modeling adsorption-desorption processes in porous media

thin slice of a porous glass known asVycor. The black patches are the glass(SiO2) and the white patches are thepores. The pores are nearly cylindrical,with a mean diameter of approximately70 Å. Natural porous materials tend tohave a more intricate pore structureand a wider distribution of pore sizes.What is common among all porousmaterials, natural or man-made, is a3D interconnected pore space. These

interconnected pores make porous ma-terials important in many scientific andindustrial processes—for example, oilrecovery, catalysis, and filtration.

Crucial to many of these processes isa knowledge of the pore-space geom-etry and an understanding of how flu-ids interact with porous materials. Flu-ids in porous media have also been auseful paradigm for studying problemsin statistical physics, including invasion

percolation, the random-field Ising model, and pat-tern formation. In recentyears, these systems haveserved as model systems for examining the role thatquenched randomness (ran-domness that is fixed in po-sition) and impurities playin phase transitions andcritical phenomena.

An important experimen-tal tool for probing fluids inporous materials is the ad-sorption-desorption isotherm.As we will see, such mea-surements provide structur-al information about theporous materials themselves.However, despite extensiveresearch in the past several

decades, inferring the pore-space geom-etry and the size distribution fromisotherm measurements remains diffi-cult. How to address the connectivity ofthe pore space (that is, its network struc-ture) appears even more daunting. Thedifficulty arises largely from the com-plexity of real porous materials and thelimited understanding of the physics as-sociated with these complicated systems.

Given this situation and the increas-ing importance of porous materials, wehave undertaken a comprehensive studyof adsorption-desorption processes in abroad range of porous materials thatcombines isotherm measurements withcomputer modeling. As part of thisstudy, we have developed a modelingprogram that lets us design variousporous media of interest and simulateadsorption-desorption processes.

Adsorption-desorption isothermsAn isotherm consists of a series of mea-surements of the amount of gas ad-sorbed (the adsorbate) on the surfaces ofa material (the substrate) as a function ofthe equilibrium vapor pressure of thegas at a fixed temperature. To charac-terize a complete adsorption-desorp-tion process, we increase the pressurefrom zero to its bulk saturated value,then decrease it back to zero. The re-sulting isotherm is commonly calledthe adsorption-desorption isotherm.

Figure 2 shows an adsorption-de-sorption isotherm of N2 in Vycor. Theisotherm is characteristic of those ob-served in many randomly intercon-nected porous materials. A striking fea-

84 COMPUTING IN SCIENCE & ENGINEERING

Editors: Harvey Gould, [email protected], and Jan Tobochnik, [email protected]

MODELING ADSORPTION-DESORPTION PROCESSESIN POROUS MEDIAGrigori V. Kapoustin and Jian Ma, Amherst College

POROUS MATERIALS ARE “SPONGELIKE” MATERIALS. THEY

ARE UBIQUITOUS IN OUR LIVES—ROCKS, CERAMICS, BONES,

AND HUMAN LUNGS ARE JUST A FEW WELL-KNOWN EXAMPLES.

FIGURE 1 SHOWS A TRANSMISSION ELECTRON MICROGRAPH OF A

C O M P U T E R S I M U L A T I O N S

Figure 1. A transmission electron micrograph of a35-nm slice of Vycor.1 The pore spaces are white.

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JANUARY–FEBRUARY 1999 85

ture of the isotherm is the pronouncedhysteresis loop. The more gradual risein the amount adsorbed results from thenonuniformity of the pore sizes, where-as the abrupt turning point and subse-quent rapid drop during desorption arecooperative events caused by the connec-tivity of the pores. Investigating howdifferent pore-space geometries andnetwork structures affect the isothermsis our modeling program’s main goal.

The basic modelOur model represents a porous me-dium as a 3D network of voids (porebodies) interconnected by necks (porethroats), both of varying sizes (see Fig-ure 3). We let the voids be sphericaland assign them radii from a charac-teristic size-distribution function. Thenecks are cylindrical; their radii obey adifferent size-distribution function,and their lengths are determined by thelocations of the voids to be connected.The voids can be either distributedrandomly or placed on a regular lattice,and the number of necks intersectingat a void can vary. We designate a set ofthe voids on the outer surface as thereservoir through which the gas entersor leaves the system.

