Small Saccharides as a Blanket around Proteins: A Computational

Publication Date (Web): June 27, 2018. Copyright © 2018 ... We find that trehalose serves as an excellent molecular insulator over a wide range of te...
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Small Saccharides as a Blanket around Proteins: A Computational Study Hari Datt Pandey and David M. Leitner* Department of Chemistry, University of Nevada, Reno, Nevada 89557, United States

J. Phys. Chem. B 2018.122:7277-7285. Downloaded from pubs.acs.org by ST FRANCIS XAVIER UNIV on 08/18/18. For personal use only.

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ABSTRACT: Saccharides stabilize proteins exposed to thermal fluctuations and stresses. While the effect of a layer of trehalose around a protein on the melting temperature has been well studied, its role as a thermal insulator remains unclear. We report calculations of thermalization in small saccharides, including glucose, galactose, lactose, and trehalose, and thermal transport through a trehalose layer between water and protein and between gold, such as a gold nanoparticle, and its cellular environment. The thermalization rates calculated for the saccharides provide information about the scope of applicability of approaches that can be used to predict thermal conduction in these systems, specifically where Fourier’s law breaks down and where a Landauer approach is suitable. We find that trehalose serves as an excellent molecular insulator over a wide range of temperatures.

1. INTRODUCTION Recent interest in thermal transport through molecular interfaces and junctions1−9 has been motivated in part due its critical role in technological applications, e.g., avoiding high concentrations of heat in small devices,10,11 thermoelectric applications,12 the possibility of thermal rectification at the nanoscale,3,6,13 and the contribution of thermal gradients to electron transfer.14−16 Thermal transport at the molecular level also plays a role in processes in the cell, e.g., the response of proteins to optical heating of functionalized gold nanoparticles (GNPs)17−19 and stabilization of proteins by saccharides against thermal stress.20,21 Predicting thermal transport in molecules remains challenging as there is no one picture, such as Fourier’s heat law in macroscopic systems, which we can apply as a framework to address this problem. The relative contributions of contact with leads,22,23 elastic scattering, controlled by molecular structure, and inelastic scattering, which gives rise to thermalization and is mediated by anharmonic coupling within the molecule and coupling with the leads, vary considerably with the system.24−26 These properties all control thermal transport through molecules, and different methods for calculating thermal conduction through the interface account for some of these factors more readily than others. We therefore address the mechanism of thermal transport as we quantify thermal conduction through the junction. In this work, we focus on thermal transport in small saccharides and its role as a thermal insulator around proteins in an aqueous environment and near GNPs. We examine thermalization through two monosaccharides, glucose and galactose, and two disaccharides, lactose and © 2018 American Chemical Society

trehalose. We also consider two structures of the monosaccharides, linear and cyclic, to examine how these structures might influence thermalization in small saccharides. After addressing thermalization in saccharides and its dependence on structure and size, we considered thermal conduction through trehalose, which is well-known to stabilize proteins undergoing thermal stress.20,21 For example, lysozyme protected by trehalose retains one-third of its activity when heated to 90 °C, compared to 18% for just wild-type lysozyme and over 80% when conjugated with trehalose glycopolymers.20 Long-range dynamic coupling between disaccharides and water has been measured,27,28 but little is known about thermal transport through saccharides in an aqueous environment. We therefore consider thermal conduction between water and a protein through a trehalose monolayer (Figure 1). Since there is also great interest in application of GNPs to controlled denaturation of proteins,17−19 we also address how trehalose acts as an insulator between gold and a cellular environment. Overall, we find saccharides to serve as excellent insulators compared to other molecules we have examined in the past. Contributions to thermal conduction through an interface due to bonding between the two materials and the molecules that bridge them have been examined in previous studies. For example, thermal conduction through ∼1 nm alkane chains bridging gold and quartz has been found experimentally to be diminished by almost a factor of 2, from 65 to 36 MW m−2 K−1 at Received: May 15, 2018 Revised: June 25, 2018 Published: June 27, 2018 7277

DOI: 10.1021/acs.jpcb.8b04632 J. Phys. Chem. B 2018, 122, 7277−7285

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The Journal of Physical Chemistry B

