.

6 Comment

Appendix A: The CoLoS group

References and background information.

1. Taken from an experiment in NMR spectroscopy under development in the Physical and Theoretical Chemistry Laboratory at Oxford University.

2. Taken from an on-line experiment in error analysis under development in the Physical and Theoretical Chemistry Laboratory at Oxford University. Data are generated by connecting to an optical rig through the Internet.

3. Data from Computer simulation of the Belousov-Zhabotinsky reaction, Chi Ho Lam, Chemistry Part II thesis, Oxford, 1996

4. A java applet showing a vibrating molecule:

5. Screen shot from a simulation written using X, C and Motif. Pete Bennett, Mathematical modelling and computer simulation of aspects of surface science, Chemistry Part II thesis, Oxford University, date.

6. An experiment under the control of the CoLoS program xyzet, developed at Kiel University, Germany.

7. Screen shot from an experimental CoLoS program on fractal growth, developed at Oxford University.

8. A screen shot from a CoLoS demonstration program on the thermodynamics of simple liquids. Andy Armstrong, Computational modelling and simulation of molecular phase dynamics. Chemistry Part II thesis, Oxford University, June 1995.

9. A screen shot for a simple simulation of gas-phase molecules.

10. The virtual lab in Oxford is at

11. An interactive computer simulation of collisional potential surfaces, Russell Strevens, Chemistry Part II thesis, Oxford University, England, 1995