2.3.1 Models of heat transport

There are many analytical models for predicting heat transport between filler particles used in thermally conductive adhesives. According to literature reviews,⁴⁵ the most common and important analytical models, mostly taking into account spherical particles, can be listed as follows:

a conductive network can be formed even when the content of particles is lower than the percolation threshold estimated conventionally.¹² With increasing temperature, self-organization of particles helps in the formation of a conducting path throughout the matrix.

Apart from more general models taking into account the filler particles in the form of spheres or ovals, there are analytical models predicting thermal conductivity of composites with carbon nanotubes, even with their various orientation distributions in a matrix.¹³

2.3.2 Thermal resistance of adhesives

If free electrons accomplish conduction (Equation 2.3), then the formulas for thermal and electrical conduction are the same. Assuming that electrical and thermal flow lines are geometrically equal, the concept of a thermal resistance can be introduced

[2.5]

where Θ_λ is the thermal resistance of a thermal conductor (K/W), l is its length (m) and A is the area of the conductor cross-section (m²). The total thermal resistance in ‘a three-dimensional network’ of conducting filler particles can be summed up using the concepts of serial and parallel resistances, as in an electrical circuit. The thermal resistance Θ_p of the singular contact path between substrate and component in the steady state conditions can be expressed in a simple equation

[2.6]

where Θλ is the thermal resistance of a filler particle (bulk resistance) and Θ_TC is the thermal contact resistance between particles making a chain in the direction of the temperature gradient. The total thermal resistance of the adhesive Θ_TA is

[2.7]

Equation 2.6 points out that the less contact between particles, the greater the thermal conductivity of the adhesive. For the same distance between joining surfaces, it requires larger particles of a filler. Practically, the experiments show¹⁴ that at an alumina filler with a fixed loading level of 50 wt%, the thermal conductivity increases about 1.7 times when filler of < 10 μm average particle size is replaced by a much larger one - not higher than 149 mm (− 100 mesh). This way of thermal conductivity improvement (by enlarging filler particles) is not good because the miniaturized packaging scale, and the technology of adhesive dispensing, generally require that the dimension of filler particles should not be higher than a few micrometers. According to Equation 2.7 it is natural that the more conducting paths there are, the lower is the thermal resistance of the composite. This can be ensured by higher filler loadings. In Fig. 2.3 the thermal conductivity of the epoxy adhesives as a function of the volume loading of AlN filler is presented. The nonlinear conductivity increases with increase of the filler content, suggesting that additional paths for heat transport are being formed not only by new chains, but also by multiplication of thermal contacts between filler particles in the whole network of particles. This method of conductivity improvement is limited because of a need for usable viscosity of the formulation and for sufficient mechanical strength of adhesion of the adhesive.

For a single particle of filler, its bulk conductivity is determined by the material used. From this point of view, diamond, with a conductivity of 2000 W/mK, is a better material than alumina (slightly better than 30 W/mK). However, with adhesives formulated in a similar way to electrically conductive adhesives, which contain silver flakes with average particle dimensions of several micrometers and a thermosetting epoxy resin as polymer matrix, the thermal conductivity value does not exceed 3 W/mK.¹⁷ This enormous difference between the thermal conductivity of pure silver (about 420 W/mK) and a thermally conductive adhesive with this filler is due to constraint by the existing thermal contact resistance between filler particles. It is the main ‘bottleneck’ for heat transport inside composite formulations.

The problem of thermal constriction resistance is well-known for the region where two solid contact members touch each other. The real contact area is only a small fraction of the nominal or apparent area. This is mainly due to the non-flat shape of the contacting surfaces and surface roughness (typical for filler particles of an adhesive), the hardness of the contacting materials, or additional materials between the contacting surfaces. It would be difficult to analyze the accidental contact area between two flake-shape filler particles of an adhesive. Thus, for understanding the problem of thermal constriction resistance, let us assume two cylindrical metallic units P₁ and P₂, ‘representing filler particles’ contacting flat surfaces (Fig. 2.4). When the contact areas are nominally flat and hard, then they touch each other in maximum of three points. In fact, these points become the small areas (a-spots) since contact member materials are deformable.¹⁸ When the contact surfaces are perfectly clean, then the total metallic contact area A_c is the sum of all a-spot areas. When cylindrical unit P₁ has a higher temperature (r₁) then unit P₂ (temperature T₂), the steady state heat flow between them is constricted through small conducting a-spots.

In the case of contacting adhesive filler particles, the number of a-spots is unknown. The total thermal resistance of such units is the sum of the bulk thermal resistances Θ_λ1and Θ_λ2 of both contact members, the constriction resistances Θ_c1 and Θ_c2 of contact members, and the real thermal resistance of the a-spot (or A_c - the sum of all a-spot areas) bump Θ_a (if a-spot is treated as a 3D structure).

The thermal constriction resistance depends significantly on both the area of the a-spot and the gap between contact members. Increasing the contact area and lowering the distance between filler particles cause a decrease of the thermal constriction resistance. As an example, the results of calculating the gap distance influence on this resistance are presented in Fig. 2.5. The thermal constriction resistances of two cylindrical copper particles, with radius of 10 mm and 40 mm length each (Fig. 2.4), were calculated using the finite element method.¹⁹ The gap between both contact members ranged from 0 up to 10 mm. Due to symmetry, only one quarter of the system was analyzed. The computer calculations were performed assuming that the thermal conductivity of the polymer matrix could be neglected and that the contact surface was perfectly clean, without oxides and alien films. In fact, the thermal constriction resistances can be treated as the thermal contact resistance Θ_TC from Equation 2.6.

For the analyzed two-particle unit at the temperature T₁ of 393 K (T₂ = 293 K), the bulk resistance of both filler particles is about 0.7 K/W, while the thermal contact resistance Θ_TC = 221 K/W for both the radius of a-spot and the gap equal to 10 mm. This means that in such a configuration, the total resistance of a thermally conductive adhesive depends mainly on the values of thermal contact resistance and numbers of contacts between filler particles making a chain in the direction of temperature gradient (Θ_TC and n in Equation 2.6). The bulk resistance plays a minimal role.

2.3.3 Bond thermal resistance

Thermal conductive adhesives usually are applied between two surfaces at different temperatures, e.g. between a heat generating component and the eventual heat sink. In such a case of heat transport, the thermal resistance Θ_T can be expressed by the formula

[2.8]

where Θ_BR’ and Θ_BR" stand for the bond thermal resistance of contacts between the adhesive layer and the joined elements, while Θ_TA represents the total thermal resistance of the adhesive (Equation 2.7). In such a case, the bond thermal resistance influence on Θ_T cannot be treated as the thermal contact resistance between solid surfaces analyzed previously. The surface of the joined element is glued by a polymer composite with a conductive filler, and macroscopic irregularities such as flatness deviations and waviness are of no importance; however, surface roughness may influence the bond thermal resistance to a large extent.

The value of the bond thermal resistance depends on the shapes and dimensions of both the surface irregularities and of the particles of the adhesive filler. As this contact is difficult to specify, an imaginary 3D material layer representing the bond thermal contact with thickness of l_BR (including all these irregularities) and an unknown thermal conductivity 1_BR, can be introduced.²⁰ With such an assumption, the heat flow Q causes the temperature gradient ΔT on every layer between the joined surfaces; two bond thermal contacts and adhesive (Fig. 2.6).

The value of thermal conductivity λ_BR can be extracted from the result of an experiment when the temperature gradient ΔT_T can be measured.¹⁷’²¹ For the steady state of the heat flow, it is possible to measure and calculate^17,22 the thermal resistance of the adhesive Θ_TA and bond thermal resistance Θ_BR with the assumption that

[2.9]

Knowing the contact area and thickness of layers from Equation 2.5, the Bond the conductivity of the adhesive λ_TA, as well as the thermal conductivity of the imaginary ‘bond’ layer λ_BR can be established.

In fact, the thickness of the ‘bond’ layer is unknown and it would be much more valuable to define a new parameter describing the bond thermal conductivity, which will not depend on the l_BR thickness. This parameter can be referenced as the relative thermal conductivity of the thermal contact and can be defined as the thermal conductivity on a unit thickness λ’_BR and then

[2.10]

It has been calculated²² that the participation of the adhesive bond resistance Θ_BR in the total resistance depends on the adhesive layer thickness l_TA. Figure 2.7 shows such dependence, plotted according to equation

[2.11]

with the following values resulting from measurement:²² λ_TA = 1.46 W/m·K, λ_BR = 0.0474, and λ’_BR = 4.74·10³ W/m²·K (with assumption that λ_BR = 10 μm).

