A NOVEL DENSITY FUNCTIONAL THEORY M O D E L I N G o n S W E L L I N G B E N T O N I T E : INTERACTION FORCES and ION EXCHANGE Guomin Yang (guomin@kth.se), Ivars Neretnieks, Susanna Wold Department of Chemical Engineering and Technology and Department of Chemistry, Royal Institute of Technology (KTH), S-100 44 Stockholm, Sweden

Background Theoretical studies of the interaction forces and ion exchange play an important role in explaining the mechanisms governing the swelling of bentonite. To that end, a novel density functional theory (DFT) approach1 of a planar electrical double layer with the primitive model is applied to calculate the interaction forces between smectite particles (montmorillonite) and the Gaines-Thomas selectivity coefficient2,3 of the Ca\Na ion exchange equilibrium.

Theory

Conclusions •  The novel DFT approach that is robust enough to be applied to model the swelling behaviour at atomic scale and ion exchange equilibrium of montmorillonite clay. • The mole fraction in bulk, surface charge density, and ionic diameter play a significant role on swelling of bentonite when saturated with groundwater.

Results

ρi (r) = ρib(r)exp −β(zieψ(r)+Δui

hs (r)+Δci(1),hs (r)+Δci

(1),el (r){ }

ci(1),el (r;ρi ) = ci

(1),el (r;ρiRFD )+ ds ρ j (s)− ρ j

RFD(s)( )cij(2),el∫j∑ (r, s;ρi

*)

cij(2),el (r, s;ρi ) = cij

(2),el r;ρi ( ʹr )( )ωij ( ʹr )d∫ ʹr ωij ( ʹr ) =κ 2 ( ʹr )Θ(| ʹr − rm |−dij )d ʹr κ 2 ( ʹr )Θ(| ʹr − rm |−dij )∫

Posmnet = kBT ρ j

j∑ (dj / 2)−

σ 2

2ε0εr−Posm

bulk

Γi = (ρi (x)0

∞

∫ − ρib )dx ezj

j∑ Γj = −2σ

12Ca2+ + Na-X↔ 1

2Ca-X2 + Na

+

KGT =[Na+ ][Ca-X2 ]

0.5

[Ca2+ ]0.5[Na-X]=[Na+ ]βCa

0.5

[Ca2+ ]0.5βNa

Figure 1. Net osmotic pressure as a function of separation. d = 0.4 nm, σ = -0.14 C/m2, CNa = 100 mM, CCa

is varied as indicated in the graph. The symbols are Monte Carlo data4. The curves show the DFT calculations.

Figure 2. The DFT calculations of net osmotic pressure as a function of separation. σ = -0.14 C/m2, CNa = 100 mM, CCa = 5 mM. The monovalent ion size is kept constant at 0.4 nm, while the divalent ion size is varied as indicated in the plot. εr = 78, T = 298 K

Table 1. Comparison of the selectivity coefficient from DFT calculations and experimental results6 for calcium-sodium exchange in Wyoming montmorillonite under confined conditions. σ = -0.11 C/m2, εr = 78, T = 298 K. ISE and ICP/AES indicates the methods used to determine the ion concentrations.

References (1)  G. Yang and L. Liu. J. Chem. Phys. 142, 194110 (2015). (2)  G. L. Gaines Jr.1 and H. C. Thomas. J. Chem. Phys. 21, 714 (1953) (3)  C. A. J. Appelo and D. Postma. Geochemistry, Groundwater and Pollution, Second Edition. Taylor & Francis 2005. Pages 241–309. (4)  M. Segad, B Jonsson, T. Åkesson, B. Cabane. Langmuir. 26, 5782–5790 (2010). (5) Henderson D, Blum L, Lebowitz JL. J Electroanal Chem. 102:315-319. 1979. (6)  M. Birgersson, L. Börgesson, M. Hedström, O. Karnland, and U. Nilsson. Bentonite erosion. SKB Technical Report TR-09-34. (Swedish Nuclear Fuel and Waste

Management Company, Stockholm, Sweden, 2009).

Samples CNa (mM)

CCa (mM)

DFT

KGT ( M)

Experiments

KGT ( M) Average interlayer

separation h h =

10 Å h =

15 Å h =

20 Å ISE ICP/AES

WyNa 01 29.3 2.0 5.7 3.8 3.3 2.1 2.0 18.7

WyNa 02 35.8 2.1 5.7 3.8 3.3 2.8 2.3 12.2

WyNa 03 47.9 4.5 5.6 3.9 3.4 2.8 2.0 8.1

WyCa 04 28.4 3.1 5.7 3.9 3.4 1.3 2.6 16.4

WyCa 05 41.6 5.3 5.7 3.9 3.4 2.1 2.8 11.5

WyCa 06 48.2 6.9 5.7 3.9 3.4 2.5 2.6 8.3

Samples CNa (mM)

CCa (mM) CNa/(CNa+CCa)

DFTKGT ( M)

Experiments

KGT ( M) Average interlayer separation

(Å) h = 15.0 Å h = 20.0 Å ICP/AES ISE

WyNa 01 29.3 2.0 93.6% 2.6 2.0 2.0 2.1 18.7

WyNa 02 35.8 2.1 94.5% 2.6 2.0 2.3 2.8 12.2

WyNa 03 47.9 4.5 91.4% 2.6 2.1 1.9 2.8 8.1

WyCa 04 28.4 3.1 90.2% 2.6 2.1 2.6 1.3 16.4

WyCa 05 41.6 5.3 88.7% 2.6 2.1 2.8 2.0 11.5

WyCa 06 48.2 6.9 87.5% 2.6 2.1 2.6 2.5 8.3

dmono = ddi = 0.425 nm dmono = 0.425 nm, ddi = 0.72 nm

CNa (mM)

CCa (mM)

mole fraction

47.9 1.0 97.9%

26.8 11.6 69.7%

62.5 21.9 22.2%

Figure 1. DFT Selectivity coefficient KGT for WyCa as a function of the separations. The sodium mole fraction in the bulk is indicated in the plot. d = 0.425 nm, εr = 78, T = 298 K.

Figure 2. DFT Selectivity coefficient KGT for WyNa as a function of the separations. The calcium ionic diameter ddi is indicated in the plot. dmono = 0.425 nm, εr = 78, T = 298 K. CNa=29.3 mM, CCa=2 mM.

1.   DFT Theory

2.   Applications of DFT modeling Net osmotic pressure:5

Surface excess:

Ca\Na Ion Exchange reaction:

Gaines-Thomas selectivity coefficient: