Heat transfer at a solid/liquid interface

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2.1 Introduction . . . . 50 2.2 Modelling . . . . 51 2.2.1 Potentials . . . . 51 2.2.2 Details of the systems studied and the simulations . 52 2.3 Effect of the curvature on the gold/water interface 54 2.4 Interface Resistance calculation . . . . 55 2.4.1 Thermal resistance of the bare gold/water interface. 55 2.4.2 Effect of polymer grafting on the thermal resistance 56 2.5 Physical analysis of the interfaces. . . . 58 2.6 Qualitative analysis by a transient non-equilibrium

method. . . . 61 2.7 Generalization of the results . . . . 62 2.8 Conclusion . . . . 65

2.1 Introduction

Metallic nanoparticles are used in several studies ranging from physics to biol-ogy due to their optical and morphological properties [148,38,39,6,139]. Ap-plications can be mentioned in molecular imaging [24] and Raman spectroscopy [112,71]. More recently, metallic nanoparticles have attracted an even greater interest due to their ability to absorb radiations and to turn them into heat through plasmon resonance [123,54,125,96,72,16,83,95,153]. This feature makes nanoparticles promising candidates for the development of drug deliv-ery systems [96, 29] and photo-induced hyperthermia to destroy tumor cells [177,158]. In this context a precise control of the local heat release into the en-vironment is sought. Measurements of the heat at the vicinity of nanoparticle is a difficult problem that has been addressed by experiments where thermosen-sitive polymers are attached to the surface of the nanoparticles allowing the measure of the temperature field [139,99,140,129,46]. Some physical models have been proposed [54,8, 56,113,140, 175, 68] to describe this temperature field at the interface between a gold nanoparticle and water and only a few microscopic models [113] but, while bare nanoparticles are rare in applications, the effect of the polymer covering has not been addressed. We seek to tackle this problem as it has great potential for the engineering of nanosystems for bio-logical applications. The thermal resistance of a gold/water interface is studied and how it is affected when a polymer is grafted on it. To answer this question, a Molecular Dynamics model of the interface between a gold nanoparticle and water has been developed.

The first step of this study is the development of the Molecular Dynamics model. As mentioned here, our solid/liquid interface is a gold/water interface.

It has been chosen to model a DHLA-Jeffamine for the grafted polymer. This is a thermosensitive polymer, just like PNIPAAm [29], and its lower critical solution temperature (LCST) can be tuned by its length. Furthermore, this polymer can make thiol bonds with the gold atoms of the solid. The process of experimentally grafting this polymer using this bond is also well handled [37]. These properties make DHLA-Jeffamine a good candidate to design self-assembled nanoparticles for photo-induced hyperthermia [37].

This first part of this chapter describes how the Molecular Dynamics model is implemented with these different subsystems and specifically how the potentials to make the different elements of the system interact together have been chosen.

In a second part, it is shown that modelling only a plane of water to address the problem of the heat transfer at the gold/water interface and that it will still be a relevant model for nanoparticles of sizes above at least 14 nm of diameter.

This is done by comparing the structure of water at the interface with gold of different radii of curvature.

Then, we detail our numerical calculations and results using the ∆T method to compute the thermal conductance of the gold/water interface and how the grafted polymer increases this conductance. We present our interpretation of

this result by computing the transmission function at the gold/polymer inter-face and the density of states of each element of our model.

2.2 Modelling

We present here the modelling procedure of our gold/water/polymer system.

These three subsystems will be modelled with three different potentials and we will first describe these potentials and how they interact with the each other.

Then we describe the five different geometries we have investigated. Three of them are nanoparticles of different radii of curvature to study its influence on the water structure and two of them are planar gold surfaces with water, one of those surfaces has a polymer grafted on it.

2.2.1 Potentials

We model systems made of gold, water and a DHLA-Jeffamine polymer. The skeletal formula of this polymer is given on Figure 2.1. We chose to use the TIP3P model for the water (the flexible version) [75,27,41], the DREIDING model for the polymer [110] and the MEAM potential for the gold [11]. These potentials are presented in Chapter1. However, here we have to combine those potentials. Between the atoms modelled by DREIDING and TIP3P, we use the DREIDING model. This is not difficult as those two models involve the same potentials only with a different set of parameters. It is more complicated between the water and the gold but gold/water systems have already been modelled in Molecular Dynamics and so potentials have already been developed and tested [113,147]. It is a Lennard-Jones potential:

E = 4

• = 0.25584406 kcal.mol−1 et σ = 3.6 Å for the interaction between oxygen and gold atoms.

• = 0.0etσ = 0.0Å for the interaction between hydrogen and gold atoms.

About the interactions between the gold and the polymer atoms, there is a thiol bond between the gold and the sulfur atoms for which the parameters can be found in the literature [76] for a harmonic bond:

E=K(r−r0)2,

Figure 2.1: Skeletal formula of DHLA-Jeffamine polymer. This polymer ends with sulfur atoms that are bonded to Au atoms of the gold surface by a thiol bond.

To model the long range interaction (van der Waals) between the polymer and the gold, we used the Lennard-Jones potential and the method proposed in the DREIDING model to compute the parameters and σ (Chapter 1). For the gold, we used the parameters from the Universal Force Field [135] giving R0(Au) = 2.934Å andD0(Au) = 0.039kcal.mol−1.

The last part is the computation of the local charges of the atoms of the DHLA-Jeffamine polymer which is done using the Gasteiger algorithm [49] as seen in Chapter 1. We indicates those charges in Table2.1.

2.2.2 Details of the systems studied and the simulations In this chapter, we will study 5 different systems:

• 3 nanoparticles of respectively 6.5 nm, 17 nm and 28 nm of diameter to study the effect on the curvature of these particles on the structure of the water at the interface.

To reduce the computational costs, we exploit the symmetries of nanopar-ticles and we model only an 8thof the particles. To build them, we create a cubic gold crystal which the edge is half the diameter of the nanoparti-cle, then we delete all atoms outside of the largest inscribing 18th of sphere within the gold cube. We then add 50 Å of water molecules around the nanoparticle to complete the system. We don’t use periodical conditions in these systems, we thus used reflective walls as described in Chapter 1.

• a planar gold surface in contact with water.

To build this system, we use the same method as for the nanoparticles but we keep the cubic gold crystal (about 125 nm3) and add 5 nm of water above. Here we used periodical conditions in the directions parallel to the gold/water interface plane. The area of the contact between the gold and the water is 28 nm2.

• a gold surface with a DHLA-Polymer grafted on it, in contact with water.

To build this system, we use the previous system and added the polymer on the gold surface using, at first, an harmonic potential for the Au-S bond. The charges computed with the Gasteiger algorithm are given in

Table 2.1: Atoms of the DHLA-Jeffamine polymer with their partial charges computed with the Gasteiger algorithm [49] and the atoms they are bonded to.

Bonded to Bonded to

ID Type Partial charge 1 2 3 4 ID Type Partial charge 1 2 3 4

1 H 0.085516632 2 59 H 0.085866632 57

2 C -0.218687368 1 3 4 5 60 H 0.085866632 57

3 H 0.085516632 2 61 H 0.085866632 57

4 H 0.085516632 2 62 O -0.028597368 56 63

5 O -0.052519368 2 6 63 C -0.170242368 62 64 65 66

6 C -0.182448368 5 7 8 9 64 H 0.103807632 63

7 H 0.102002632 6 65 H 0.103807632 63

8 H 0.102002632 6 66 C -0.155060368 63 67 68 72

9 C -0.180900368 6 10 11 12 67 C -0.224421368 66 69 70 71

10 H 0.102092632 9 68 H 0.121653632 66

11 H 0.102092632 9 69 H 0.085866632 67

12 O -0.039567368 9 13 70 H 0.085866632 67

13 C -0.171776368 12 14 15 16 71 H 0.085866632 67

14 H 0.103715632 13 72 O -0.028597368 66 73

15 H 0.103715632 13 73 C -0.170242368 72 74 75 76

16 C -0.155119368 13 17 18 22 74 H 0.103807632 73 17 C -0.224422368 16 19 20 21 75 H 0.103807632 73

18 H 0.121652632 16 76 C -0.155063368 73 77 78 82

19 H 0.085866632 17 77 C -0.224421368 76 79 80 81

20 H 0.085866632 17 78 H 0.121653632 76

21 H 0.085866632 17 79 H 0.085866632 77

22 O -0.028598368 16 23 80 H 0.085866632 77

23 C -0.170242368 22 24 25 26 81 H 0.085866632 77

24 H 0.103807632 23 82 O -0.028666368 76 83

25 H 0.103807632 23 83 C -0.171608368 82 84 85 86

26 C -0.155060368 23 27 28 32 84 H 0.103699632 83 27 C -0.224421368 26 29 30 31 85 H 0.103699632 83

28 H 0.121653632 26 86 C -0.168484368 83 87 88 92

29 H 0.085866632 27 87 C -0.225854368 86 89 90 91

30 H 0.085866632 27 88 H 0.119221632 86

31 H 0.085866632 27 89 H 0.085792632 87

32 O -0.028597368 26 33 90 H 0.085792632 87

33 C -0.170242368 32 34 35 36 91 H 0.085792632 87

34 H 0.103807632 33 92 N -0.074401368 86 93 94

35 H 0.103807632 33 93 H 0.102721632 92

36 C -0.155060368 33 37 38 42 94 C -0.057589368 92 95 96 37 C -0.224421368 36 39 40 41 95 O 0.025455632 94

38 H 0.121653632 36 96 C -0.182861368 94 97 98 99

39 H 0.085866632 37 97 H 0.101644632 96

40 H 0.085866632 37 98 H 0.101644632 96

41 H 0.085866632 37 99 C -0.198523368 96 100 101 102

42 O -0.028597368 36 43 100 H 0.100813632 99

43 C -0.170242368 42 44 45 46 101 H 0.100813632 99

44 H 0.103807632 43 102 C -0.199291368 99 103 104 105

45 H 0.103807632 43 103 H 0.100871632 102

46 C -0.155060368 43 47 48 52 104 H 0.100871632 102

47 C -0.224421368 46 49 50 51 105 C -0.198619368 102 106 107 108

48 H 0.121653632 46 106 H 0.100776632 105

49 H 0.085866632 47 107 H 0.100776632 105

50 H 0.085866632 47 108 C -0.180728368 105 109 110 112

51 H 0.085866632 47 109 H 0.101361632 108

52 O -0.028597368 46 53 110 S -0.018744368 108

53 C -0.170242368 52 54 55 56 111 C -0.198036368 108 112 113 114

54 H 0.103807632 53 112 H 0.100670632 111

55 H 0.103807632 53 113 H 0.100670632 111

56 C -0.155060368 53 57 58 62 114 C -0.180885368 111 115 116 117 57 C -0.224421368 56 59 60 61 115 H 0.101346632 114

58 H 0.121653632 56 116 H 0.101346632 114

117 S -0.018754368 114

Water

Gold

Water

Water Water

Gold Gold Gold

Figure 2.2: Gold/water interfaces with different radius curvature of the gold surface.

The first three systems are 8th of a nanoparticle of the indicated diameter. We only represent slices of the whole system here. The fourth system is a planar surface (with periodic boundary conditions in the directions parallel to the gold plane). All those system are simulated at 310 K.

Table 2.1.

After a first simulation to equilibrate this structure, we change this bond into a Morse potential and then add water on the structure avoiding the polymer by 3 Å.

Due to the full atomistic model, where hydrogen atoms are represented, we need a time step always under 1 fs in the simulations. We used a 0.5 fs time step during the equilibration run under canonical ensemble. We used a 0.1 fs time step in microcanonical runs. To ensure reaching the equilibrium of temperature of 310 K and pressure of 1 atm in the canonical ensemble, we ran the system for 800 ps. Only after this equilibrium is reached, we switch to the microcanonical ensemble simulation and all the results we present are produced we data collected during this stage.

2.3 Effect of the curvature on the gold/water inter-face

We use our four first systems to study the effect of the radius of curvature of the gold surface on the structure of the water. Those different systems are presented on the Figure 2.2 (we represent only slices for the nanoparticles).

The larger the radius of curvature, the closer the gold/water interface will be to the plane model. The simulations allow us to characterise this convergence in terms of water density profile.

We have calculated the density profiles of the water surrounding the gold to characterise those interfaces and the results are presented on Figure2.3. We see that the water density is maximum a few ångströms away from the gold surface and is structured up to 1 nm from it. This is true near the plane of gold and the nanoparticles of at least 17 nm of diameter. Around our smallest nanoparticle, 6.5 nm of diameter, there is no structure in the water and no density pick. A

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