ANALYTICAL APPLICATIONS OF MONTE CARLO TECHNIQUES Oscar A. Giiell and James A. Hdcombe Department of Chemistry Universily of Texas at Austin Austin. TX 78712
The physical sciences deal with discrete events. The properties of a system as a whole are determined by the average behavior of its componentsgalaxies, molecules, or quarks. Most of the chemist’s world can be described by discrete random (stochastic) events between atoms, molecules, and photons. Sampling procedures, molecular diffusion, absorption/emission of radiation, and adsorption/desorption processes in a chromatographic column are just a few examples of everyday events that involve random processes ( I ) , albeit with a known statistical probability that enables one to measure a related variable (e.g., intensity, gas flow). Monte Carlo techniques study chemical systems by looking at the random character inherent to many chemical and physical processes, and they often present a much more accurate picture of nature with a much simpler mathematical approach than is realized with more conventional theories. This REPORT deals with analytical applications of the theory of random processes, in particular with numerical solutions obtained by Monte Carlo 0003-2700/90/0362-529A/$02SO/O
@ 1990 American Chemical Society
techniques. There is no unambiguous definition of “Monte Carlo methods” or “Monte Carlo simulations,” but it is often assumed that the approach applies to any use of random numbers. This is very nearly correct, because Monte Carlo methods comprise that branch of experimental mathematics that is concerned with experiments on random numbers (2). Experimental mathematics has had a long history. The term Monte Carlo method was introduced in 1949 (3)and, as the name implies, refers to random numbers--roll of the dice, spin of the
-
wheel, and so forth. In a chemical context, Monte Carlo can be referred to as the study of an object of interest by matching random numbers, corresponding to a sample of its elements, with probabilities derived from its physical and chemical macroscopic properties. Table I contains a simplified classification of Monte Carlo tecbniques in analytical chemistry. We will attempt to recognize the different modes of Monte Carlo approaches presented in the literature and also to unify them. These examples and references have been selected (randomly in some cases) from the large number
available. They typify the many overlapping modes of Monte Carlo techniques found in the chemical literature. Monte Carlo approaches offer simple and straightforward applications and make much information available, but care must be taken to avoid correlations between sequences of pseudorandom numbers. The stochastic simulation is most useful for complex situations and probably not well suited to simple cases. In general, the techniques present poor precision but good accuracy for a number of samples obtained from different randomization seeds. The expected variance decreases in inverse proportion to the number of trials. In spite of their general applicabiity, Monte Carlo approaches have been criticized (relative to other numerical solutions) as having slow convergence and as being suitable only in cases where reduced accuracy is acceptable (4, 5). However, the increasing availability of faster microcomputers and supercomputer networks is providing the necessary means to neutralize these concerns and satisfy requirements for high accuracy. Computers have become an integral part of the analyst’s toolbox, with options ranging from micro- and minicomputers to parallel processors and supercomputers. The latter, which are accessible by network from several National Science Foundation centers (S),
ANALYTiCAL CHEMISTRY, VOL. 62, NO. 9, MAY 1. 1990
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REPORT
Table 1.
Monte Carlo technkpes In analytical chemistry Monte Carl0 taChn1qw
Comm
sanpiilbg (random &ice) Renl-aianal emuiation (noise seneration)
Simple use of randomnumbers. Simple use of random numbers. Used also for statistical Studies.
Cimpleuseofrandomnumbers. Used ai80 for signaishape simulation and expllcll simulation. impie use of ranm numbers. Used also fw statistical studies end explicit
simulation. Annealing (freesnergyminimlzatlon
sed for signal-shape
simulation, explicit simuiatlon, and optimization and experlmatai design.