vibrational quasiparticles, i.e., the vibrational excitations of the molecule akin to phonons in a solid, which carry heat. When thermalization occurs rapidly, however, alternative approaches are needed. If local temperature is well-defined at the atomic scale, we can sketch a temperature profile between T1 and T2, where for T1 > T2 we would observe a drop in temperature from one lead to the molecule, across the molecule, and again from the molecule to the other lead. The boundary resistance between the leads is the sum of these resistances in series. Since temperature can always be defined as proportional to the mean square atomic velocity in a classical system, this kind of a temperature profile can be calculated using data from a classical molecular dynamics (MD) simulation. Classical MD simulations, such as reverse nonequilibrium MD simulations,9,34−36 are widely used to estimate thermal conduction through molecular interfaces. Some studies have compared results of calculations of thermal conductance in molecular junctions using a Landauer picture with the results of MD simulations as a check on the assumption that, over the length of the junction, thermalization effects on thermal conduction in the molecule are likely small. For example, a recent computational study using both a nonequilibrium Green’s function approach32 and nonequilibrium MD simulations of thermal transport in a polyethylene junction33 found good agreement between the two calculations over a wide range of temperature, including room temperature, suggesting little contribution of thermalization. However, MD simulations have in other studies been found to exaggerate the role of anharmonicity and thermalization,29 as is apparently also the case at lower temperature for polyethylene.33 We thus prefer to calculate the rate of thermalization quantum mechanically as a check on the validity of a Landauer-like picture, where diffusive transport and thermal resistance may still arise by elastic scattering. In the following section, our approach toward calculation of thermalization rates in molecules and thermal boundary conductance is presented, as are the structures and anharmonic constants of the saccharides. Results of calculations of thermalization in saccharides and thermal conduction between water and protein and gold and water via a saccharide layer are presented and discussed in section 3, followed by conclusions in section 4.

Figure 1. (Top) Two objects at different temperature, T1 and T2, with a molecular interface made up of saccharides. In this illustration, one object is water and the other a protein, surrounded by a layer of trehalose. (Bottom) Thermalization is calculated in the monosaccharides glucose (cyclic and linear) and galactose (cyclic and linear) and the disaccharides lactose and trehalose.

room temperature, when the chemical bond to the gold via an SH end-group is replaced by a van der Waals contact via a methyl end-group.23 Additional thermal resistance at the interface may arise from elastic and inelastic scattering within the molecules themselves. It is this latter contribution that we consider in this study, where our focus is the thermal resistance in saccharides introduced by molecular structure and anharmonicity. The nature of the bond between the saccharide and the materials it bridges may also influence thermal conduction through the interface, but we assume that it is sufficiently strong that this contribution to resistance is relatively small. Thus, the insulation provided by the saccharide is no smaller than what we report here and may be slightly greater if coupling between the saccharide and the substrates is sufficiently weak. The role of thermalization within the molecules at the interface in thermal transport through molecules has recently also been been examined, with a focus on alkane chains, fluorinated alkanes,24,26 and polyethylene glycol (PEG) oligomers.25 If thermalization occurs sufficiently slowly, thermal conduction may be calculated adopting a quantum mechanical approach based on the Landauer formalism, which neglects effects of inelastic scattering.2,29,30 The Landauer approach was introduced to quantify electrical conductance in mesoscopic systems1,31 and can be applied to calculate thermal conduction through molecules between two leads at different temperatures.2,32,33 It can incorporate effects of elastic scattering of

2. THEORETICAL AND COMPUTATIONAL METHODS Thermalization in a molecule mediates the resistance and its temperature dependence in the junction. In recent work, we have developed and detailed an approach to determine the rate of thermalization in molecules and the length over which it occurs.24−26 Here we summarize that approach. A starting point to estimate the rate of thermalization is Fermi’s golden rule (GR). However, for the GR estimate to be valid, the product of the local density of vibrational states and the matrix elements that couple them needs to be significantly larger than 1. When this condition is not met a more general approach is needed,37 which we summarize here. We find it convenient to address thermalization in two steps, starting with processes within the molecule itself, followed by contributions of coupling between the molecule and its environment. The vibrational Hamiltonian of the isolated molecule is H = H0 + V, where the zero-order Hamiltonian, H0, is expressed as a sum of the energy in each of the N N vibrational modes of the molecules, i.e., H0 = ∑α = 1 εα , where εα 7278

DOI: 10.1021/acs.jpcb.8b04632 J. Phys. Chem. B 2018, 122, 7277−7285

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The Journal of Physical Chemistry B is the energy of mode α. The vibrational modes of the molecule are coupled by anharmonic interactions, V. The zero-order states are product states labeled by the number of vibrational quanta in each mode. The coupling, V, includes terms to third order in the anharmonicity; higher order anharmonic coupling could be included but these terms typically decrease exponentially in magnitude38,39 so we neglect them. If the molecule, H, is initially excited to one of the eigenstates of H0, energy may be redistributed due to the anharmonic interactions, rearranging the population of the vibrational modes until the molecule has equilibrated. For an isolated molecule, H, however, thermalization may not occur,37,40−42 regardless of size, an example of a many-body localized (MBL) system.43−46 Localization in the vibrational state space of an isolated molecule occurs when the product of the average anharmonic matrix element and the local density of states coupled by anharmonicity is less than of order 1. Defining the transition parameter as T (note the distinction in notation between T and temperature, T), the criterion for localization is 37 2π T (E) ≡ 3 ⟨|Vαβγ |⟩2 ρ2 (E) < 1, where ⟨|Vαβγ|⟩ is the average size of the matrix element coupling the triple of modes α, β, and γ and ρ is the local density of states coupled by the cubic anharmonic interaction. This localization criterion is the result of a self-consistent analysis of the distribution of rates at which vibrational states on the energy shell relax toward equilibrium. The most probable rate in a large molecule is always 0 when the transition parameter is less than 1, T < 1, despite finite coupling between vibrational states due to anharmonic interactions. Though localization only occurs in isolated molecules, when the molecule is not isolated the most probable rate at which excess vibrational energy in a mode relaxes toward equilibrium, which we refer to as the relaxation rate, nevertheless remains in practice slow if T < 1. Interactions between vibrational states may be separated into two relaxation processes that can occur while conserving energy. In one process, decay, a vibrational excitation in a mode decays into two of frequency ωβ and ωγ. In the other process, collision, a vibrational excitation in a mode of frequency ωα combines with another of frequency ωβ to create a vibrational excitation in a mode of frequency ωγ. One then has T = Td + Tc, where24−26 2π 2 Td(ωα) = ⟨|Φαβγ |nα(nβ + 1)(nγ + 1)⟩2 ρres (ωα) 3