It can be concluded from calculation that the bond thermal resistance may influence the thermal resistance of the joint significantly. Similar results were observed in experiment.²³ Three thermally conductive adhesives were used to bond together two thick platelets of silicon carbide-reinforced aluminum (AlSiC) composites. The adhesive thickness was kept to a minimum, ca. 40–60 μm. The results of thermal resistance measurements at room temperature are listed in Table 2.1.

Certainly, bond thermal resistance depends strongly on the conductive filler content in adhesive formulations and on the bonded materials. It has been calculated²⁴ for different silver-filled epoxy adhesives and combination of Si – Al bonded materials, that 2Θ_BR /Θ_T may change from 36 % to over 90 % (with thickness of adhesive layer in the range 26–54 μm).

Participation of bond thermal resistance in total thermal resistance depends on joint temperature. It has been measured²³ that for all types of adhesive formulations described in Table 2.1, an increase of operating temperature from 23 °C to 300 °C causes a decrease of the ratio Θ_BR /Θ_T of about 18 %.

Based on calculation and results of experiments, it can be generally concluded that the adhesive bond resistance in comparison to the total resistance of thermally conductive adhesive joints is an important factor, especially in the case of microelectronic packaging where the adhesive layer is in the order of micrometers.

2.4.1 Micrometer-sized fillers

Since the 1990s, isotropically conductive adhesives have been widely used in electronic packaging. Such composites consist of a polymer resin and electrically conductive fillers, mostly silver, although gold, nickel and copper are also used. Silver is unique among all of the cost-effective metals by nature of its oxide, which unlike other metal oxides, is a good conductor. Silver, with thermal conductivity 420 W/m·K, has been considered to be the most suitable material also as the thermally conductive filler. In fact, typically formulated electrically conductive adhesives with fillers of micronized silver particles can be also used as thermally conductive adhesives. The particles usually have the form of very thin flakes (Fig. 2.8a), but producers may also use particles of different shapes (e.g. more spherical, Fig. 2.8b), which influences the thermal contact resistance.

As was stated earlier, larger particles of filler cause higher thermal conductivity of composites because of the lower number of contact points between particles, which generate thermal contact resistances Θ_TC (Equation 2.6). This means that with the use of nano-sized silver particles as fillers, the summarized value of Θ_TC increases significantly and the thermal resistance of the singular contact path Θ_p between two surfaces also increases.

2.4.2 Nanometer-sized fillers

Nanosilver as a base material of composites for ink-jet technology is used in today’s miniaturized electronics, mostly for conductive microstructures and contacts with dimensions in the range of tens of micrometers.²⁷ Nevertheless, the results of experiments show that nanosilver particles of 3–7 nm diameter added to a micro-sized filler in thermally conductive adhesives may improve the adhesive thermal conductivity, even by 2.55 times.²⁸ This is explained by making the contact resistance between basic (micro-sized) particles lower, owing to the fusion process of nano-particles at temperature of 200 °C.

Theoretically, according to Equation 2.7, the multiplying of the conductive paths may decrease thermal resistance of adhesive Θ_TA. Because of this, there are efforts to enhance the electrical conductance of adhesives containing micro- sized silver particles by adding silver nanowires^29,30 or carbon nanotubes.³¹ In a system composed of silver particles and either nanowires or carbon nanotubes, these slender particles are excellent materials to contact the particles together. As an example, Fig. 2.9 shows that the carbon nanotubes are spanning over the gap between two flake silver particles and forming a conductive pathway. Unfortunately, for the heat transport the role of such a single pathway is not important and only a much denser network of such additional ways of heat transport may improve the thermal conductivity of an adhesive.

Carbon nanotubes were discovered in 1991 by Sumio Iijima,³² firstly as multi-wall structures with outer diameters of 4–30 nm and a length of up to 1 μm which consisted of two or more seamless graphene cylinders concentrically arranged. Single-wall carbon nanotubes, which are seamless cylinders each made of a single graphene sheet, were reported two years later. Their diameters range from 0.4 to 2–3 nm, and their length is usually of the micrometer order, i.e. with a very high aspect ratio. At present, carbon nanotubes can be obtained using various techniques: arc discharge, pyrolysis of hydrocarbons over catalysts, laser and solar vaporization, and electrolysis. These nanoscale materials can exhibit different morphologies such as straight, curled, hemitoroidal, branched, spiral, and helix-shaped, with different numbers of walls.^33,34

The high value of thermal conductivity of carbon nanotubes is the most desirable feature that can improve heat transport in adhesives. The thermal conductivity of individual multi-wall carbon nanotubes is estimated to fall in the range of 2000 to over 3000 WIm-K.^35–37 It is also reported that the l of a carbon nanotube depends on both its chiral vector and length. Non- equilibrium molecular dynamic modeling has clearly shown³⁸ that the correct chirality may increase the conductivity by tens of percent, and the longer is the nanotube, the higher is the thermal conductivity.

As the measurement of the thermal properties of a single nanotube is extremely difficult, and conventional methods cannot be used for this purpose, the values of λ presented in the literature vary. Additionally, experimental results show that the thermal conductivity of a carbon nanotube depends on both temperature and tube diameter. The λ value increases with decreasing tube diameter and increases with increase in temperature, appearing to have an asymptote near 320 K.³⁷

The first composites with carbon nanotubes were proposed in the last decade of the twentieth century. Various materials have been used as matrices, firstly polymers – mostly thermoplastics based on epoxy resin, but also polymethyl methacrylate (PMMA) and some gels.³⁹ The dispersion process requires a crucial technology to develop a nanocomposite because the specific surface area of carbon nanotubes is too large to well disperse in the matrix resin. According to theoretical calculations,⁴⁰ the specific surface area of single-wall nanotubes may reach a value of 1315 m²/g. For multi-walled tubes the value is lower, but even for higher tube diameters the specific surface area amounts to a few hundreds of m²/g. If the filler is not properly dispersed, the thermal property of the composite cannot be significantly increased. Several processing methods are used for this reason, many of them based on improving nanotubes/matrix interactions,^41–44 such as an ultrasonic bath, ultrasonic finger, melt-mixing, speed mixer, in-situ polymerization or solution processing. To prepare carbon nanotubes for proper dispersion on a macroscopic scale, some preprocessing is required. They can be purified to eliminate non-nanotube material, followed by deagglomeration for dispersing individual nanotubes. As the phonon scattering across the interface between particle fillers can greatly affect the thermal conductivity of a polymer composite, to improve heat transport, functionalization of the carbon nanotubes surfaces is necessary. This is usually done by a chemical treatment, consisting of many steps.^41,45

In fact, even using complicated processes for improving nanotube surfaces and interactions with matrix, it is impossible to achieve filler content higher than a few weight percent. It is reported that the dispersion of 5 wt% carbon nanotubes in an epoxy resin has been obtained by an ultrasonic treatment. Although the carbon nanotubes were well separated, they remained poorly distributed.⁴⁶ At such a filler content, the viscosity of the composite is more then ten times higher in comparison to the pure matrix.³⁹ It is desirable to disperse this filler as uniformly as possible, because the thermal conductivity of a composite rises almost linearly with rising content of carbon nanotubes in the polymer matrix.⁴⁴

Nevertheless, the thermal conductivity of composites filled with single-or multi-walled carbon nanotubes is lower than for composites with micro-sized fillers. The measured l value usually reached less then 1 W/m-K.^47–50 This is lower than the calculated value based on the model of the heat transport with neglecting all interactions between the nanotubes.⁵¹ For contents of ‘average shape’, multi-wall nanotubes at a single weight percent, the model predicts the thermal conductivity of a composite at the level of 1.8 W/mK. It has been calculated that effective conductivity increases linearly with increasing nanotube loading; the conductivity of the adhesive is not very sensitive to the nanotube length (increases marginally with increase in length) and the conductivity can change drastically with a change in the diameter – a smaller diameter of the nanotubes can significantly increase the overall conductivity.

Adding of carbon black with nano-sized particles may improve the thermal conductivity of epoxy-based composites with 2 wt% carbon nanotubes.⁴² by about 20 %. Nevertheless, thermal conductivity of such composites is reported as only ca. 0.6 W/mK.