We model the adsorption-desorp-tion processes as follows:

During adsorption, the relative pres-sure, p = P/P0, increases from 0 to 1. Pis the vapor pressure, and P0 its value atbulk saturation. At a given p, we com-pute the thickness h of the adsorbedfilm in each pore—void or neck—of ra-dius r by solving the local chemical-potential equilibrium equation:2

(1)

The left-hand side of Equation 1 rep-resents the chemical potential of the

absorbed film; the right-hand side rep-resents the chemical potential of thevapor. The first term in Equation 1

comes from the van der Waals attrac-tion between the substrate and the ad-sorbate; α characterizes the strength of

−

−−( ) =α γa

h n r hT p

3

k ln

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180Adsorption

Desorption

Relative pressure (P/P0)

Am

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ST

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0 1.00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Figure 2. An adsorption-desorption isotherm of N2 in Vycor (a porous glass) at 65 K.The desorption branch shows a sharp turning point at P/P0 = 0.6. (P is the vaporpressure, and P0 its value at bulk saturation.)

Department focus & editors

This department presents new ideas and developments in computer simulations.Columns will discuss new applications and techniques, new physics that hasemerged from the use of simulations, and techniques that supplement computersimulations. The emphasis of the department is on tutorials for students and pro-fessionals who are not expert in the particular application or technique.

Harvey Gould is a professor of physics at Clark University, Worcester, Mass. Hismain interests are in computer simulation methods with application to con-densed-matter physics and pedagogy. He is coauthor with Jan Tobochnik of Intro-duction to Computer Simulation Methods, 2nd ed. (Addison-Wesley, 1996). He re-ceived his AB and PhD in physics from the University of California at Berkeley. Heis a Fellow of the American Physical Society and a member of American Associa-tion of Physics Teachers. Contact him at Clark Univ., Dept. of Physics, Worcester,MA 01610-1477; [email protected]; http://physics.clarku.edu/~hgould.

Jan Tobochnik is an associate professor of physics and computer science at Kala-mazoo College. His research involves Monte Carlo and molecular-dynamics simu-lations of a variety of systems including structural glasses, granular materials, bi-nary alloys, and phospholipids. He is coauthor (with Harvey Gould) of An Intro-duction to Computer Simulation Methods, 2nd ed. (Addison-Wesley, 1996). He re-ceived his BA in physics from Amherst College and his PhD in physics from CornellUniversity. He is a member of the American Physics Society, the American Associa-tion of Physics Teachers, the American Association of University Professors, andPhi Beta Kappa. He is on the executive committee of the Division of Computa-tional Physics of the APS. Contact him at the Departments of Physics and Com-puter Science, Kalamazoo College, 1200 Academy St., Kalamazoo, MI 49006;[email protected]; http://cc.kzoo.edu/~jant/.

.

this interaction. The second term isdue to the surface tension γ of the ad-sorbed film-vapor interface. a is the ef-fective size of an adsorbed molecule, n= a−3, and k is Boltzmann’s constant.The values α/k = 200 K, γa2/k = 126 K,and a = 3.8 Å model N2 in porous ma-terials at T = 65 K (the condition atwhich our experiments are performed).

As p increases, capillary condensationoccurs, first in small pores and then inlarge pores. Consider adsorption in asingle pore. As the vapor pressure in-creases from zero, molecules from thegas first form a film coating the walls ofthe pore, due to the van der Waals at-traction between the walls and themolecules. As the film thickens, main-taining the adsorbed film-vapor inter-face becomes increasingly expensive.Eventually, condensation forms insidethe pore at a pressure lower than thatat saturation. This phenomenon isknown as capillary condensation. Theoccurrence of capillary condensation ina given pore is indicated by Equation 1no longer having real roots for h.

Thus, Equation 1 captures the entirefilling process in a single pore, fromequilibrium film growth to eventualcondensation. Knowing the status ofeach pore (that is, either covered by afilm of thickness h or filled with a con-densed fluid), we can compute the to-tal volume of the adsorbed fluid for agiven value of p.

During desorption, p decreases from1 to 0. Capillary evaporation initiatesfrom the reservoir. Evaporation of thecondensed liquid in a pore depends ontwo criteria. First, at a given p, the pore

radius r must be greater than or equalto that given by the Kelvin equation

.(2)

(The Kelvin equation is derived byequating the chemical potential of a gasof pressure P to that of a coexisting liq-uid meniscus whose mean radius ofcurvature is r.3) The second conditionis that the pore must be connected to avapor cluster (a group of emptied poresleading to the reservoir). The abruptturning point during desorption in Fig-ure 2 corresponds to the formation ofa connected vapor cluster that spansthe entire medium, a process analogousto invasion percolation.4,5

So, unlike adsorption, where local ef-fects alone govern pore filling, in de-sorption, the connectivity of the porousnetwork plays a crucial role. The pro-gram must be able to anticipate ava-lanches of pore emptyings (that is, theemptying of a pore leads to simultane-ous emptyings of a group of largerpores). Once capillary evaporation hasoccurred in a pore, we calculate the filmthickness again (using Equation 1) asduring adsorption.

Implementing the modelWe use graphs to represent porous me-dia. This representation is a naturalchoice, given that a graph is an abstractdata type consisting of a set of verticesinterconnected by edges. Thus, a graphcan conveniently represent a porousmedium, with the vertices correspond-

ing to the voids and the edges corre-sponding to the necks. To the verticesand edges, we then add fields consist-ing of variables that specify the geo-metrical attributes of the voids and thenecks (for example, their radii andlengths). With this approach, making aporous model becomes equivalent todesigning a good graph. The input pa-rameters are the locations of the voids,the connectivity (characterized by theaverage number of the necks converg-ing to a void), and the size distributionsassigned to the voids and the necks.

Making the graph. We first gener-ate a point-cloud, a set of 3D coordi-nates that specifies the locations of thevertices of the graph. Point-clouds ofdifferent internal structures (random orotherwise) can coexist in the same ma-terial to create spatial heterogeneity inthe system. We produce 3D point-clouds by combining 1D (linelike)point-clouds and 2D (planar) point-clouds. This process might seem un-necessary, given that we could write afunction that places points in space di-rectly. However, this approach lets uscreate a variety of 3D point-clouds byperforming a few simple operations onpoint-clouds in lower dimensions,rather than writing a separate code foreach porous model to be designed.Later we will show a porous model builton a jungle gym graph, in which thevertices are situated on a cubic lattice.

We interconnect the vertices of apoint-cloud by the edges to make acomplete graph, which then serves asthe skeleton for the porous model beingconstructed. To make the connections,we choose a characteristic length l and aprobability q. We then examine eachpair of vertices. If the separation be-tween the two vertices is less than l, wegenerate a connection with probabilityq. This method is sufficient for building

r

aT

p= = −[ ]−2 31γ

kln



VRML

The wide availability of Virtual Reality Markup Language tools and browsers makesVRML an ideal format for visualizing porous models. VRML allows a porous modelto be presented as a 3D virtual world, which can then be viewed by a VRML brow-ser (for example, Netscape 3.0). Because this format is simple, portable, and well-supported, it provides an effective 3D visualization tool.

For basic VRML information, visit the VRML Consortium Web site, http://www.vrml.org/. For VRML software and other resources, visit the VRML Repository, http://www.sdsc.edu/vrml/. For VRML tutorials, visit http://www.ncsa.uiuc.edu/General/VRML/tutorial.html.

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random or jungle gym porous networks.For example, we make a jungle gymgraph by setting l equal to the spacingbetween the two nearest lattice sites andq equal to unity. This method can bemodified to create more complicatedconnections.

The resulting graph has the topologyof the desired porous network. To makethe model more realistic, we need toadd shape and size to the vertices andthe edges in the skeleton. We choosespherical vertices and cylindrical edges,and assign different size distributions totheir radii. We ensure that each result-ing void’s radius is larger than or equalto the radius of the largest neighboringneck coming to its site. Last, we desig-nate a set of voids on the system’s outersurface as the reservoir.

Adsorption-desorption algorithms.To facilitate modeling the adsorption-desorption processes, three searches areundertaken, which produce graph col-oring. The first locates the pores thatbelong to a connected cluster leading tothe reservoir, because only those poresare accessible in an adsorption-desorp-tion experiment. These pores are thencolored. (This search is not needed fora jungle gym porous model.) The sec-ond, performed during adsorption,identifies the pores in which capillarycondensation has taken place (the filledpores are then colored differently). Thethird, performed during desorption,monitors the growth of the vapor clus-ter (and produces a different coloringfor the pores that would be accessible).

