SUPPORTING INFORMATION FOR. Mass Spectrometrically-Detected Statistical Aspects of Ligand Populations in Mixed Monolayer Au 25 L 18 Nanoparticles

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1 SUPPORTIG IFORMATIO FOR Mass Sectroetrically-Detected Statistical Asects of Lig Poulations in Mixed Monolayer Au 25 L 8 anoarticles Aala Dass,,a Kennedy Holt, Joseh F. Parer, Stehen W. Feldberg, Royce W. Murray,* Kenan Laboratories of Cheistry, University of orth Carolina, Chael Hill, C An Identical--Indeendent-Site Model for the Kinetics of Lig Exchange Reactions at Gold anoarticles Coarison with Exeriental Data. An abbreviated version of the inetic odel was resented in the ain aer. This is a ore detailed version. The basic assution in this analysis is that each gold nanoarticle (P has identical indeendent sites that are caable of binding a lig. In the resent case of Au 25 L 8 nanoarticles, 8 thiolate lig sites. The binding sites are always occuied by thiolate lig or that is, the exchange rocess is considered to roceed by an associative not a dissociative echanis. For conciseness the ligs are referred to as, where designates the starting lig (i.e., the starting nanoarticle is M designates the incoing lig.. The generalized reaction of interest is: off M M Production of [M occurs only when relaces in the oiety [M or when relaces in [M ; loss of [M occurs only when

2 relaces or relaces in [M. With the added assution that the rate of loss of an (or for a given oiety will be roortional to how any 's (or 's are to that oiety, the following exression obtains: d [M - ( dt ( [ [ [M [M - ( [ ( [ [M off bul off bul bul bul 2 If the bul concentrations of ([ bul [ bul are large enough to be considered constant throughout the course of the reaction, n. 2 becoes d [M dt ( [M ( ( [M off off ( [M - 3 where seudo-first-order rate constants off are defined by: off off [ bul 4 [ bul Siulations of the inetic uilibriu distribution were carried out with double recision quic basic using an exlicit finite difference algorith with the following exressions: 5 (for [M t[m off t[m 6 (for [M t[m off t[m 7 ( for [M ( ( t[m ( t ( t [M off 2 off t[m 8

3 At uilibriu ( ( off ( off [M [M ( [M 9 where the subscrit "' denotes the uilibriu concentration of the subscrited seces. Therefore: [M ( off [M off ( ( [M The uilibriu distribution (as a function of couted iteratively using n. 3 for any chosen values of, off gives exactly the sae distribution as the failiar binoial distribution. The reason for this is easily rationalized. For indeendent identical sites the robability that any given site will be occuied by lig is defined by: off The corresonding robability that the site will be occuied by lig is off off defined by: 2 Cobining ns. -2 gives: [M ( ( [M ( ( ( [M 3 At uilibriu the fraction of any given secies [M exression for the binoial distribution: c P will be given by 3

4 4 4 where P c is the total P concentration: 5 The self-consistency of ns. 3 4 is easily deonstrated. Eqn. 3 is rearranged to give: 6 The ratios [M [M [M [M can be couted with n. 4: 7 8 Cobining ns. 6-8 gives: 9 ( ( [M [M ( [M [M ( (!!( (!!( ( ( [M [M!!( (!!( ( ( [M [M ( ( ( ( ( ( ( ( (!!(! (!!(! [M P c P [M c

5 This identity confirs the uivalence of the uilibriu conditions couted by siulation (ns. 68 by the binoial distribution exression (n. 4. If only the uilibriu condition is of interest, the seudo-first-order ruireent can be relaxed off can be redefined as: off off [ bul, 2 [ bul, 2 where [ bul, [ bul, are the bul concentrations of when uilibriu has been attained. A surrising observation is that an irreversible or quasireversible relaceent rocess ( is also exactly described by the binoial distribution (n. 4 at any tie t if is now defined in ters, the fraction of the total lig resent: [ M c P - θ 22 where θ is the fraction of Ps having the coosition M - θ [ M c P - 23 The non-uilibriu distribution for any given value of can be exactly dulicated (within the liits of coutational accuracy by n. 4. Eqn. 22 is alicable to uilibriu as well as non-uilibriu conditions. At uilibriu (see ns. 2:,, off 24 5