2.4.3 Diamond fillers

Adhesives filled with diamond seem also to be very promising for heat transport composites because specially purified synthetic diamonds could have the highest thermal bulk conductivity, 2000 W/mK or higher. Unfortunately, thermal measurements using micron-sized synthetic diamond powder as filler showed a relatively low thermal conductivity in comparison to one with micron-sized silver. This is believed to be due to the presence of nitrogen and other impurities in the synthetic diamond.⁵² Additionally, the thermal conductivity of diamond is strongly dependent on the microstructure of the material⁵³ and is correlated to grain size, even for the bulk material.⁵⁴ Probably, there are effects of structural defects such as stacking faults, twins or dislocations from synthetic processes. Typically, where the grain size is in excess of 30–50 μm, the effects of grain boundaries on thermal conductivity become insignificant, whereas submicron grain sizes can reduce the thermal conductivity by more than two orders of magnitude. As a result, adhesives with diamond filler have lower thermal conductivities than those with micro-sized silver as a filler.⁵²

2.4.4 Electrically insulating fillers

Thermally conductive but electrically insulating adhesives are increasingly important for electronic packaging in applications where a composite ought to meet at least three requirements, namely mechanical bonding, heat dissipation and acting as an insulating layer. Usually, composites with insulating fillers are characterized by lower coefficients of thermal expansion than conducting adhesives. Beside diamond, boron nitride, aluminum nitride, alumina and other materials can be used as electrically insulating fillers.

Aluminum nitride (AlN) with its relatively high thermal conductivity of 110–200 W/mK^36,52,55 is often used as a conducting filler. Generally, the thermal conductivity of an adhesive with this filler increases with increasing filler particle size and its volume fraction.^15,16 The same influence of alumina (Al₂O₃) particle size on the thermal conductivity of adhesives has been observed.^14,56 The possible maximum volume content of such fillers in a polymer, which results in a higher value of composite thermal conductivity, can be obtained when a mixture of particles with different diameters is used.⁵⁷ Generally, the thermal conductivity of the polymer composites can be enhanced by applying a hybrid filler.⁵⁸

2.5.1 Types of polymer base materials

The main role of the polymer base material of an adhesive is to form mechanical bonds at interconnections. This is done after the polymerization process, when an adhesive has the form of a solid body. Before hardening by polymerization, it should have a suitable viscosity and provide the right characteristics for printing, stenciling, and other industrial dispensing methods.

Polymers are commonly classified as either thermoplastics – typically able to be melted or softened with heat, or thermosets – which resist melting and cannot be re-shaped. Thermoset epoxies are by far the most common adhesive matrices and for electrically conductive adhesives they have found use since the early 1950s. Thermosets are crosslinked polymers and generally have an extensive three-dimensional molecular structure. The polymerization process is strongly accelerated at higher temperatures, although thermosets cured at room temperatures are commercially available when the correct catalyst or hardener is added. Crosslinks are chemical bonds occurring between polymer chains that prevent substantial movement, even at elevated temperature. Silver–epoxy can be considered the base line for isotropic conductive adhesives used for electronic component assembly.⁵⁹ The curing of an epoxy is complex and the whole reaction consists of several steps. The chemistry of the cure begins with the formation and linear growth of the polymer chains, which soon begin to branch, and then to crosslink. This sudden and irreversible transformation from a viscous liquid to an elastic gel marks the first appearance of the infinite network. Gelation typically occurs below 80 % conversion, and beyond the gel point the reaction continues with substantial increase in crosslink density towards the formation of one infinite network.⁶⁰

Vitrification is another phenomenon of the growing chains or network. This transformation from a viscous liquid or an elastic gel to a glass begins to occur as the glass transition temperature of these growing chains or networks become coincidental with the cure temperature. Further curing in the glassy state is extremely slow and, for all practical purposes, vitrification brings an abrupt halt to curing.⁶⁰

Polymeric materials have thermal conductivities about 2000 times lower than that of silver and much lower than those of other fillers. Thermal conductivities of all polymeric materials, epoxy or other types, thermoset or thermoplastics, range from 0.2 to 0.3 W/mK and so these practically do not take part in the heat transport. Nevertheless, the thermal conductivity of resins can be improved by introducing a high-order structure having microscopic anisotropy while maintaining macroscopic isotropy. In the case of diepoxy monomers with a biphenyl group or two phenyl benzoate groups as mesogens, the thermal conductivities are up to five times higher than those of conventional epoxy resins.⁶¹ This is possibly due to these molecular groups, because mesogens form highly ordered, crystal-like structures which suppress phonon scattering.

In commercial applications, the filler is responsible for virtually all the heat transport, but the thermomechanical properties of the polymer matrix may strongly influence the thermal conductivity of adhesives for microelectronic packaging, mostly by shrinkage during polymerization and then relaxation processes.