Experiment (explicit simulation)
ves the most infwmatlon.
placing complex experiments, with cheaper and less time-consuming computer simulations based on accurate models. The simulation of chemical and physical processes enables optimization of analytical methods and instrumentation and allows a better understanding of the mechanisms underlying the technique. Why should one bother with random numbers if one can use numerical solutions? Are Monte Carlo techniques difficult to apply? Interestingly, the use of stochastic methods can be more accurate and precise than other techniques. Many researchers who use Fourier transform algorithms or even just take an average without fear of the statistical errors underlying the approach disregard the use of random determinations because of their appearance of uncertainty. In general, Monte Carlo applications are computationally intensive, but the computer code is significantly simpler than that needed for other numerical approaches. (The computational systems used in this REPORT include a laboratory-based Macintosh SE microcomputer and a Cray X-MP124 supercomputer located at the University of Texas Center for High Performance Computing.) Determination of ff An example of the application of Monte Carlo techniques is the determination of T by using a random target (e.&, a dart board) consisting of a circle inscribed in a square with unit area. The value of u is obtained choosing random points ( x , y coordinates: R,,, R,) on the board and using the ratio of points falling inside the circle ( N C i dto those falling within the square
4 Flaure 1. Stochastic determination of f f with a microcomouter.
offer the highest speed, largest memory, and best precision. Currently, supercomputers are defined as computers with 20-loo0 million floating-point operations per second (MFLOPS); designa emphasizing extremely rapid processing of vectors (arrays); exceptionally large, rapidly accessible, active memory; and high cost. Although most stochastic applications can be accommodated by smaller computational 5SOA
* ANALYTICAL CHEMISTRY, VOL.
systems such as microeomputera (4, a, certain simulations require the speed and architecture of a supercomputer. Because of the general iterative construct of the computer code for Monte Carlo-based simulations, the arrayvectorizing ability of the supercomputer makes it “the machine of choice” in these casea. Several companies (e.g., Du Pant [6]) have saved millions of dollars by re62, NO. 9, MAY 1. 1990
-- 4.0 T
= 4.0-
N&&
(1) (2)
N,,, where Neircle represents all points within the circular area Acircie,satisfying the inequality: (3) The computer code is very simple (see box), and the results of several trials can produce a good approximation to the real value, as can be seen in Figure 1. This figure presents the mean and one standard deviation of 10 determinations, each with a different initialization “seed” for the random generator, and represents a random walk toward the true value of u with a variance in-
versely proportional to the number of trials (statistical “noise.” Z 0: N-I) as predicted by probability theory (7,8). A Monte Carlo approach offers pqor precision but good accuracy on the auerage of several determinations, similar to optical polarimetry. One would expect that the estimated value approaches the true value of T as the sample size goes to -; however, computer uncertainty and round-off error make random series start over again for a very large sample (e.g., lo9 for a microcomputer or 10’4 for a supercomputer). Figure 2 shows that for large samples the average of 10samples behaves as a random walk about the true value of T with a monotonic reduction of standard deviation. Analytical applkatkm
Monte Carlo sampling (random choice). The simplest intuitive application of random numbers is for random sampling (e.g., random choice from a production batch for quality control). The variance of the population is immediately available from random samples. Systematic sampling (nonrandom) is simpler and can give similar precision in the absence of unsuspected periodicity. However, no trustworthy method is known for estimating the variance of the population from nonrandom samples (9,lO).If the sample is to be used to deduce the properties of the population, it must be random (11). Thus, all the members of the population should have an equal (or well-known) opportunity of beiig in a sample. Stratified random sampling can be used to reduce sampling bias among different batches (9,lZ).Several 88mples may be taken from each batch, and then some of them are randomly chosen for analysis. The application of variable sampling frequency may also be significant considering, for example, the sampling of animals in a forest. Owls are nocturnal and do not have the same chance of being in a sample wllected during the daylight. Other discussions of sampling theory can be found in the literature (e.g., 9, IO). Monte Carlo real-signal emnlation (noise generation). Another elementary stochastic application is the generation of noise to study the characteristics of real signals. The noise is added to a signal produced by a referencefunction (e.g., Beer’slaw). Asignal produced by a random distribution also can he generated. White noise is produced with Gaussian random numbers, and other sources of noise can be produced as described in the box (p. 532 A) in accordance with the distribution expected (e.g., llf noise). Noisy
signals generated by Monte Carlo simulations can be used to study correlations among different sources and to solve complex systems such as multicomponent “matrix correction’’ equations (13).In other examples, simulated data with different levels of superimposed noise have been used to evaluate curve-fitting algorithms (14, 15) and to study error propagation in analytical procedures (16). Confidence intervals for a set of previously estimated parameters (e.g., by least squares or x2 minimization) can be estimated by Monte Carlo statistical studies (5). The fitted parameters are used to generate several hypothetical data seta. The same intervals and variance of the experimental data set employed in modeling are ysed to simulate each signal. After several possible seta have been generated. statistical
criteria can be applied to estimate the uncertainty in the parameters. Monte Carlo integration. One might apply an integration method similar to the Monte Carlo determination of T to obtain the area under any function f ( x ) just by counting the fraction of random values that falls under the curve and multiplying it by the range (5). However, a more efficient method is desirable. Since the integral is proportional to the average value of the function over the range a 5 x 5 b
r(x) =
1.6
fwdx
(4’
the average of f(x) from N random samples of x can be used to approximate I ( x ) . Using Equation ii (p. 532 A) to obtain a uniform random x(i) between a and b
Fbure 2. Stochastic determination of T with a supercomputer. Tlm avaage and one standard deviation 01 10 sarc.1~at i n m l s ot 20,000 Irlals each are shown. The value determined l a 10’ irlals Is T = 3.1416 i 0.0003(0.01% RSD). An arbnrw 6tBndard deviation (r i 1.63/N“2) Is also pesentedas me p i r of m l h symmsblCal C W ~ L
huKtlon PiMC (Ntotal)
radius2 = 0.25 Ncircle = 0 do 1000 i = 1. NtOtal x = rendom - 0.5 y = random 0.5
..((x’x + y’y)
-
(bop of trials1 ([-0.5,0.5] deviates)
.le. radlus2) Nclrcle = Ncircle -k 1 {point inside circk
PiMC = 4.0’NcirclelNtotal end
ANALYTICAL CHEMISTRY, VOL. 62, NO. 9, MAY 1, 1990
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REPORT
-.
\’
where Ax = b a. This integration technique is easier to apply than conventional numerical integration methods, although it converges slowly (u u N-”2). Thismethod is not recommended for simple functions, but it is very efficient for the evaluation of multidimensional integrals (7) because u is nearly independent of the number of dimensions of the function. Many problems in chemistry and physies involve averaging over several variables (e.g., position, velocity, charge, freeenergy). For afunction Fof D dimensions, one c h m e s N random points (from N*D random numbers) and sums F over them using an appropriate Ax; range for each random dimension. Equation 5 becomes:
)
(6)
For example, considering the function: F(x, y, z ) = (x2e-*)~2e-Wz*e-z) (7) one might be interested in the value of the integral:
(z2e-’)dxdydz
(8)
Applying Equation 6, the execution time is just three times that for one dimension (i.e., N*D and not ND, as would be expected with numerical integration techniques), and the computer code is very simple (about 15 lines). The results of several trials are presented in Table 11. A very precise value would require many more trials, although the convergence toward the true value is obvious. The method of importance sampling can be applied to improve the convergence of the algo-
~intewatbnd Equation 8 with Trials
J(X. Y. z)
I 7 . 6 8 2 5 i 1.0312 7.8691 i 1.5775 7.9032 i 0.4457
532A
rithm (7). For example, random numbera are chosen from a nonuniform distribution that resembles the integration function (e.g., ex for Jer2dx). Thus, the function is sampled more often in the regions that contribute most to the
ANALYTICAL CHEMISTRY. VOL. 62, NO. 9, MAY 1. 1990
area. The precision is generally increased. Monte Carlo ensemble (cluster). A Monte Carlo ensemble is a random collection of particles at equilibrium. Most analytical signals one measures in
d e s on the vertices of a lattice (7). Monte Carlo ensembles are found in the determination of thermodynamic properties (e.g., enthalpy, adsorption isotherms), percolation (e.g., diffusion in porous media),production of fractals by diffusion-limited aggregation (7),classical molecular dynamics (18). “cold fusion” experiments (19), quantum applications GO), fundamental interactions of elementary partides (20, and neural networks (22). Gorbunov and eo-workers (23)have employed a Monte Carlo ensemble to study the adsorption of proteins on an absorbent pore model and their resulting behavior in a chromatographic column. Using a random distribution of particles between two plates, they could study the effect on the observed distribution coefficient from changes in the distribution and size of the active sites on the surface of the proteins as well as the activity of the adsorbent. Monte Carlo annealing (free-energy minimization).This technique is based on a modified Monte Carlo ensemble, better known as simulated annealing by association with ita thermodynamic analogy. Annealing is the physical process of obtaining an organized crystal lattice by heating a solid until it melts and cooling it down carefully until it crystallhs. If the cooling is slow enough, the free energy of the solid is minimized to a global optimal lattice (i.e., the ground state)and a perfectcrystalisobtained (22).Thephysical annealingcan be modeled by Monte Carlo techniques using the Metropolis algorithm (24). This simulation algorithm uses a sample of particles at a given initial pmition; it transforms a current state i , with energy E,, to a subsequent statej , with energy EJ,by displacing a randomly chosen particle by a random amount in a random direction (26).The new state is accepted if the energy of the system is lowered
E, - E , 5 0 (9) or if the Metropolis criterion is satisfid
the laboratory represent the auerage behavior of molecular populations. These average values can be obtained from the partition function derived from statistical thermodynamics (17). Monte Carlo techniques have been in-
tensively applied to perform “ensemble” averages enabling the derivation of partition functions in cases of complex systems with many degrees of freedom. The approach is often based on the random distribution of mole-
where R J i ) is a uniform random number, ks is the Boltwnann constant, and Tthe ateolute temperature (Le., Boltzmann distribution). This latter acceptance criterion Dermita the state of the system to “w&der” away from an apparent local minimum and ensures the location of the global minimum. The right-hand side of the inequality in Eguation 10 is similar to the exponen-
tialdecreaseshownbycainFigure A in the box. After several trials, the temperature is reduced. The process is
ANALYTICAL CHEMISTRY, VOL. 82. NO. 9, MAY 1. 1880
5SSA
REPORT repeated until convergence is achieved. The implications of the Metropolis “acceptance test” are that unfavorable states are possible, but that the probability of accepting one as the next starting state decreases as E, - Ei gets larger or as T gets smaller. Monte Carlo annealing can be used for combinatorial (discrete) optimization of any system if E is replaced by the appropriate response or “objective” function (e.g., signal magnitude) and kBT is replaced with a control parameter c that governs the probability of accepting a state of the system if it does not represent a movement toward an optimum. After a series of moves (e.g., >lo), the control parameter c is reduced, for example by lo%, thereby narrowing the acceptance criteria for “unfavorable” responses. The process is repeated until no more transitions occur within an acceptable time ( 5 ) . The method has unique features when compared with other optimization techniques ( 5 ) because it converges asymptotically to the global maximum ( 2 2 ) . Several algorithms have been proposed to apply Monte Carlo annealing to continuous functions (e.g., least-squares modeling) with multiple local minima (26, 27).