These searches are implemented us-ing a depth-first search with a depth-bound algorithm. (The depth bound isuseful because a DFS is recursive.)That is, a DFS starts at multiple nodes(voids); when it reaches the depthbound, it puts the last node on a queueto serve as a starting node for the next

recursive search. The search continuesuntil no more nodes are on the queueof the starting nodes.

The following procedures summa-rize the algorithm for simulating an ad-sorption-desorption experiment.

For adsorption, perform a DFS tofind the connected pores, and colorthem as reachable. For p changing from0 to 1 in n equal increments,

(1) Go through all reachable pores andcompute the film thickness, usingEquation 1. Identify those that arecapillary-condensed, and color themas filled.

(2) Compute the total volume of thefilled pores V1 and the total volumeoccupied by the adsorbed film (bysolving Equation 1) in the unfilledreachable pores V2. The total vol-ume of the amount adsorbed isVfilled = V1 + V2.

(3) Write one entry of (p, Vfilled/Vtotal)for the adsorption isotherm, whereVtotal is the volume of the entirepore space.

For desorption, for p changing from1 to 0 in n equal increments,

(1) Do a DFS to find thepores that would beaccessible at the cur-rent p, and color them.Accessible pores arethose that belong to aconnected path of un-filled pores leading tothe reservoir.

(2) Go through all accessi-ble pores. Empty themif the pore radius r isgreater than or equalto that given by Equa-tion 2 for the currentvalue of p. Computethe film thickness im-

mediately after capillary evapora-tion.

(3) Compute the total volume of thefilled pores (V1) and the total vol-ume occupied by the adsorbed filmin the emptied pores (V2), whereVfilled = V1 + V2.

(4) Write one entry of (p, Vfilled/Vtotal)for the desorption isotherm.

The resulting adsorption-desorptionisotherm is stored as a list of 2D coor-dinates that can be later analyzed anddisplayed.

The software. Our program con-sists of a series of libraries written inC++ and several utilities that use theselibraries to perform particular tasks.The libraries consist of classes that im-plement specified abstract data types(for example, graphs and queues); thatperform operations on abstract datatypes; and that provide functions fordata conversions, display, and analysis.(A class is C++’s implementation of theconcept of an object. Objects encapsu-late data and functions that operate onthese data.) Although the libraries weredeveloped using Borland C++ 5.xx andBorland C++ Builder 1 & 3 on Win-

Void Neck

Figure 3. A 2D representation of a “void-neck”network.

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Figure 5. A sequence of snapshots taken during adsorption in the porous model in Figure 4. Only the filled necks are shown. (a)P/P0 = 0.68, (b) P/P0 = 0.72, and (c) P/P0 = 0.76.

(a) (b) (c)

Figure 6. A sequence of snapshots taken during desorption in the porous model in Figure 4. Only the emptied necks are shown.(a) P/P0 = 0.76, (b) P/P0 = 0.66, and (c) P/P0 = 0.64.

(a) (b) (c)

Figure 4. (a) A corner of a jungle gym porous model with a normal neck size distribution. The light-gray voids designate thereservoir (through which the gas enters and leaves). (b) The adsorption-desorption isotherm for this model.

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dows 95, they are compatible with stan-dard C++. By using preprocessor direc-tives, they can be quickly ported toUnix or other operating systems thatsupport standard C++.

The utilities are short programs con-sisting of various libraries. The utilitiesperform several categories of tasks:

• Generate abstract data types thatmodel porous materials.

• Implement functions that solveEquations 1 and 2.

• Perform the procedures for simulat-ing adsorption-desorption processes.

• Analyze and display the resultingdata.

In the object-oriented-programmingparadigm, these tasks can be handlednaturally by managing a set of objectsthat represent the porous models con-structed, the functions implemented,and the data produced.

The utilities are executed in com-mand lines, which lets us control thevirtual experiments through batch filesthat call the appropriate utilities andmake the intermediate data readily ac-cessible. Moreover, this way of runningthe utilities makes the execution moreportable and efficient.