Tc(ωα) =

2π 2 ⟨|Φαβγ |nαnβ (nγ + 1)⟩2 ρres (ωα) 3

Wd(ωα) =

∑

|Φαβγ |2 (1 + nβ + nγ ) ωβ ωγ

β ,γ

(Γα + Γβ + Γγ) (ωα − ωβ − ωγ )2 + (Γα + Γβ + Γγ)2 /4 Wc(ωα) =

(3a)

ℏ 8ωα

∑

|Φαβγ |2 (nβ − nγ ) ωβ ωγ

β ,γ

(Γα + Γβ + Γγ) (ωα + ωβ − ωγ )2 + (Γα + Γβ + Γγ)2 /4

(3b)

Here, Γβ is the rate at which a vibrational excitation in mode β relaxes due to coupling to other modes and to the environment. For most vibrational modes of large molecules Γβ is of the order 1 ps−1,49 as will be seen for the saccharides below. For a large molecule often (ωα − ωβ − ωγ)2 < (Γα + Γβ + Γγ)2/4, i.e., the spacing between resonances is typically much smaller than the line width, and in practice, the GR rate does not depend much on the value of Γβ. A value of 1 ps−1 has been used for each Γβ, and as in previous calculations,47,48,50 we have checked that the GR value remains essentially unchanged upon varying it. However, for some molecules this condition may not be met, as we now consider. The GR is valid when the transition parameter, T, is substantially greater than 1.37 Close to the localization threshold the average rate is greater than the most probable rate; they approach each other only at large T. In general, above the localization threshold, the most probable relaxation rate becomes37 Wdmp(ωα) = Wd(ωα)(1 − T −1(ωα))1/2 ,

T≥1

(4a)

Wcmp(ωα) = Wc(ωα)(1 − T −1(ωα))1/2 ,

T≥1

(4b)

The thermal average is

(1a)

W (T) = Q−1 ∑ (Wcmp(ωα) + Wdmp(ωα))e−ℏωα / kBT α

(1b)

(5)

Estimates for the relaxation rates in molecules that form a thermal bridge between two substrates when T > 1 are given by eqs 4 and 5. The molecules are of course not isolated; they are in contact with thermal baths. We account for energy transfer between the thermal baths and the molecules bridging them by assuming vibrational quasiparticles enter and leave the molecule in steady state. MBL is lost with dephasing,37 which occurs due to coupling to the thermal baths. The time for a vibrational quasiparticle to traverse the junction is mediated by the diffusion time due to elastic scattering (see below). We calculate the dephasing rate, η, using the transport time calculated as L2/2D, where L is the length of the junction and D is the energy diffusion coefficient in the absence of inelastic scattering, discsused below. Approaches to calculate effects of dephasing on thermalization within molecules when T < 1, as well as the rate of thermalization, in practice small, are given by eqs S1 and S2.51

Φαβγ is the coefficient of the cubic terms in the expansion of the interatomic potential in normal coordinates, and for nα we use ⟨nα⟩, where ⟨nα⟩ = (eℏωα / kBT − 1)−1. Averaging over the modes at a given temperature (internal energy), the transition parameter, T, at temperature T, is given by24−26 T (T) = Q−1 ∑ T (ωα)e−ℏωα / kBT α

ℏ 16ωα

(2)

where Q is the partition function. The GR provides an estimate to the average relaxation rate for excess energy in a mode. There are then decay and collision contributions to the decay rate, with rates Wd and Wc, respectively. For a vibrational mode α with frequency ωα and excess vibrational excitation the average rate can be written as Wavg = Wd + Wc, where47,48 7279

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The Journal of Physical Chemistry B The thermal boundary conductance, hBd, between two materials, the inverse of the thermal boundary resistance or Kapitza resistance,52 is expressed as the ratio of the rate of heat transfer, Q̇ , to the temperature difference, ΔT, across the boundary and the area of the interface, A hBd =

