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Capillary Condensation of Associating Fluids in Slit-Like Pores: A Density Functional Theory

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Capillary Condensation of Associating Fluids in Slit-Like Pores: A Density Functional Theory
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  Capillary Condensation of Associating Fluids in Slit-Like Pores:A Density Functional Theory Krzysztof Stepniak,* Andrzej Patrykiejew,* Zofia Sokołowska,† and Stefan Sokołowski* ,1 *  Department for the Modelling of Physico-Chemical Processes, Maria Curie-Skłodowska University, 20031 Lublin, Poland; and  †  Institute of Agrophysics, Polish Academy of Sciences, Dos´wiadczalna 4, 20290 Lublin, Poland  Received November 10, 1998; accepted February 17, 1998 Adsorption of associating particles interacting via the Lennard– Jones potential in slit-like pores is studied using the density func-tional theory. A model of association with one site per particle isconsidered. Capillary phase diagrams for three values of the as-sociation energyare evaluated. We showthat the association shiftsthe entire diagram toward higher temperatures. The critical aver-age densities of the confined fluid, however, are almost indepen-dent of the association energy. Our results also demonstrate thatthe association significantly influences the layering transitions. © 1999 Academic Press 1. INTRODUCTION The development of a theory that accounts for highly direc-tional interactions between particles has been a major chal-lenge in the theory of fluids for many years (1, 2). A big portionof this area of study is the limit of strong polar interactionssuch as those that occur in hydrogen bonding. Typical exam-ples here are water and alcohols. The first attempts to describesuch bulk systems have been associated with application of perturbation theories. However, a common problem with per-turbation theories for strongly interacting molecules has been alack of bond saturation, which occurs due to steric effects (3).Since all molecules are allowed to interact with the samepotential, theories predict that such molecules will bond withmore particles than can possibly be within range of the bondingpotential. A theory that prevents this has been developed byWertheim (4, 5). His theory allows only nonbonded sites onmolecules to form new bonded states. Wertheim’s system of equations describing the structure and thermodynamics of thesystem can be written for models with one, as well as withseveral, associating site per particle. The models of bulk fluidswith one (appropriate, e.g., for carboxylic acids or for dimer-ization of nitrogen dioxide) and with four associating sites(appropriate to model methanol, for example (6)) have been themost extensively studied.The structure of associating fluids near adsorbing surfaceshas been intensively studied by computer simulations, as wellas by various theories of inhomogeneous fluids. Theoreticalapproaches have been most often based on the application of integral equations for the density profile (see, for example,Refs. (7–11)). Recently, this problem has been also addressedby researchers using one of the most powerful theoreticalapproaches, widely used in cases of simple nonuniform fluids(cf. Ref. (12) and the references quoted therein), namely, thedensity functional (DF) theory (13–20).Recently, Segura  et al.  (17, 18) have presented a new andsuccessful version of the DF theory that describes associatingfluids adsorption. This theory is based on the Evans–Tarazonaweighted density functional theory (12, 21) and on Wertheim’s(4, 5, 22, 23) perturbation treatment of associative contributionto the free energy functional. The tests of this theory haveshown that in the case of associating hard spheres near a hardwall a generally good agreement between theoretical predic-tions and Monte Carlo data is obtained. This theory has beenalso applied to the study of adsorption of fluids near partiallypermeable walls (19) and to investigations of the influence of association on wetting transitions (20).Simple fluids adsorbed in narrow pores have been the sub- jects of a great deal of interest in recent years (24–31). Mosttheoretical works have been focused on the nature of the phaseequilibrium of fluids confined in single pores of idealizedgeometry. This topic has been examined with a number of different methods, including density-functional theories, grandcanonical ensemble Monte Carlo simulations, and moleculardynamics computer simulations. More recently, attention hasbeen also paid to the pores with energetically and/or geomet-rically nonuniform walls (32–35). The influence of associationon the properties of bulk uniform systems has been also asubject of numerous theoretical works, e.g., (36–42). It hasbeen established that the increase in the association energyleads to increase in the fluid critical temperature and to thewidening of the coexistence region. However, the effect of association on the phase behavior of confined fluids has notbeen studied so far.The principal aim of this work is to study a simple model of the Lennard–Jones (LJ) associating fluid confined by slit-likepores with Lennard–Jones adsorbing walls. We focus the at- 1 To whom correspondence should be addressed.Journal of Colloid and Interface Science  214,  91–100 (1999)Article ID jcis.1999.6170, available online at http://www.idealibrary.com on91 0021-9797/99 $30.00Copyright © 1999 by Academic PressAll rights of reproduction in any form reserved.  