StrMctermg Effects i n Biiisiry NecleatieMo
l ^ / i i o i i e f M i a r
By
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StmESaüoiüis
Coarse-Grained Nucleation Theory
Stephan Braim", Thomas Kraska'' and Vitaly Kalikmanov^'*^
"Institute for Physical Chemistry, University of Cologne, D-50939 Cologne, Germany ''Twister Supersonic Gas Solutions BV, Einsteinlaan 20, 2289 CC, Rijswijk, Netherlands '^Department of Geosciences, Delft Universit}', Stevinweg 1, 2628 CN, Delft, NetherlandsAbstract. Binaiy clusters formed by vapor-liquid nucleation are frequently nonhomogeneous objects in which components are not well mixed. The stmcture of a cluster plays an impor-tant role in nucleation and cluster growth. We demonstrate stnicturing effects by studying high-pressure nucleation and cluster growth in the n-nonane/niethane mixture by means of molecular dynamics simulation. It is found that methane is squeezed out from the cluster core which is rich with n-nonane molecules. At typical simulation conditions - pressure 60 bar, temperature 240 K, nucleation rate J ~ 10^^cm~^s~' - the mole fraction of methane in the critical cluster reaches 80%, being much higher than its equilibrium bulk liquid fraction at the same pressure and temperahire. These observations are supported by the recently formulated coarse-grained theory of binary nucleation.
Keywords: binaiy nucleation, adsorption, molecular dynamics, coarse-graining PACS: 64.60.Q-, 82.60.Nh, 05.20.Jj, 61.20.Ja
l i N T R O D I I C T I O N
Nonuniform stmcture o f nuclei fomied in the process o f binary nucleation i n a variety o f m i x -tures plays an important role i n nucleation and cluster growth. U p to date the main theoretical tool describing binaiy nucleation is the binaiy classical nucleation theory ( B C N T ) [1] which con-siders a cluster to be an object w i t h a rigid boundary and uniform intensive bullc liquid properties inside it. Meantime, it has been known for quite a while that predictions o f B C N T can differ by many orders o f magnitude ü'om the experimental data and even lead to unphysical results v i o -lating the nucleation theorem (see e.g. [2] and references therein). One o f the explanations o f this discrepancy is the inability o f B C N T to take into account adsoiption effects resulting i n a nonuniform cluster stmcture. Wilemski [3] proposed the revised B C N T based on the combination o f capillarity approximation w i t h the Gibbs concept o f dividing surface. For a binary (and more generally, a multi-component) system the Gibbs construction is not just a usefiil tool (as i n the single-component case) but a necessity, since it is impossible to choose a dividing surface f o r a mixture i n such a way that adsoiption o f all species on it vanish. Taking into account adsorption
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43
within the phenomenological approach does not resolve another deficiency o f the capillarity ap-proximation: the latter assumes that the surface energy o f a cluster o f any size can be described i n terms o f the plain layer surface tension. Obviously, for small clusters this concept loses its meaning.
In the present study we analyze stnicturing and size effects i n binaiy nucleation using molecu-lar dynamics simulation and theoretical considerations based on the recently foimulated coarse-grained theoiy of binaiy nucleation [2, 4 ] . The system under study is the mixture o f n-nonane and methane at high pressures.
S I M U L A T I O N
To simulate the binaiy mixtiu'e o f n-nonane/methane we employ an interaction potential based on the Transferable Potentials for Pliase Eqitilibriiiin - United Atom model developed by Siep-mann et al. [ 5 ] , in which the n-nonane molecule has 9 Lennard-Jones sites each representing a
CH2 or CH2 group. It has a fixed bond length, hannonic angle bending potentials and dihedral
torsion potentials. Simulations are perfonned using the Moscito [6] software package. To induce supersaturation we employ the eariier developed approach based on the expansion o f the system utilizing the Joule-Thompson effect [7]. Expansion is simulated via a stepwise enlargement o f the simulation box with short equilibration mns i n between. The box is expanded by approximately 0.1 n m i n all three dimensions until the desired density and pressure are reached (usually around 100 expansion steps are sufficient).