Q̇ AΔT

each molecule, analogous to the treatment of the mean free path of an electron in a conductor.31 The thermal boundary conductance may be calculated more directly within the Landauer method, without separately estimating the mean free path in harmonic approximation, using, e.g., nonequilibrium Green’s function approaches.55−57 We note that if thermalization leads to well-defined local temperature we can instead calculate the thermal boundary resistance across the molecular interface using a series model, 1/ hBd = 1/hL‑mol + 1/hR‑mol + 1/hmol. The boundary resistance Bd Bd within the molecule is 1/hmol, and 1/hL‑mol and 1/hR‑mol are, Bd Bd respectively, the boundary resistance between the left substrate and the molecule and right substrate and molecule. An approach to calculating these terms has been given previously.25 We note only that these quantities depend to some extent on the crosssectional area of contact between the molecule and heat bath, which is somewhat arbitrary,58 but an estimate for that contact can be made by matching the thermal conductance with that given by the Landauer formalism in the limit of low temperature. For sufficiently long molecules the resistance through the molecule, 1/hmol, dominates the resistance in the junction. We have constructed the initial structure of each of the saccharides from the visualization software Avogadro. Subsequent electronic structure calculations were carried out using the Gaussian 09 package. The initial geometries were optimized at the level of molecular mechanics with the General Amber Force Field (GAFF) followed by the PM6 semiempirical method. The minimum structure obtained from PM6 is further minimized using the Hartree−Fock method (HF) with basis sets 3-21G, 6-31G, 6-31G*, and 6-31G** sequentially. The final minimum structure obtained from HF/6-31G** was used as an initial structure for the DFT calculations. The frequencies, normal modes and third order coupling constants were calculated from DFT/6-31G**. In the DFT calculations, the ultrafine integration grid and a two-electron integration calculation accuracy of 10−13 were applied for all the molecules. The force constants were calculated at each step of minimization throughout the calculation. For the calculations presented below, a representative structure for each saccharide, the lowest energy structure we found, was adopted, though of course the molecule could be in a number of low energy structures at many of the temperatures considered here. Nevertheless, based on experience with other sizable molecules for which we have calculated rates of internal energy redistribution for several structures,59 the rates of thermalization that we calculate are not expected to vary significantly among the various low energy structures of the molecule.

(6)

Thermal boundary conductance can be expressed in terms of the quasiparticles, e.g., phonons in a solid, with energy ℏω. Let the vibrational mode density per unit volume on side j be ρ̅j (ω) and the phonon speed on side j be vj (ω). In the limit T2 = T1 + dT52 hBd =

1 d 4 dT

∫ dωvLρL̅ (ω)ℏωn(ω)τ(ω)

(7)

where τ is the transmission coefficient. Assuming a flat surface one obtains the factor 1/4 by accounting for all incident angles of vibrational quasiparticles striking the interface;52 different geometries may change somewhat the conductance.53,54 In any case, we absorb the 1/4 into a parameter below that can be adjusted for different structures. Equation 7 is the Diffuse Mismatch Model (DMM) result for thermal boundary conductance.52 Accounting for detailed balance between the left and right sides of the interface gives one contribution to the transmission coefficient, τ0 τ0(ω) =

vR ρR̅ (ω) vLρL̅ (ω) + vR ρR̅ (ω)

(8)

The transmission coefficient also may need to account for scattering within the molecules. Let L be the length of the molecule and l be the mean free path. As a vibrational excitation crosses the molecule it scatters roughly N times, where N ≈ L/l. Assuming for simplicity that upon scattering the vibrational excitation reverses direction 50% of the time and otherwise continues in the same direction, the transmission coefficient, τ, neglecting interference effects, is τ(ω) =

τ0(ω) 1 + (L /l)τ0(ω)

(9)

If only the number of phonons, n, depends on temperature, eq 7 can be written as hBd = B

∫ dωvLρL̅ (ω)C(ω)τ(ω)

(10)

where a factor, B, has been introduced which allows, e.g., for incomplete surface contact and includes the factor of 1/4 in eq 7. We shall simply fit B to experimental data for the results presented below. Though not necessary, it may be convenient to adopt a Debye model for the phonon densities, where

3. RESULTS AND DISCUSSION To address thermalization in saccharides we begin by calculating relaxation rates with eqs 3−5. We thus first need to calculate the transition parameter, T, with eq 1 and 2. Values of T are plotted in Figure 2a for the monosaccharides glucose and galactose in both ring and linear structures and for the disaccharides lactose and trehalose at internal energies corresponding to temperatures up to 600 K in intervals of 100 K. The temperature that is indicated is used in the thermal averaging in eq 2. Consider first 100 K. We see that at this temperature all the saccharides lie below or near T = 1, with only lactose slightly above this threshold. Thus, all small saccharides at low temperature exhibit effects of MBL. By 200 K the disaccharides are above T = 1, and by 300 K none of the saccharides exhibit effects of localization, with the possible exception of the cyclical monosaccharides, for

3ω2

ρ ̅ (ω) = 2 3 and v is a representative value for the speed of 2π v sound. The standard form for the Landauer expression for thermal ∂n(ω , T ) 1 conductance is 4π ∫ dωℏω ∂T τ ̅(ω),55−57 where the transmission coefficient is τ̅(ω) and can be related to eq 10 by τ̅(ω) = 4πBvLρ̅L(ω)C(ω)τ(ω). We refer to the boundary conductance given by eq 10 also as a Landauer model to emphasize the absence of inelastic scattering through the junction while allowing for elastic scattering, with scattering length, l, within 7280

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At low temperature we find substantial differences in the thermalization rates for the saccharides. This is due to the sensitivity of the thermalization rate to the value of the transition parameter, T, when T is less than 1, as observed earlier for other molecules.25 At low temperature there is an order of magnitude variation in the thermalization rate, which by 300 K diminishes to only about a factor of 2, a variation that holds at higher temperature. At 300 K the thermalization rate varies between about 1 and 2 ps−1 for all the saccharides, values that roughly double by 600 K. The thermalization rate should not much depend on the size of the system when T is significantly larger than 1,25 which is consistent with the similar values observed in all systems at temperatures 300 K and higher. Thermalization is due to inelastic scattering arising from anharmonic interactions in the molecule and interactions with the leads. Elastic scattering also mediates energy transport. The energy diffusion coefficient for trehalose was computed in the harmonic approximation via the propagation of wave packets, which were expressed in terms of the normal modes (SI) of trehalose,60 as done for other molecules in earlier work.24−26 The wave packets were initiated around each of the middle seven carbon and oxygen atoms of trehalose and averaged. We plot the time dependence of the average variance of the wave packets in Figure 3, where we find a slope of 11.2 Å2 ps−1, yielding a

Figure 2. (a) Transition parameter, T, plotted against temperature for glucose (cyclic structure is gray circle, linear structure is X), galactose (cyclic black circle, linear +), lactose (triangle) and trehalose (square) from 100 to 600 K. The MBL threshold, T = 1, is also shown. (b) Thermalization rate is plotted against temperature for the same systems.

which T appears above but close to 1. We expect that at higher temperature the transition rates should be well represented by the GR estimate for the average rate. The value of T and the onset of facile thermalization, where T is at least 1, appears to be influenced by the size and shape of the molecules. The value of T is greater for the disaccharides lactose and trehalose than for the monosaccharides glucose and galactose, which is not surprising, as the former have a larger local density of states than the latter. The value of T also appears to be larger for the linear structures of the monosaccharides than the cyclical structures. This is due to the greater number of low frequency modes of the linear structures. For both glucose and galactose there are 2 modes below 60 cm−1 for the linear structures, and none for the cyclic structures, which gives rise to a larger local density of states for the linear structures and thus the larger value of T. The full set of frequencies for all the saccharides are listed in the SI. We have calculated the most probable relaxation rates with eqs 3−5 where T > 1, and with eqs S1−S2 where T < 1. The results for the mono- and disaccharides with internal energies corresponding to temperatures up to 600 K in intervals of 100 K are plotted in Figure 2(b). The inverse of the relaxation rate, the relaxation time, is an approximation to the thermalization time. We expect these times to be similar if T1 and T2, the temperatures of the substrates, are not very different, which is the linear response regime. For this reason we refer to the rates calculated with eq 5 as the thermalization rate.

Figure 3. Variance in energy as a function of time is plotted. The variance was calculated as an average over seven wave packets propagated using the normal modes of trehalose. A linear fit to the data gives 2D = 11.2 Å2 ps−1. Inset: The length of propagation of the wave packet is plotted as a function of time. This length was obtained as an average over seven wave packets propagated using the normal modes of trehalose. A linear fit gives a vibrational energy propagation speed of 6.0 Å ps−1.

diffusion coefficient of D = 5.6 Å2 ps−1. This value is on the smaller side of what has been computed for biomolecules in the past. For example, there has been much interest in computing and measuring energy diffusion in proteins,49,61−72 for which values of D between 10 and 20 Å2 ps−1 have been computed.70 A smaller value for a disaccharide is unsurprising, as trehalose lacks the low-frequency heat-carrying modes of proteins. The speed of sound in trehalose was computed by propagating wave packets commencing from the same 7 atoms and calculating the position of the center of the wave packet as a function of time. We plot the result in Figure 3, where we find a slope corresponding to the propagation speed of v = 6.0 Å ps−1. The mean free path is 2D/v, yielding a mean free path for elastic scattering of 1.9 Å, comparable to values we have found in the past for protein molecules.67 The elastic mean free path obtained 7281

DOI: 10.1021/acs.jpcb.8b04632 J. Phys. Chem. B 2018, 122, 7277−7285

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We consider first Au−trehalose−water. To examine the effect of trehalose on the boundary conductance, we start by calculating the thermal boundary conductance between the gold and water, for which we use eqs 8−10, with temperatures ranging from 100 to 400 K. We note that while 400 K may not appear realistic for water, even higher temperatures are reached transiently when GNPs are heated by lasers,18 so it is worthwhile considering such temperatures. To calculate the thermal conductance using eq 10 it is convenient to use a Debye model for the density of states. The density of states is then expressed in terms of the speed of sound of the two materials, which for gold and water is, respectively, 3240 and 1484 ms−1. Equation 10 needs to be integrated to the lower Debye temperature, which is 170 K for gold. We take the length of the junction to be the length of trehalose, which is 1.0 nm. We use for the elastic mean free path the value 1.9 Å, calculated above, in the Landauer model to estimate thermal conduction in the absence of thermalization. The results for the thermal boundary conductance for Au−water and Au-trehalose-water are plotted in Figure 5. The boundary conductance of Au−water has been reported to be 150 MW m−2 K−1,75 so we introduce a factor B = 0.25 in eq 10 to approach this value at higher temperature.

from the simulations of the saccharides, and for previously studied proteins and fluorinated alkanes,26 originates from aperiodicity in structure and is analogous to the mean free path for diffuson modes in amorphous materials.73 We refer to the length over which thermalization occurs in the molecule as the thermalization length. It depends on the time for thermalization, the inverse of the rate, W, and the length over which energy propagates during the time for thermalization to occur. Because the mean free path in saccharides, 2 Å, is quite short, we estimate the thermalization length in terms of the energy diffusion coefficient, D. We calculate the thermalization length as (2D/W)1/2, where D = 5.6 Å2 ps−1, as noted above, and the rates are plotted in Figure 2b for the mono- and disaccharides. We plot the thermalization lengths in Figure 4

Figure 4. Inelastic scattering lengths in glucose (cyclic structure is gray circle, linear structure is X), galactose (cyclic black circle, linear + ), lactose (triangle), and trehalose (square) from 100 to 400 K. The black line indicates the mean free path for elastic scattering.

from 100 to 400 K. We see that the length varies between about 7 and 40 Å for the various saccharides at 100 K and reaches about 3−5 Å at 300 K and 2−4 Å at 400 K. For all of the saccharides in this temperature range, the length over which thermalization occurs is no shorter than the 2 Å mean free path for elastic scattering. To illustrate thermal transport through saccharides we consider two systems. One includes a layer of trehalose between gold, which could be a GNP, and a second substrate, taken to be water, which is representative of a cellular environment. For the other, we consider a layer of saccharides between a protein and water to examine the extent to which saccharides act as an insulator. Since trehalose has been widely studied as a stabilizer around protein molecules, we take trehalose as the saccharide for all the calculations, so that the length of the junction is about 1 nm. We found (Figure 4) that, at least to 400 K, the elastic mean free path of 0.2 nm is shorter than the length over which thermalization occurs, which is on the order of 1 nm. Because thermalization does not occur locally within the junction we use a Landauer model to calculate the thermal conductance. We note that we encountered a similar situation for fluorinated alkanes,24 where the thermalization length at 400 K was comparable to the length of the junction, about 2 nm, and the computational results matched those of the experiments74 very well.

Figure 5. Thermal boundary conductance between gold and water (black) and between water and a protein (blue) calculated using eq 10 without any molecules at the interface (solid curves) and the thermal boundary conductance with a layer of trehalose in between (dashed). Even with strong coupling between trehalose and each material, its presence at the interface enhances thermal insulation by a factor of 4−5.

We observe a marked resistance due to the 1 nm trehalose junction between gold and water. Thermal conductance is not very sensitive to temperature, due to the low Debye temperature of gold, saturating around about 150 MW m−2 K−1 by around 200 K. However, for the trehalose junction the conductance reaches a value of only 30 MW m−2 K−1, a factor of 5 smaller than the conductance without the trehalose interface. Merely one layer of trehalose, about 1 nm thick, introduces an insulating layer that markedly reduces thermal conductance between gold and water or a cellular environment, and that is assuming that coupling between trehalose and each lead is strong. If it were weak thermal resistance would be somewhat greater. The insulating effect of trehalose is much greater than what we have calculated for other molecular junctions in the past. For example, for PEG oligomers, which have been adopted as capping agents on solvated GNPs,17 even a length of 4.0 nm only reduced the thermal conductance to 60MWm−2 K−1. 7282

DOI: 10.1021/acs.jpcb.8b04632 J. Phys. Chem. B 2018, 122, 7277−7285

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The Journal of Physical Chemistry B Finally, we consider a layer of trehalose around a protein in water. For the protein we have chosen myoglobin, since we have previously calculated the thermal boundary conductance between myoglobin and water.3 We found it to be comparable to the thermal boundary conductance between other proteins and water using eq 10 as the basis for the calculation and those results were comparable to results of molecular simulations of the boundary conductance.76 Our focus here is the effect of introducing a layer of trehalose between the protein and water. For the speed of sound in the protein we use 2000 m s−1, a value obtained both in previous calculations67 and deduced from experiments on myoglobin.77 The results for the thermal conductance for water−myoglobin and water−trehalose− myoglobin are plotted in Figure 5 over a wide range of temperatures. Since the vibrational spectrum of water and myoglobin extends far above the phonon band of gold we find the thermal boundary conductance to increase gradually with temperature, rather than approach a limiting value by about 200 K, as we observed for the system with gold. Around 300 K the thermal boundary conductance for water-myoglobin is about 300 MW m−2 K−1, a value that is fairly typical for proteins and water.76,78 With a layer of trehalose the thermal boundary conductance drops to about 80 MW m−2 K−1, almost a factor of 4 smaller. The 1 nm layer of trehalose thus acts as an effective insulator to sudden changes in ambient temperature. We see that for the water−protein system thermal transport within trehalose makes a somewhat smaller contribution, diminishing the resistance at the boundary by about a factor of 4, than it does for gold and water, where it reduces conductance by about a factor of 5. The reason is that the contribution to the transmission coefficient, τ0, due to detailed balance in the former case is about 0.5, i.e., the density of states for the two systems over the thermally accessible modes is about the same, as is the speed of sound in the two systems, yielding τ0 roughly 0.5 (eq 8). On the other hand, for the gold-water system τ0 is larger, about 0.8. In that case, the resistance arising from the mean free path for elastic scattering in trehalose makes a somewhat larger contribution to the transmission coefficient, τ.

either of these two materials. The excellent insulating properties of trehalose, in terms of the reduction of thermal conductance compared to the boundary conductance between the leads without the molecular layer, is more pronounced than has been observed in computational work on other molecular layers,22−26 and may contribute to its stabilizing effect toward change in temperature, at least over short times. It will be important to measure energy and thermal transport through polysaccharides. Time-resolved thermoreflectance experiments could measure thermal conductance through layers of saccharides between solid-state leads, as has been carried out for a variety of molecular interfaces.29,30,74 Particularly important would be time-resolved studies of vibrational energy transport of saccharides in solution, as this would provide information about both energy relaxation and transport in saccharides. Relaxation assisted 2D IR studies,79,80 as carried out, e.g., for PEG oligomers,81−83 would provide valuable information about these molecules. On the length scale of the saccharides studied here, about 1 nm, thermalization was not found to be sufficiently rapid to generate local temperature, and a Landauer-like picture, which includes elastic scattering but not inelastic, was appropriate for calculating thermal conductance. We found a similar situation in an earlier study of perfluoroalkanes about 2 nm in length, where the results of calculations24 using a Landauer-like approach yielded results in good agreement with experiment.74 Future work will examine thermal conduction through larger saccharides, for which the resistance will increase with the length, but thermalization will begin to influence thermal conduction. In that case, we expect thermal conductance at the interface to be more sensitive to change in temperature,25 particularly for the case where gold is one of the leads. In that case phonons that enter can up-convert to higher frequency modes of the molecule as thermalization occurs and transfer to the second lead, as has been observed in computational work carried out on larger molecular systems.25,35

4. CONCLUSIONS We have examined the origin of thermal resistance in small saccharide molecules, including monosaccharides galactose and glucose, and dissacharides lactose and trehalose. For these systems, which extend up to 1 nm, thermal resistance is predominantly due to elastic scattering, as inelastic scattering due to anharmonic interactions and contact with the surrounding environment is not fast enough to compete with elastic scattering within the molecule. We compared thermalization in monosaccharides that are linear and cyclic, and found that the onset of facile thermalization occurs at lower temperature in linear monosaccharides compared to cyclic monosaccharides. At lower temperatures, then, where T is less than 1 for some structures but not for others, the thermalization rates vary considerably depending on the saccharide and the structure. At higher temperature, including room temperature, thermalization rates in mono- and disaccharides are comparable, but they are still too small to give rise to appreciable inelastic scattering on the 1 nm length scale of these molecules. We have calculated thermal conductance between gold and water and between protein and water through a layer of trehalose, accounting for elastic scattering within the saccharide. We find that trehalose provides a strong insulating layer between

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.8b04632.

