Molecular Mechanisms of Gas Diffusion in CO 2 Hydrates

Transcription

1 Supportig Iformatio Molecular Mechaisms of Gas Diffusio i CO Hydrates Shuai Liag, * Deqig Liag, Negyou Wu,,3 Lizhi Yi, ad Gaowei Hu,3 Key Laboratory of Gas Hydrate, Guagzhou Istitute of Eergy Coversio, Chiese Academy of Scieces, Guagdog Key Laboratory of New ad Reewable Eergy Research ad Developmet, Guagzhou, Chia Laboratory for Marie Mieral Resources, Qigdao Natioal Laboratory for Marie Sciece ad Techology, Qigdao, Chia 3 The Key Laboratory of Gas Hydrate, Miistry of Lad ad Resources, Qigdao Istitute of Marie Geology, Qigdao, Chia Correspodig Author: liagshuai@ms.giec.ac.c

2 CO trasport through a edge that was shared by two rigs I the sectio, we preset aother example of a CO molecule trasport from a 56 cage to a eighborig 56 cage by passig through the shared 5-membered water rig. At some poit withi this trajectory, two water molecules from oe edge of the origial shared 5-member rig leave their host lattice (Figure S, b ad d). We ca see that oe water molecule was completely off its origial lattice positio ad the other water molecule was also sigificatly deviated from its origial lattice positio. The CO molecule passed through a edge that was shared by the 5-member rig ad its eighborig 6-member rig. Figure S. (a-c) Molecular cofiguratios showig the mechaism of a CO molecule trasport from a 56 cage to a eighborig 56 cage by passig through the shared 5-membered water rig, as observed from a MD trajectory at 30 K. (d) The rotated rig structure at the poit of the CO passig through, showig that oe water molecule was completely off its origial lattice positio ad the other water molecule (circled) sigificatly deviated from its origial lattice positio. The gree dash lies i (d) represet the origial cage structure, where oe water molecule was circled to help compare its origial lattice positio (dashed circle) ad the curret positio (solid circle). The time idexes for each cofiguratio are give i the legeds. The molecular cofiguratios are represeted ad colored as i Figure i the mai text.

3 Details of the calculatio of the kietic parameters I this sectio, we give the details of the calculatio of relevat kietic parameters of the CO diffusio i si hydrates at lower temperatures. We first detect the cage-to-cage hoppig evets by coutig the umber of CO molecules that moved more tha 5 Å i the 0 s simulatios. The radial distributio fuctios (RDF) of CO molecules withi si hydrates are show i Figure S. We ca see the valley betwee the st ad d peaks i the RDF correspods to a CO -CO molecular distace of about 9 Å. Gas molecules that moved more tha half of this distace should be trasported ito aother cage. Therefore 5 Å is a reasoable threshold distace here for trackig the cage-to-cage hoppig of the CO molecules. Figure S. The radial distributio fuctio of CO molecules i si hydrates at 300 K ad 00 MPa. 3

4 Withi the 0 trajectories performed at each temperature, we detected a total of 5, 6, ad 3 hoppig evets at 30, 35, ad 30 K, respectively. This correspods to a hoppig rate of 5 0 7, ad hops/s at 30, 35 ad 30 K. We the used trasitio state theory (TST) to estimate the hoppig rates of the CO molecules at lower temperatures. I TST, the hoppig rates ca be calculated usig the equatio, kt ΔG k = exp h kt () Where k is the hoppig rate, T is the temperature, k is oltzma s costat, h is Plack s costat, ad ΔG is the Gibbs free eergy of activatio. It ca be rewritte as, kt ΔH TΔS k = exp h kt () where ΔH is the ethalpy of activatio, ad ΔS is the etropy of activatio. This equatio ca be further rewritte i a liear form, k ΔH k ΔS l = + l + (3) T k T h k where l is the atural logarithm. Note these three forms of the TST equatio have bee preseted i the mai text. We ca the solve Eq. (3) usig a weighted liear regressio method, θ ( T X WX ) T = X WY (4) where θ is a parameter vector of the slope ad itercept of the liear regressio, X is a matrix of iput sequece /T, ad Y is a matrix of the output sequece l(k/t), ad W is the weight matrix which is determied by the ratio of the umber of hoppig evets observed at this temperature ad the total umber of the observed hoppig evets, 4

5 k ΔS l 30 + h k θ = X = ΔH 35 k l Y = l W = l (5) The weighted liear regressio geerated a ethalpy of activatio (ΔH ) of 38 ± 3 kj/mol ad a etropy of activatio (ΔS ) of 348 ± 0 J/(K mol). The stadard deviatio of the slope, S k, of the regressio is determied usig the equatio, ( y yˆ ) i i i= Sk = x x i= ( i ) (6) ad the stadard deviatio of the itercept, S b, of the regressio is determied usig the equatio, S b ( y ˆ i yi) xi i= i= = x x i= ( ) i (7) where x i represets the idepedet variable ( T i Eq.(3)), x is the mea value of the x i, y i is the depedet variable ( l k T i Eq.(3)), y ˆi is the expected value of y i (here replaced by the fitted value), ad is the umber of data poits for the liear regressio aalysis ( = 3 here). The results of the weighted liear regressio aalysis are preseted i Figure 5 i the mai text. 5

6 Refereces () Truhlar, D. G.; Garrett,. C.; Klippestei, S. J. Curret Status of Trasitio-State Theory. J. Phys. Chem. 996, 00, () Seber, G. A. F.; Lee, A. J. Liear Regressio Aalysis; Joh Wiley & Sos, 0; Vol