Their performance was shown to be competitive with the best algorithms currently available (26)and unique for the most complex functions (27).Kalivas et al. (28)showed that the generalized simulated annealing algorithm of Bohachevsky et al. (27) is more effective in locating a global maximum in the response surface of an analytical system than is SIMPLEX optimization, although the latter converges in a smaller number of steps if the response surface is not exceedingly complex (several local maxima or minima). Monte Carlo experiment (explicit simulation). The goal of modern ana-
lytical instrument development is not to simplify the physical and chemical nature of the system to permit modeling, but rather to provide an easy-touse, sensitive, and selective analytical tool. Because of this objective, experimental verification of the utility of a technique generally precedes any pursuit of a detailed model of the system. As a consequence, the functioning form of the instrument or system may be quite elaborate and far from conducive to models that are both simple and accurate. However, with the computational power now available, the explicit simulation of complex chemical sys-
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ANALYTICAL CHEMISTRY, VOL. 62, NO. 9, MAY 1, 1990
tems through mathematical experiments is possible and recommended. Physical and chemical processes are stochastic in nature because all the molecules in a given state have the same probability of reacting at some time t . Deterministic approaches (e.g., solving a system of differential equations by integration) are only approximations because the time evolution of a chemical system is discrete, not continuous-the molecular populations change only by integral amounts. The theory of stochastic processes accounts for the discrete and random nature of chemical reactions and also enables the estimation of the variance in the model. However, the solution of the master equation in stochastic formulations can only be solved numerically (Le., by a Monte Carlo experiment) for complex cases. The numerical solution is formally exact and accounts for the inherent fluctuations and correlations (29). Complex reactions (29) and heat transfer problems (30) were treated by kinetic Monte Carlo experiments. Several thermodynamic approaches have been used for Monte Carlo experiments. Monte Carlo annealing, discussed above, was proposed for explicit simulations (24).Microcanonical Monte Carlo (i.e., an ensemble) is an efficient modification based on the addition of an extra degree of freedom: a Monte Carlo “demon” (31). The demon travels around the system exchanging energy with the particles in an attempt to change the dynamic variables of the system. The procedure is repeated until the demon and the system reach equilibrium (7). A Monte Carlo explicit simulation is based on the principle that any complex process can be broken down into a series of simpler independent events (Le., a Markov chain [30]),each represented by a probability distribution. Future events are determined by the present state and are independent of the past. This fractalization approach is accurate if a “small” time interval At, relative to the system, is employed. Thus, At should be large enough to introduce uncertainty in any particular component event (randomness) but small enough to satisfy the requirement of event independence. The stochastic experiment starts by locating, in the computer memory, a “sample” of several thousand particles in a geometric representation of the instrument. The actual experimental conditions are emulated gradual heating, change of pH, gashiquid flow, and so on. Every At during the simulation each particle is monitored for displacement from its position (positions stored in X ( i ) , Y(i),Z(i) arrays) and
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for reactions that would convert it (Type ( i ) array) to another species (e.g., A B) or change its state (e.g., gas to solid). Reactions can be accommodated from either a thermodynamic or kinetic probabilistic approach. The evolution of the particles is followed until the sample is “consumed” or a steady state is reached. Monte Carlo experiment codes are relatively simple and demand little memory but are computationally intensive and often require the use of fast supercomputer processors to achieve accuracy for the simulation within an acceptable execution time. Monte Carlo program complexity increases almost linearly with the complexity of the problem. This can be compared with the complexity of conventional solutions (i.e., analytical equations), which increase approximately with the square of problem complexity (30). Electrothermal atomization. Electrothermal atomization for atomic absorption spectroscopy (ETA-AAS) is a good example of a difficult-to-model analytical system. The complex chemistry and geometry associated with the graphite furnace have made the theoretical application of conventional analytical solutions of equations nearly +
impossible. The unique features offered by Monte Carlo techniques have been shown (e.g., 32, 33). A simple atomization mechanism in ETA-AAS can be described by:
M (surface) + M (g, in)
-
M (g, out) (11)
where M represents the analyte on the surface or in the gas phase, inside or outside the furnace. The simulation algorithm (33) starts by randomly distributing the initial number of particles over a small region at the bottom of the furnace center. For every At (under 0.5 ms) each surface particle has a chance to desorb into the gas phase and each gas particle moves a distance Ad until it hits the surface or leaves the furnace. The effect of desorption on the number of analyte atoms in the gas phase N,,, can be described by:
udesN:;i] exp( -
$)
(12)
where t represents time, Nwallis the number of analvte ” .Darticles on the
graphite surface (i.e., proportional to e, the fractional surface coverage), Udes is a preexponential factor for desorption from the surface, rides is an order of release, and E$,, is an activation energy for desorption. Assuming that 8 a Nwall,rearranging, and integrating Equation 12 between t and t At (Le., Tis almost constant), one finds the auerage fraction of particles desorbed during At. This is also equivalent to the probability of desorption Pdes for a large number of particles during the small time interval At. For first-order desorption:
+
(
exp udes exp
(- 2)
At)
(13)
The adsorption of gaseous particles back onto the wall can be handled differently. Because adsorption is determined for euery collision by the fraction of particles having energy larger than activation energy for readsorption Eids (Le., from Boltzmann distribution), and because the interaction time between particle and surface is much smaller than At, the average probability of readsorption for a particle colliding with the wall during t At is:
+
Q.
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ANALYTICAL CHEMISTRY, VOL. 62, NO. 9, MAY 1, 1990
(14) where S* is the sticking coefficient for readsorption upon collision with the surface (Le., the probability of hitting an active site) and fe(0)represents the fraction of empty sites on the surface (close to unity at all times because of the small sample size used). For every At, a uniform random number R,, is assigned to each particle i on the surface or colliding with it. It is compared with the average probability given by Equation 13 or 14. If R,, is less than the average probability of reaction, then the particle will react; it will be removed from or placed onto the surface. Diffusion of the particles in the gas phase can also be accounted for from a semimicroscopic point of view. The vectorial displacement of each particle in a 3D coordinate system (AxL,Ayt, Az,) is deduced from the random walk. The magnitude of the movement is given by:
AdL= R,2(6DT,At)”2
+
+
(15)
where Ad, = (Ax: Ayp A Z : ) ~and ’~ R,, is a Gaussian random number calculated as described in the box on p. 532 A. D T is the temperature-depen-
c
L w w w
"W""
i"'
''LE
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dent diffusion coefficient for the gas temperature at the particular ( x i , yi, zi) position of the particle i at time t. In the gas phase: "dir
&,=DO( Ti(x, y, z (16) To )) where TO= 273 K and ndif = 312 from kinetic gas theory. The direction of the move (Ax, Ayi, A d is chosen from a
3D random direction from Equation iv (33). Figure 3 presents cross-sectional views of the furnace during simulated atomization (34).The position of every particle can be monitored not only in the gas phase but also on the graphite surface. Figure 4 shows additional information that can be extracted from the simulation. The relative rates of
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Figure 3. Explicit simulation of Cu atomization. Early stage: ( 8 )abso"+time
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ANALYTICAL CHEMISTRY, VOL. 62. NO. 9, MAY 1. 1990
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Flgure 5. Model macromolecule. Each active slle tss Its indlvldualinteractlw snergy and actlon radius. Sphalcal Madinates lr. 8.9) are used IO lxallze me patches mi mi^^) tome OBnfer of mass.
collision, desorption, and adsorption are also available. Experimental conditions can be modified with ease using this stochastic approach. A simulation with 10,OOO particles runs in less than a minute on the Cray X-MP. Chromatography of biopolymer< High-performance liquid chromatography is an accepted method for separation and control of purity in biochemistry and biotechnology. “However, until now, no paper has appeared descrihing the prediction of chromatographic data for high molecular weight proteins based on their amino acid composition.” (35) This can be explained by the complex physical and chemical processes occurring inside the column, including solubility and stability (pH, T,buffers), distribution of active (hydrophobic or hydrophilic) “patches” on the molecular surface, diffusion and sedimentation, electrical mobility, aggregation, conformational changes, and the role of displacing agents. A Monte Carlo explicit simulation algorithm has been suggested to deal with such a complex system (36). Molecular cartography can be used to quantitate the topgraphic structure of a protein surface (37) and, thus, to 10a t e the active sites on its surface. This information can be incorporated into the simulation by using a spherical (or other shaped) particle with a series of spherical patches on ita surface (Figure 5). Eacb of the sites baa a given radius, defining an action radius, and an energy representing the relative strength of its interactions with the mobile and stationary phases. Diffusion in the column can be considered a random walk employing Equation 15, and the angular m i t i o n of the molecule can be considered random in absence of fields. A 540,.
Figure 6. Expllclt slmulatlon of biopolymer separation. dlmutlon mefflcients(&)are
(a) 0.10. (b)1.0, and (c) 10.0.
I
I
The case shown mnasponda lo cywt c in Figvs E. Dimlbutionof n-mcmpamclss(a) mchsd Io the smIionary phase. (b)In mobile phase. and (c) I w b g Ibcapillary. The re& mites of (d) desorption. (e) colliaiw.and (ladsoption ) (Ye aka ahown.
me
simple example is presented in Figure 6 for a sample of 2000 particles in an open-tubular capillary column, with each particle having 10 equal sites randomly distributed on ita surface. No solvent gradient was assumed and no interaction between the partides was allowed. A pseudo-equilibrium has been assumed for the interaction between the biopolymer active sites and the stationary phase upon collision us-
ANALYTICAL CHEMISTRY, VOL. 82. NO. 9, MAY 1, 1990
ing a site-dependent distribution coefficient: fq---exp(-s)
* ’-
fmob,
(17) where fSat, v d f,,,b represent the fraction of partdes ahsorbed to the stationary phase and those in an unbound condition in the mobile phase when
considering sites J on the molecule, respectively. The prohability of adsorption (i.e., binding) foreachsitej"touching" the wall is:
(18)
and the probability of breaking the bond is:
The chromatographic behavior shown in Figure 6 is noticeably affected by changes in the distribution coefficients of the sites. The output from the Monte Carlo experiment also includes information that is not easily available experimentally, as shown in Figure I. The interaction algorithm can he changed from thermodynamic to kinetic approaches with relative ease. Considering more complex phenomena, such as cooperative kinetics, conformational changes and aggregation can also be accommodated with minimal effort. This simulation technique might prove
fundamental in the feasibility evaluation and ultimate development of very high speed HPLC.
investment of the analyst's time because of the inherent simplicity in constructing the necessary algorithms.
ConcluSiMls
This work was supported by the National Science Foundation under grant CHE-8704024 and by Cray Research, Inc. O.A.G. acknowledges the supplemental support provided by the University of
It seems appropriate when dealing with analytical systems that behave randomly, at least at the molecular level, to employ techniques that also are based on the utilization of random numbers. Likewise, when random methods are to be used, such as for sampling, it seems equally obvious to make unbiased, legitimate random selections. All these approaches fall within the range of stochastic methods and can employ Monte Carlo techniques. Monte Carlo methods traditionally have been associated with theoretical chemistry and physics where often no other approach was available. With the increasing availability of computing power, either at the micro- or supercomputer level, Monte Carlo techniques should be viewed just like any other tool useful in solving analytical problems. In many instances it should prove to be not only the most convenient means of solving a particular problem, but also one that offers the most accurate answer with the smallest
Costa Riea.
References (1)Cfiell, 0. A.; Holcombe, J. A. FACSS XV Annual Meeting, Abstracts; Boston, 1988;No. K12. (2) Hammersley, J. M.; Handscomb, D. C. Monte Carlo Methods; Methuen: London, 1964. (3) Metropolis, N.; Ulan, S. J. Am. Statist. Assoe. 1949,44(247),335. (4) Lindfield,.C. R.; Penny, J.E.T. Mierocomputers
Numerrcal Analysis; Ellis
Horwood Chiehester, England, 1989. ( 5 ) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterlmg, W. T. Numerical Recipes; Cambridge University: Cambridge, 1986. ( 6 ) Borman, S. Chem. Eng. News 1989,
69(29),29. ( 7 ) Could, H.; Tohoehnik,J. An Introduetion to Computer S"datron Methods; Addison-Wesley: Reading, MA, 1988; Part 2. (8) Crandall,R.E.Pascal Applicationsfor the Scrences; Wiley: New York, 1984. (9) Cochran, W. B. Sampling Techniques, 3rd ed.; Wiley: New York, 1977. (10) Kratochvil, B.; Wallace, D.; Taylor, J. K. Anal. Chem. 1984.56.113R.