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ASSOCIATED CONTENT

S Supporting Information *

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Expressions accounting for effects of the dephasing rate on the most probable thermalization rate, expressions for the calculation of vibrational energy propagation and diffusion in terms of the normal modes of the saccharides, and the vibrational modes computed for the saccharides (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

David M. Leitner: 0000-0002-3105-818X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Support from NSF Grant No. CHE-1361776 is gratefully acknowledged. 7283

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The Journal of Physical Chemistry B

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(25) Pandey, H. D.; Leitner, D. M. Influence of Thermalization on Thermal Conduction through Molecular Junctions: Computational Study of Peg Oligomers. J. Chem. Phys. 2017, 147, 084701. (26) Pandey, H. D.; Leitner, D. M. Vibrational Energy Transport in Molecules and the Statistical Properties of Vibrational Modes. Chem. Phys. 2017, 482, 81−85. (27) Heugen, U.; Schwaab, G.; Bründermann, E.; Heyden, M.; Yu, X.; Leitner, D. M.; Havenith, M. Solute Induced Retardation of Water Dynamics: Hydration Water Probed Directly by Thz Spectroscopy. Proc. Natl. Acad. Sci. U. S. A. 2006, 103, 12301−12306. (28) Heyden, M.; Bründermann, E.; Heugen, U.; Niehues, G.; Leitner, D. M.; Havenith, M. The Long Range Influence of Carbohydrates on the Solvation Dynamics of Water − Answers from THz Spectroscopic Measurements and Molecular Modelling Simulations. J. Am. Chem. Soc. 2008, 130, 5773−5779. (29) Majumdar, S.; Sierra-Suarez, J. A.; Schiffres, S. N.; Ong, W.-L.; Higgs, C. F.; McGaughey, A. J. H.; Malen, J. A. Vibrational Mismatch of Metal Leads Controls Thermal Conductance of Self-Assembled Monolayer Junctions. Nano Lett. 2015, 15, 2985−2991. (30) Hopkins, P. E. Thermal Transport across Solid Interfaces with Nanoscale Imperfections: Effects of Roughness, Disorder, Dislocations and Bonding on Thermal Boundary Conductance (Review Article). ISRN Mechanical Engineering 2013, 2013, 682586. (31) Datta, S. Electronic Transport in Mesoscopic Systems; Cambridge University Press: Cambridge, 1995. (32) Bürkle, M.; Hellmuth, T. J.; Pauly, F.; Asai, Y. First-Principles Calculation of the Thermoelectric Figure of Merit for [2,2]Paracyclophane-Based Single-Molecule Junctions. Phys. Rev. B: Condens. Matter Mater. Phys. 2015, 91, 165419. (33) Buerkle, M.; Asai, Y. Thermal Conductance of Teflon and Polyethylene: Insight from an Atomistic, Single-Molecule Level. Sci. Rep. 2017, 7, 41898. (34) Stocker, K. M.; Gezelter, J. D. Simulations of Heat Conduction at Thiolate-Capped Gold Surfaces: The Role of Chain Length and Solvent Penetration. J. Phys. Chem. C 2013, 117, 7605−7612. (35) Kuang, S.; Gezelter, J. D. Simulating Interfacial Thermal Conductance at Metal-Solvent Interfaces. J. Phys. Chem. C 2011, 115, 22475−22483. (36) Eslami, H.; Mohammadzadeh, L.; Mehdipour, N. Anisotropic Heat Transport in Nanoconfined Polyamide-6,6 Oligomers: Atomistic Reverse Nonequilibrium Molecular Dynamics Simulation. J. Chem. Phys. 2012, 136, 104901. (37) Leitner, D. M. Quantum Ergodicity and Energy Flow in Molecules. Adv. Phys. 2015, 64, 445−517. (38) Gruebele, M.; Bigwood, R. Molecular Vibrational Energy Flow: Beyond the Golden Rule. Int. Rev. Phys. Chem. 1998, 17, 91−145. (39) Bigwood, R.; Gruebele, M.; Leitner, D. M.; Wolynes, P. G. The Vibrational Energy Flow Transition in Organic Molecules: Theory Meets Experiment. Proc. Natl. Acad. Sci. U. S. A. 1998, 95, 5960−5967. (40) Leitner, D. M.; Wolynes, P. G. Statistical Properties of Localized Vibrational Eigenstates. Chem. Phys. Lett. 1996, 258, 18−24. (41) Leitner, D. M.; Wolynes, P. G. Quantization of the Stochastic Pump Model of Arnold Diffusion. Phys. Rev. Lett. 1997, 79, 55−58. (42) Leitner, D. M.; Wolynes, P. G. Vibrational Relaxation and Energy Localization in Polyatomics: Effects of High-Order Resonances on Flow Rates and the Quantum Ergodicity Transition. J. Chem. Phys. 1996, 105, 11226−11236. (43) Nandkishore, R.; Huse, D. A. Many-Body Localization and Thermalization in Quantum Statistical Mechanics. Annu. Rev. Condens. Matter Phys. 2015, 6, 15−38. (44) Burin, A. L. Many-Body Localization in a Strongly Disordered System with Long-Range Interactions: Finite-Size Scaling. Phys. Rev. B: Condens. Matter Mater. Phys. 2015, 91, 094202. (45) Bar Lev, Y.; Reichman, D. R. Dynamics of Many-Body Localization. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 89, 220201. (46) Burin, A. Localization and Chaos in a Quantum Spin Glass Model in Random Longitudinal Fields: Mapping to the Localization

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