The n-nonane/methane mixture is characterized by a very low vapor molar fraction o f n-nonane, ~ 10"'', so that the methane vapor fraction is w 1 while its fraction in the liquid phase at the given pressure p and temperature T varies between 0.05 and 0.45 - see Fig. 1. Calculations were performed w i t h approximately 10^ methane and 343 n-nonane molecules. The expansion starts at a supercritical density p{CHjs) = 25.9mol/dm^, 300 K and supercritical pressure o f 15-20 MPa. To detect clusters we use the Slillinger distance criterion combined w i t h a life-time criterion [ 7 ] : two molecules belong to the cluster i f their distance is less than 1.5(7^^ (where (J^^ is the size o f the n-nonane molecule) over the period of 2.5 ps. This is a residence time o f a molecule i n the Stillinger sphere and is larger than the time between collisions o f two molecules. Nucleation rates are deteimined using the Yasiioka-Matsitmoto method (for a detailed discussion see R e f [ 2 ] , Ch. 8) are found to be: J ~ lO^^ - lO^^cm-^s"'.
The question is: how the clusters, and especially the critical cluster, are stmctured. So far we are aware o f only one experimental data on the stmcture o f water/butanol nano-droplets [9]. These droplets exhibit a core-shell stmcture with the less volatile component (water) i n the core and the more volatile one (butanol) on the surface. I n this sense our simulation results are consistent w i t h the experimental findings, though f o r a different system [10]. Figure 2(a) shows a typical snapshot o f a cluster Its core consists o f n-nonane (gi'een) sunounded by methane molecules (blue) located mainly on the outer shell. The cluster shown i n F i g . 2(a) is beyond the critical size, but the critical clusters look similar The reason for this core-shell stmcture is likely the minimization of the surface energy. This has a straightforward impact on the modeling o f binary nucleation since the surface tension o f the binary cluster is lower than that o f the homogenous binaiy mixture i n the bulk phase due to cluster stnicturing. Effectively, a lower surface energy leads to a higher
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methane/n-nonane, corn: P C - S A F T
0.0 0.2 0.4 0.6 0.8 1.0 x(methane)
F I G U R E 1: Phase diagram of n-nonane/methane system at js—.T(methane) plane. Labels are the tem-peratures of the corresponding isotherms. Small cir-cles: MD simulations; curves are con-elations with PC-SAFT equation of state [8]. The circle labeled "MD 240 K" shows the methane fraction in llie
criti-cal cluster at p = 60 bar, T = 240 K and nucleation
rate J ~ lO^^cm^-'s"' found in M D simulations. Hor-izontal arrow points at the equilibrium liquid frac-tion of methane at approximately the same p and T:
x^^ [p = 60bar, T = 248.15K) Ri 0.36.
nucleation rate. Hence, nucleation theories using the bulk surface tension are expected to predict too l o w nucleation rates at least because o f cluster structuring.
In the phase diagram Fig. 1 we indicate the methane fraction in the critical cluster (green circle), for the nucleation conditions: p = 60 bar, T = 240 K , J 10^''cm~-^s~' which appears to be w 0.8 whereas the corresponding equilibrium molar fraction at these p and T is much lower:
Xcm,eq ~ 0.36.
THEORY
For theoretical study o f the effects of adsorption we consider an arbiti'ary cluster o f a (n-nonane) and b (methane) molecules with the bulk content {n\, n\) molecules. The bulk composition is characterized by the znolar fraction o f the more volatile component (b): x\. Within the general foimalism o f Gibbs dividing surface we present the total number o f molecules o f each component as «; = n] + nj'"^, = a,b. The excess numbers n^'^'^ are found by choosing the eqidinolar surface
for the mixture, known also as the K-surface [11], which is defined through: E;=fl,6 = 0,
where v} is the partial molecular volume o f component; i n the liquid phase. Its combination w i t h the Gibbs adsorption equation yields the excess numbers n^'^,nf''^. The Gibbs fi-ee energy o f cluster fomiation g = AG/ICBT (/CB is the Boltzmann constant) is a fimction o f total numbers as w e l l as the bulk composition x l [2, 4 ] :
,>u;xb)
= - X
ln'5
'/-X"'
Ini=a,b i
+ 0micro,Jn^„-l] (1)
g«i(na,'Urx\)
Here the first teim is the nonequilibriuni part o f g containing the supersaturations S; o f compo-nents. The equilibrium part geq consists o f the bulk and surface contributions. The bulk contribu-tion (second term i n (1)) reflects the difference i n the bulk composicontribu-tion between the given cluster and the bulk liquid at p and T; p™'^'^{x\, T) is the total pressure above the bulk binary solution w i t h the composition A|,;'™'"'(x[,,r) is the coiresponding vapor fi-action. The surface contribution (last term i n (1)) is obtained using coarse-graining o f the configuration integral o f the binaiy (na,nb)-cluster It is based on tracing out the degrees o f freedom o f the more volatile component (b).
45
0.0
log,„ J (cm'^E-'')
a b
F I G U R E 2: Clii.ster structure, (a) Snapshot o f a big cluster in M D simulations. The core consists mainly of n-nonane (green) while methane is located mainly on the outer shell, (b) Methane fraction in the critical cluster at /; = 60 bar, T = 240 K at various nucleation rates predicted by CGNT [4] (open circles). Also shown is the result of M D simulations (closed circle). Arrow points at egi/ilibriiim methane liquid fraction:
•TCfl4,eq ~ 0.36
As a result one is left w i t h the configuration integral o f the single-component cluster o f pseudo-o molecules with the interaction energy implicitly depending on the fraction o f ö-molecules i n the original binary cluster This 77a-cluster is treated by means o f the mean-field kinetic nucleation theory [2]:
0micro,n(4)
the free energy per surface particle, « ^ ( 7 ? a ; x j ) is the number o f surface particles in the «^-cluster Critical cluster corresponds to the saddle point o f g i n the space o ftotal numbers n„, « i (for details see R e f [2]). Predictions o f this model, teimed Coarse-Grained Nucleation Titeory (CGNT), f o r the methane fraction i n the critical cluster, x^^^^, at = 60 bar,
T = 240 K and various nucleation rates are shown i n Fig. 2(b). A t high rates J ~ 10^''cm~^s~'
'^CHi consistent w i t h the values found i n our M D simulations.
R E F E R E N C E S
[ I ] H. Reiss, J. Chera. Phys. 18, 840 (1950); D. Stauffer, I Aerosol Sci. 7, 319 (1976). [2] V.I. Kalikmanov, A'i/cfefl/'/o;; theoiy. Springer, Dordrecht, 2013.
[3] G. Wilemski, J. Chem. Phys. 80, 1370 (1984); J. Phys. Chem. 91, 2492 (1987). [4] V.LKalikmanov, Phys. Rev. E 81, 050601(R) (2010).
[5] M.G. Martin, J.I. Siepmann, J. Phys.Chem. B 102, 2569 (1998).
[6] D. Paschek and A. Geiger, MOSCITO 4, Department of Physical Chemistiy, University of Dortmund, 2002.
[7] R. Römer and T. Ki-aska, J. Phys. Chem. C 113, 19028 (2009). [8] J. Gross, G. Sadowski, Ind. Eng. Chem. Res. 40,1244111260 (2001).
[9] B. E.Wyslouzil, G.Wilemski, R. Strey et al., Phys. Chem. Chem. Phys. 8, 54 (2006). [10] S. Braun and T. Kraska, J. Chem. Phys. 136, 214506 (2012).
[ I I ] A . Laaksonen, R. McGraw and H . Vehkamaki, J. Chem. Phys., I l l , 2019 (1999).
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structuring effects in binasy nucleation: Molecular dynamics simuli
and coarse-grained nucleation theory
s t e p h a n Braun, Thomas Kraska, and Vitaly Kalikmanov
Citation: AIP Conf. Proc. 1527, 4 3 (2013); doi: 10.1063/1.4803200 V i e w online: http://dx.d0i.0rg/l0.1063/1.4803200
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