6 Table shows the relative errors indicating the difference in a distribution couted by siulations based on n. 3 a distribution couted with n. 4 where or has been couted fro n. 22. The relative error, ε rel, is ore sensitive to errors in the wings of the distribution is defined by: ε rel [ M [ M siulation [ M cobinator ial cobinatorial 25 For 8, 6 off t t, ε rel is given for 6 (colun #, 2 (colun #4, 56 (colun #5. Colun #2 siulated data were couted with off 5 colun #3 siulated data were couted with 6 off t 6 t with other araeters the sae as for colun. The values of the rel ε in coluns #2 are about ten ties as large as the values in colun # indicating that the ε rel aroxiately roortional to off t. This relationshi can vary with the design of the algorith used for the nuerical siulation for exale the coutational error could be reduced ( coutational efficiency enhanced by using an ilicit instead exlicit finite difference algorith; however the code would be ore colicated its rearation ore tie consuing. The accuracy seed of the exlicit algorith were sufficient for the resent uroses. The iortant observation is that ε rel values are very sall over the entire distribution strongly indicating that the correlation is atheatical not aroxiate or fortuitous. While this suggests that there is a straightforward atheatical roof, we have not found it. Also resented in the Table are the noralized tie, t, ruired to ( off achieve the secified values of. The oerative rate exression is a first-order rocess (assuing seudo-first-order conditions where [ [, the concentrations of the ligs, are defined by: [ [ M

7 [ ( [ M - 27 Then: d [ dt off [ [ 28 7

8 Table : Accuracy of the Binoial Distribution as a Descritor of Irreversible uasi-reversible Lig Exchange Distribution. Relative error: ([M - siulation [M- cobiniatorial [M- cobinator ial # #2 #3 #4 #5 8 ( off off t t t 8 ( off off t t t 8 ( off off t t t 8 ( off off t t t 8 ( off off t t t -6.49e-4-6.5e e-4 -.3e e e e e e e e e e e-5.6e e e e-4-4.7e e e e e e e e e e e e e e e e e e e e-4.38e-6 -.6e e e e e e e e e e-6-4.3e e e e e e e-5 -.3e e e e e e e e e-6.68e e e e e-5.272e e-6-3.7e e e e e e e e e-5 -.3e-6-6.4e-5-2.3e e e e e e e e e-5 -.9e e-4 8

9 Defining, the fraction of lig reaining at any tie (see n. 22: [ [ M - cmpc cmpc 29 n. 28 becoes: d dt off off ( ( off 3 The solution to this differential uation is well nown: ( ln ln ( off, off off, t 3 where, off 32 Results of siulations lotted as ln[ ( ( vs ( t for a range,, of values of K ( are in exact agreeent with n. 3., off off The Inverse Proble: Fitting Exeriental data. The analysis has shown that the shae of the distribution can be deterined by the binoial distribution given in n. 4. It does not atter if the syste is at uilibriu is defined by n. or 2 or if the syste is not at uilibriu is defined by n. 22 (a definition that is also valid for the uilibriu condition. The exeriental distribution is reresented by alitudes A,xtl (in arbitrary units reresenting the fraction of Ps with ligs. These data can be renoralized (see ns to define θ,xtl : θ,xtl A A,xtl,xtl 33 9

10 Then (see n. 22, xtl θ,xtl 34 In rincile, the value of is nown the distribution, couted fro the binoial exression (see n. 4: θ, theory, is straightforwardly θ,theory,xtl,xtl! (!(!,xtl,xtl!!(! 35 ote that there are no adjustable araeters ruired for the generation of θ, theory. Less obvious is that the distribution θ (for a given value of deends only on the value of but does not deend on the teoral constancy in the values of off (see ns. 4 5 in other words the rate rocesses do not have to be seudo-first order. This was easily confired by siulation. However, rate exressions describing the tie deendence of the relaceent reactions will be greatly colicated if seudo-first order conditions do not aly.

11 Figure S-: H MR sectru of Au 25 (SCH 2 CH 2 Ph x (S(CH 2 5 CH 3 y as reared using a 5:5 ixture of henylethanethiol hexanethiol in the Brust reaction. The sectru was obtained in ethylene chloride-d 2 using a Bruer 4 MHz widebore sectroeter at 3 K. The integration of the henyl rotons of the henylethanethiolate were coared with those of the terinal ethyl rotons of the hexanethiolate. The terinal ethyl roton resonances of hexanethiolate slightly overla those of the tetraoctylaoniu counterion, so only the right half of the ea was integrated. The area of the half-ea was ultilied by two to estiate the total integration a henyl:ethyl ratio of.:.42, which is indicative of an average lig coosition of.6 henylethanethiolates 7.4 hexanethiolates er P

12 Figure S-2. Reroduced fro Ref. e Figure 6. MALDI-TOF-MS sectra (Alied Biosystes Voyager-DE Pro of Au 25 (SCH 2 CH 2 Ph 8 in DCTB atrix with varying laser intensity delineating the olecular ions fro fragent ions in ositive negative linear ode. Soe sectra here are clied at the to; see Suorting Inforation (of Ref e for colete sectra exales. 2