To visualize our porous models, weuse the Virtual Reality Markup Lan-guage (for more information onVRML, see the related sidebar). Fig-ures 4a, 7a, and 8a show several VRMLrepresentations of porous models ofour design; Figures 5 and 6 showVRML snapshots taken during ad-sorption and desorption. This visual-ization capability has been very usefulin designing porous media (particularlythose with heterogeneous structures)and developing algorithms for model-ing physical processes taking place inthese complicated systems.

Similarly, data manipulation andanalysis occur outside the main simu-

lation code. By storing data in ASCIIand tab-delimited internal formats, wecan analyze the resulting data and dis-play them in any spreadsheet. More-over, this approach lets us use Perl orVisual Basic for performing tasks in-volving data conversion and format-ting, instead of writing full-blown C++programs.

Three simulationsWe have used the above modeling pro-gram to simulate adsorption and de-sorption in three porous models.

The first model is spatially homo-geneous—that is, all the necks obey asingle size-distribution function. Figure4a shows a corner of a jungle gymporous model with voids at the sites ofa cubic lattice; six necks converge ateach void. The model comprises 8,000voids and 28,000 necks. We assign theradius of each void to be that of thelargest neck coming to its site. To thenecks, we assign a normal size-distrib-ution function, ranging from 10 to 70 Åwith a mean of 40 Å. Figure 4b showsthe resulting adsorption-desorption

isotherm. The isotherm remarkably re-sembles that in Figure 2. Figures 5 and6 display snapshots taken during the ad-sorption-desorption process.

To examine the effects of the pore sizedistribution on the isotherms, we as-signed a power-law size-distributionfunction to the necks and left everythingelse unchanged. The power-law size-dis-tribution function ranges from 10 to 200Å with an exponent of −2.7 (that is, theprobability for a size r to occur dependson r in a power-law form, r–2.7). Figure 7shows a corner of this model and the re-sulting isotherm. The hysteresis regionon this isotherm is broader. We attributethe more gradual rise during adsorptionto a wider range of neck sizes. We at-tribute the turning point commencing ata lower pressure during desorption tothe presence of a larger number ofsmaller necks. Prompted by the experi-mental measurements on a shale sample,we also designed a “checkerboard” por-ous model. It disperses regions with apower-law neck size distribution (10–50Å with a −2.7 exponent) and a normalvoid size distribution (10–200 Å with a


Grigori V. Kapoustin is a junior at Amherst College, majoring inmathematics and computer science. His interests include com-puter simulations, Internet technologies, and Linux. He has beenresponsible for developing and maintaining the Amherst CollegeAlumni Web site. He is a member of the Amherst College ACMChapter. Contact him at Box 568, Amherst College, Amherst,MA 01002-5000; [email protected]; http://www.amherst. edu/~gvkapous.

Jian Ma is an assistant professor in the Department of Physics atAmherst College. Her research is in experimental condensed-matter physics, centered on phase transitions in 2D and disor-dered systems, and fluids in porous materials. She received herPhD in physics from the University of Washington. She is a mem-ber of the American Physical Society. Contact her at the Dept. ofPhysics, Amherst College, Amherst, MA 01002; [email protected]; http://www.amherst.edu/~physics/faculty. html.

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mean of 105 Å) among regions with anormal size distribution (10–70 Å with amean of 40 Å). Figure 8 displays a cor-ner of this model and the resultingisotherm.

This isotherm behaves qualitatively

differently than isotherms of homoge-neous systems; it has two distinct step-wise (percolation-like) features duringdesorption because two different groupsof necks exist. An intermediate step alsooccurs during adsorption, resulting

from capillary condensation in the nor-mal regions and subsequently in thevoids enclosed in the power-law regions.This isotherm appears to capture the es-sential characteristics of an experimen-tal isotherm taken in shale (see Figure


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Figure 7. (a) A corner of a jungle gym porous model with a power-law neck size distribution. The light-gray voids designate thereservoir. (b) The adsorption-desorption isotherm for this model.

Figure 8. (a) A “checkerboard” porous model, which disperses regions with a power-law neck size distribution and a normal voidsize distribution among regions with a normal neck size distribution. The light-gray voids designate the reservoir. (b) The ad-sorption-desorption isotherm for this model.

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9). If the size-distribution functions areallowed to extend to a very large size(instead of being truncated, as in ourmodels), the intermediate step on ad-sorption would disappear. However, thetwo stepwise features on desorptionwould remain.