tention on the investigation of the influence of the associationon the capillary condensation. We calculate the density profilesand adsorption isotherms and evaluate the phase diagrams.This is done by using the density functional theory outlined inRefs. (17–19). Moreover, to test the proposed theoretical ap-proach we perform comparisons of the theoretical predictionsat temperatures well above the bulk critical temperature withthe results of grand canonical ensemble Monte Carlo simula-tions. To our best knowledge, the studies presented here are thefirst in the literature to investigate the influence of associationon the capillary condensation of adsorbing surfaces via thedensity functional method. 2. THEORY We consider fluid particles interacting via the pair potentialcomposed of nonassociative and associative terms (36–42), u  r ,    1 ,    2   u non  r      A   A  u  AA   r ,    1 ,    2  , [1]where  r   is the magnitude of the vector  r  connecting the centersof molecules 1 and 2 and    i ,  i    1, 2, are the orientations of molecules 1 and 2 relative to the vector  r . The double sum runsover all association sites belonging to each molecule;  u non  isthe nonassociative part of the potential. We assume that  u non ( r  )is just a truncated LJ (12, 6) potential u non  r    u  LJ   r     4      /  r   12      /  r   6   r   r  cut  ,0  r   r  cut  , [2]where the cut-off distance is assumed to be  r  cut     2.5   .Without loss of generality we take     as the unit of length,      1.One associative site per particle is considered, and the as-sociative potential is given by (17–19, 36–42) u as  12   u as   x 12      a ,  x 12  a 0,  x 12  a  . [3]Here,  x 12   r 12  d (   1 )  d (   2 )  ,  d (   ) denotes the positionand orientation of the attractive interaction site,   a  and  a  arethe association energy and the range of attraction, respectively,and the distance between the center of the molecule and theassociative site is  L . In this work, we choose the following setof parameters:  L    0.5 and  a    0.1 . . . . These parametersensure steric saturation at the dimer level (18, 19).The fluid is confined in a slit-like pore. Each of the porewalls is the source of the Lennard–Jones (9, 3) potential v   ˜    z   3 3/ 2 2   s   z 0  z    9    z 0  z    3   [4]and the total adsorbing potential is v     z   v   ˜    z   v   ˜    H    z  , [5]where  H   is the pore width. In this work we assume that  z 0   0.6 and that (3  3/2)  s  /  k     10.As we have mentioned above, our theoretical approach isbased on the density functional theory outlined in details inRefs. (17, 18, 20). Thus, in order to omit unnecessary repeti-tions, we describe only the basic points of the theory. Toproceed, we write the grand potential  as a functional of fluiddensity,    ( r ) (12, 17–20),   F        r  v     z     d  r , [6]where    is the chemical potential. According to perturbationaltreatment, the Helmholtz free energy,  F  , is broken up into asum of ideal and excess parts,  F     F  id    F  ex , with F  id   /  kT        r  ln     r   1  d  r . [7]We apply the classical approach (12) and divide the excess freeenergy into parts associated with the contributions due torepulsive, attractive, and nonassociative forces acting betweenmolecules, and the contribution arising from the association(17–19),  F  ex   F  repex   F  att ex   F  assex .The first step toward the development of appropriate expres-sions is the division of the nonassociative pair potential intorepulsive and attractive terms. We implement the Weeks–Chandler–Andersen approach (43); thus the attractive part of the potential is u att   r         r   r  min u  LJ   r    r   r  min , [8]where  r  min    2 1/6   . The repulsive forces,  u rep ( r  )    u  LJ  ( r  )   u att  ( r  ), are approximated by a hard sphere potential. Thecontribution  F  repex is evaluated employing the concept of smoothed density (12, 17–20, 44),   ˜( r ), that is the local densityaveraged with a weight function  w ,   ˜   r     d  r     r  w   r  r   ,    ˜   r  . [9]The weight function  w  is described by a power series (21, 44), w  r  ,      w 0  r    w 1  r      w 2  r     2 , [10]and the coefficients  w 0 ,  w 1 , and  w 2  are written in (44).92  STEPNIAK ET AL.  The expression  F  repex thus takes the form F  repex    d  r    r   f  repex    ˜   r  , [11]where  f  repex is the free energy density of hard spheres of diameter d   and is calculated from the Carnahan–Starling equation (45)  f  repex      /  kT     4  3    /   1    2 , [12]where        d  3  /6. In principle, the effective hard-spherediameter  d   can be optimized (12), but in this work we simplyset it as equal to    .The nonassociative attractive forces are treated in a meanfield fashion; i.e., F  att ex    d  r  f  att ex  r  , [13]where  f  att ex  12    d  r     r     r  u att    r  r    , [14]whereas the associative term is given by (17–19) F  assex    d  r    r   f  assex    ˜   r  . [15]In the above,  f  assex    ˜    /  kT    ln       ˜         ˜    /2  0.5  , [16]and     is the fraction of molecules not bonded at the associativesite. According to (17–19),    (   ˜) is obtained from a quiteanalogous equation, as in the case of bulk associating fluids(36–42),      ˜    11    ˜       ˜   , [17]where   (   ˜) is approximated by    ˜     exp   a  /  kT    1      6  L 2    2  L  a 2  L  a  g  r  ;    ˜   a  2  L  r   2  2 a  2  L  r   rdr  . [18]In the above,  g  is the pair distribution function of the system of hard-spheres at density    ˜. This function is assumed to be givenby (36–42) g  r     exp  u rep  r    /  kT    y d   r   , [19]with  y d  ( r  ) being the Percus–Yevick hard-sphere cavity func-tion at the density    ˜. Although the Percus–Yevick cavity func-tion is not exact for hard spheres, Chapman (42) showed thatwhen used to approximate  g ( r  ) it produces more accurateresults than the cavity function from computer simulations.Obviously, Eqs. [18] and [19] are approximations. This meansthat the use of the approximate cavity function from the Per-cus–Yevick equation in Eq. [19] leads to cancellation of someerrors in the evaluation of the ratio of unbonded particles,    .Finally, the equilibrium density profile is obtained by min-imizing the grand potential,     /    (  z )    0, and hence weobtain  kT   ln      z   /    b   v     z    f  ex    ˜    z    f  ex    b    d  r  u att    r   r       z     b       ˜    z      z   f  repex    ˜    z    f  assex    ˜    z      z  dz     b   f  repex    b    f  assex    b  , [20]where  f   ex  is the excess free energy derivative with respect tothe density and    b  is the bulk (reference) density, correspond-ing to the chemical potential   .The calculations performed for hard associating spheres nearthe hard wall (17–19) and near the Lennard–Jones (9, 3) singlewall, as well as for Lennard–Jones associating particles incontact with the Lennard–Jones (9, 3) wall at high tempera-tures, have indicated (20) that the above-described theoryworks reasonably well. Obviously, within the region of phasetransition the theory fails; this is not surprising, given the meanfield character of the theory. Nevertheless, we can expect thatthe qualitative picture emerging from the DF calculationsshould be correct. Therefore, in the next section, the resultsfrom the DF theory are compared with computer simulationsonly at the temperatures well above the critical temperature. 3. RESULTS AND DISCUSSION To verify the theory, we have performed some grand canon-ical ensemble Monte Carlo simulations, using a method quitesimilar to that described in Refs. (19, 20, 46). The simulatedsystem consists of a cuboid box with the  z  axis bounded by thewalls (external fields) and with periodic boundary conditions inthe  x  and  y  directions. The box size was  XL    YL    10,  ZL    H     6. The particle translation parameter was selectedto assure the total acceptance ratio at the level of 25–30%. To93 CAPILLARY CONDENSATION OF ASSOCIATING FLUIDS  obtain final averages, 8  10 7 Monte Carlo steps were carriedout. During simulational runs we evaluated the density profilesof all particles, as well as of undimerized species.Figures 1a and 1b compare the results at different degrees of association and different bulk densities. Part a shows the re-sults for weak association. Although the association energy isnot so low here,   * a     a  /      10, the temperature is quitehigh,  T  *    kT   /      2, and at the configurational chemicalpotential   *    0.7374 the ratio of unbounded particles inthe bulk phase is close to 0.99; thus we have plotted here onlythe total density profile. Figure 1b is for   * a    15  T  *    2 andfor   *  0.00464. In general, the agreement of the simulatedand theoretical data for LJ associating and nonassociatingparticles is reasonable. Due to the character of the DF theoryused here and the known results for nonassociating systems(12), we did not expect to obtain fully quantitative agreementwith simulations. The observed discrepancies are quite similarto those observed previously for an associating fluid adsorbedon a flat wall (20).We now proceed to investigations of the phase behavior of associating fluids in pores. To establish the effects of changesin the association energy we have considered three situations,  * a    0.0, 7.0, and 10.0. Because we present only the resultsof DF theory below, for sake of convenience, the state of thebulk reference system is