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ANALYTICAL CHEMISTRY, VOL. 62, NO. 9. MAY 1. 1990
f
541 A
REPOR7 (11) Miller, J. C.; Miller, J. N. Statistics for Analytical Chemistry; Ellis Horwaod Chichester, England, 1984. (12) Youmans, H. L. Statistics for Chemistry; Charles E. Merrill: Columbus, OH, 1973;Chapter 7. (13)Howarth, R. J. Analyst 1973,98,777. (14) Larsson, J.A.; Pardue, H.L. Anal. Chem. 1989,61,1949. (15)Frank, I. E. Chemomet. Intellig. Lob. Syst. 1987,I, 233. (16) Efstathiou, C. E.; Hadjiioannou, T. P. Anal. Chem. 1982,54,1525. (17) Atkins,P. W.PhysieolChemistry, 3rd. ed.; Freeman: New York, 1986; Chapter 21. (18)Rahman, A,; Stillinger, F. H. J . Chem. Phys. 1971,55,3336. (19) Ichimsru, S.; Ogatn, S.; Nakano, A,; Iyetqmi, H.; Tajima, T, submitted for publication in Phys. Re". Lett. (20) Reynolds, P. J.; Ceperley, D. M.; Alder, B. J.; Lester, W. A., Jr. J. Chem. Phys. 1982.77,5593. (21) Rebbi, C. Lattice Gouge Theories and Monte Carlo Sm"otions; World Scientific: Singapore, 1983. (22) Aarts, E.; Karst, J. Simulated Annealing and Boltzmann Machines; Wiley: Chichester, England, 1989. (23) Gorbunov, A. A,; Lukyanov, A. Ye.; Pasechnik, V. A.; Vakhrushev, A. V. J. Chromatogr. 1986,365,205. (24)Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E. J. Chem. Phys. 1953,21,1087. (25) Alder, B. J. In Supercomputer Simu-
lations in chemistry: Dumis. M.. Ed.: Springer-Verlag: Bedin. 1986. (261 Vanderbilt. I).; Louie, S C.J.Compur. Phw. 1984.56.259. (27) Bohachevsky, I. 0.;Johnson, M. E.; Stein, M. L. Technometrics 1986.28,209. (28) Kalivas, J. H.; Roberts, N.; Sutter, J. M.Anal. Chem. 1989,6J,2024. (29) Guillespie, D. T.J . Phys. Chem. 1977. 25,2340. (30) Howell, J. R.Ad". Heat Transfer 1968, 5,l. (31)Creutz, M. Phys. Rev. Lett. 1983.50, 1411.
(32) Giiell, 0.A,; Holcomhe, J. A. Speetrochim. Acta, Part B 1988.43,459. (33) Giiell, 0.A,; Holcombe, J. A,, unpuhlished work. (34) Gcell, 0.A.; Holcomhe, J. A. Monte Carlo Simulation of Atomization Processes in Electrothermal Atomizers, videotape ed.; University of Texas: Austin, 1OPO A"y".
(35) Mikes, 0.High-Performance Liquid Chromatography of Biopolymers and Bioolieomers. Part B: Elsevier: Amster-
(39) Programmer's Library Reference Manual; Cray Research, No. SR-0113, 1986.
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Oscar A. Giiell receiued a B.S. degree from the Uniuersity of Costa Rica in 1985 and is currently a Ph.D. s t u d e n t at the Uniuersity of Texas at Austin. His research interests include incorporation of computational techniques s u c h as M o n t e Carlo, nonlinear dynamics, and n e u r a l networks i n t o analytical chemistry.
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J a m e s A. Holcombe (right) received his Ph.D. from the University of Michigan in 1974. H e then joined the faculty at Austin and was Program Director for Chemical Analysisat NSFin 198485. His research interests include the study of f u n d a m e n t a l processes occurring during atomization and excitation and the use of biological organisms for ultratrace m e t a l preconcentration and speciation.
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