T he primary roles of computermodeling in our study are to test

ideas prompted by experimental resultsand to suggest additional features tolook for in new experiments. In addi-tion to complementing experiments,our modeling program can be appliedto several interesting theoretical prob-lems. For example, near the onset ofdesorption (that is, the sharp turningpoint during desorption in Figure 2),pores are expected to empty in a seriesof avalanches whose size distributionobeys a power-law distribution.6,7 Weare investigating this feature.

We also plan to implement an inter-active front end that allows point-and-click design of porous models andsimulations of adsorption-desorptionexperiments. This front end will beeasy to implement, because it involvescreating hidden batch files for execut-ing these tasks. In addition, we plan todivide our large modeling package intoseveral threads and run them in paral-lel on several computers. In this way wewill be able to tackle much largerporous systems and perform varioustasks more efficiently.

AcknowledgmentsThis work is supported by NSF GrantNo. CTS-9803387. We thank the edi-tors for their helpful suggestions.

References1. P. Levitz, G. Ehret, and J. Drake, picture pub-

lished in Physics Today, Vol. 42, No. 7, July1989, p. 25.

2. S.M. Cohen, R.A. Guyer, and J. Machta, “Hy-drodynamic Modes of Superfluid Helium Ad-sorbed on Nuclepore,” Physical Rev. B, Vol.33, No. 7, 1 Apr. 1986, pp. 4664–4668.

3. S.J. Gregg and K.S.W. Sing, Adsorption, Sur-face Area and Porosity, 2nd ed., AcademicPress, San Diego, 1982, pp. 111–121.

4. D. Wilkinson and J.F. Willemsen, “InvasionPercolation: A New Form of Percolation The-ory,” J. Physics A, Vol. 16, No. 14, 1 Oct.1983, pp. 3365–3376.

5. D. Wilkinson, “Percolation Effects in Immisci-ble Displacement,” Physical Rev. A, Vol. 34,No. 2, Aug. 1986, pp. 1380–1391.

6. J.P. Sethna et al., “Hysteresis and Hierarchies:Dynamics of Disorder-Driven First-Order PhaseTransformations,” Physical Rev. Letters, Vol. 70,No. 21, 24 May 1993, pp. 3347–3350.

7. R.A. Guyer and K.R. McCall, “Capillary Con-densation, Invasion Percolation, Hysteresis,and Discrete Memory,” Physical Rev. B, Vol.54, No. 1, 1 July 1996, pp. 18–21.

Adsorption

Desorption


Am

ount

of g

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bsor

bed

(cm

3 at

ST

P)

0 1.00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

10

20

30

40

50

60

70

80

Figure 9. The adsorption-desorption isotherm of N2 in shale at 65 K. The desorptionbranch has two stepwise features, at P/P0 = 0.9 and 0.3, indicating two groups of necks.

Suggestions for further study

1. Write a software package that constructs a graph by implementing vertices inter-connected by edges. Add fields to the vertices and edges. Each vertex has threefields: a set of 3D coordinates, a radius, and a Boolean specifying if it belongs tothe reservoir. (For example, let the interior vertices be “true” and the exterior ver-tices be “false.”) Each edge’s field is merely its radius. Assign radii between 100and 200 Å to the resulting voids and the necks (assuming a uniform distribution).

2. Suppose that the porous model just constructed is initially at saturation. Pressurethen decreases, and the condensed fluid begins to evaporate in the pores. Solvethe Kelvin equation (Equation 2 in the main article) to find the pressure at whichevaporation in a pore of radius 150 Å would occur. Then start from the reservoirand locate the vapor cluster at the given pressure. Also identify the pores thatwould have undergone capillary evaporation were they not blocked by other(smaller) pores.

3. Write a code that locates all the avalanche events during desorption in the porousmodel created in Problem 1. Plot the avalanche distribution—that is, the numberof avalanches versus the avalanche size. Hint: Given that we have the “blueprint”of the porous system, we should be able to find out which pores would empty atevery stage of desorption, thus identifying all possible avalanches. There are sev-eral approaches to this problem. The code written in Problem 2 for locating thevapor cluster on desorption might be useful here, too.

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modeling adsorption-desorption processes in porous media

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