characterized by the values of the bulk fluid density,    b , and not by the values of the chemical poten-tial. The relation between the chemical potential and    b  resultsfrom the bulk counterpart of the theory.First we present some results for the pore with  H     6.Figures 2a and 2b show examples of the adsorption iso-therms,   ,    0  H  dz      z     b  , [21]at  T  *    0.7 (part a) and at  T  *    1.17 and 1.18 (part b),calculated for the nonassociating fluid,  T  *    0. The lasttemperature is higher than the capillary condensation criticaltemperature for this system, which equals 1.176. At  T  *  1.17only a very narrow hysteresis loop appears, indicating that thistemperature is very close to the critical temperature, and thetransition between the “liquid-like” and the “gas-like” adsor-bate is located at    b    0.0234. We should add here acomment on how the equilibrium density was evaluated. Foreach thermodynamic state we evaluated the grand potential,  .Then, the values of     for the adsorption and the desorptionbranches were plotted versus the chemical potential. The locusof the transition is at the intersection of the correspondingbranches of      (  ).At the low temperature  T  *    0.7 the hysteresis is verypronounced, cf. Fig. 2a, Moreover, the adsorption branch ex-hibits an additional loop connected with the layering transitionwithin the first layer adjacent to the pore walls. The first layercondensation is very well illustrated by the density profiles, cf.Fig. 2c, which confirms that the first (“inner”) loop at theadsorption–desorption isotherm is associated with adsorptionin the first layer. The lowest local density profile correspondsto the bulk density value just before the completion of the firstlayer; the subsequent profile corresponds to the filled firstlayer. The profile at the highest bulk density correspondsalready to the fluid-like adsorbate condensed inside the entirepore. The bulk fluid density at which the layering transitionoccurs is    b    0.000905 and is higher than the bulk densitycorresponding to the equilibrium condensation point,    b   0.00059. Finally, Fig. 2d shows several density profiles at  T  *   1. Note that before condensation, continuous filling of the FIG. 1.  A comparison of total density profiles (T) from theory (lines) andcomputer simulations (points). In part b the curves labeled M are the densityprofiles of monomers. The calculations have been carried out for   * a    10 and     0.7374 (part a) and for   * a    15 and      0.00464 (part b).Temperature is  T  *    2 in both cases. 94  STEPNIAK ET AL.  first layer occurs. Before capillary condensation, which takesplace at    b    0.0921, we even observe a weak signal due tothe second layer formation.Our calculations have revealed that for nonassociating fluidthe transition within the first layer occurs for  T  *    0.75. Thistemperature is close to the critical temperature of the layeringtransition for the first layer. We must stress, however, that wehave not performed calculations for  T  *    0.675, and hencewe do not know the lower temperature limit for this transition.The strength of the adsorbing potential used in our modelcalculations is about 2.4 times weaker than the strength of theadsorbing field over the graphite surface. The layering transi-tion occurs only in the situation in which the adsorbate wets thepore walls and terminates at its own critical point. For the weak substrate potential used here, one can expect finite wettingtemperature. Thus, the layering transition is likely to occurover a finite interval of temperatures.In Fig. 3 we show similar results, but for   * a    10. Part adisplays examples of the adsorption isotherms at three temper-atures. The highest temperature,  T  *  1.27, is just equal to thecritical temperature for this system. At any  T  *    0.7 thelayering transition is absent, and the condensation in the poreis always connected with an instantaneous filling of the entirepore. Part b of Fig. 3 shows examples of the density profiles at T  *    1. The transition from the gas phase to the liquid phaseoccurs at    b    0.00415. Below the capillary condensationpoint only a thin film adjacent to the pore wall is formed. Inthis system the capillary condensation transition competes nowwith the wetting transition. When we compare the densityprofiles given in Fig. 3b with those in Fig. 2d, we realize an FIG. 2.  The adsorption isotherms (parts a and b) and density profiles (parts c and d) for nonassociating fluid in pores with  H     6. The isotherm in part ais at  T  *    0.7, in part b  at  T  *    1.17 and 1.18. Density profiles in part c are at  T  *    0.7 and for (from bottom to top)    b    0.0009, 0.00091, and 0.001.Density profiles in part d are at  T  *    1 and for (from bottom to top)    b    0.03, 0.06, 0.07, and 0.0921 (two coexisting profiles, gas-like and liquid-like). 95 CAPILLARY CONDENSATION OF ASSOCIATING FLUIDS
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