2.5.2 Role of cure shrinkage of polymer base material

The level of volume shrinkage depends mainly on the type of resin, and may reach a few percent of its initial value. Figure 2.10 presents a typical volume shrinkage chart of an epoxy resin during curing at 150 °C. Of course, the time of curing depends strongly on temperature. In the case presented in Fig. 2.10, the volume shrinkage indicates the high polymerization level of the tested resin after about 25 minutes of heating at 150 °C, while temperature 60 °C needs more than 300 min to achieve the final shrinkage (7.56 %).⁶³ The role of the shrinkage process during polymerization at high temperature is easy to observe in the case of electrically conductive adhesives, which usually consist of a similar type of resin and filler (e.g. micrometer-sized silver flakes) as thermally conductive adhesives. For such composites, the curing time depends mostly on the curing temperature, the type of polymer base material and additives. The cure process produces a shrinkage of the polymer matrix, which exerts a pressure on the conductive particles, forcing them into closer contact. Independent of the dominant mechanism of current transport,⁶⁴ closer contact between filler particles strongly decreases the electrical resistance of adhesive joints. Examples of such resistance changes for two different composites are presented in Fig. 2.11.

Also in the case of thermally conductive adhesives the shrinkage of the polymer during the polymerization process makes real contact between filler particles. Let us assume that in the beginning of the shrinkage process such particles are in the form of perfect spheres, and that they touch each other at one contact point ‘A’ (Fig. 2.12). Further shrinkage results in stress between particles which may cause elastic or plastic strain of the contact members. For the epoxy resin with shrinkage characteristics presented on Fig. 2.10, the contact pressure was estimated at more than 0.16 GPa. It has been calculated that if two silver particles of 1 mm diameter (2R) were deformed by this shrinkage force contact radius r between them increases from ‘point’ to more than 0.08 μm⁶² (Fig. 2.13).

The change of the contact area influences the thermal contact resistance. To answer the question of how the contact state between particles affects the thermal conductivity of the adhesive, numerical modeling of the silver-filled formulation was used. The simulation was performed not only for two balls (Fig. 2.13) but for the more complex 3D structure called the unit cell.^66,67 The unit cell has the maximum possible packing efficiency (V_filler/V_total = 74.05 %) in 3-D for spherical particles, i.e. hexagonal close-packed (HCP) or face-centered cubic (FCC) crystal equivalent. It was assumed that particles contact each other in the ideal pure metallic contact area (the impurities, roughness etc. were neglected).

The unit cell was analyzed using ANSYS 8.0 commercial finite element method software for two different contact area radii (r) of balls (Fig. 2.13). The simulation results show that increasing the r/R ratio from 0.02 to 0.05 causes a change of the thermal conductivity of the cell from 26.45 to 49.12 W/mK. Such results of calculation show that the role of the polymer matrix may be crucial for the level of heat transfer in composites. But this state may not be stable, as the stresses in the resins may relax with time. This is because thermosetting polymers are characterized by viscoelastic properties. The viscoelastic behavior of the matrix cannot be neglected because of the stress relaxation.

2.5.3 Viscoelasticity of polymer base materials

The general linear viscoelastic equation is the basic equation for modeling the development of viscoelastic stresses (σ_ij) as a function of temperature and loading time in fully cured polymer materials⁶⁸’⁶⁹

[2.12]

where G and K denote the shear and bulk relaxation moduli, respectively (they are time (t)- and temperature (T)-dependent), and is the effective strain contribution.

Shear and bulk moduli also depend on the conversion level (degree of cure, α). Therefore, the viscoelastic stress described by Equation 2.12 can be written as

[2.13]

and consists of mechanical, thermal, and cure shrinkage parts^70,71

[2.14]

where β_L^g and β_L^r are the linear coefficients of thermal expansion measured below the glass transition temperature T_g (glassy region) and above T_g (rubbery region), respectively, and γ_L is the linear cure shrinkage. The change of the modulus during curing can be explained by the change in the molecular structure of the matrix.

If the time is relatively short (t → 0), shear and bulk relaxation moduli are treated as instantaneous shear and bulk moduli G₀ and K₀, respectively, and they can be determined from the values of the instantaneous (obtained from the high-rate tests) elastic (Youngs) modulus E₀ and Poisson’s ratio v₀

[2.15]

Poisson’s ratio is often assumed to be time-independent in viscoelastic materials; therefore Equation 2.15 can be used for the whole time-scale of the shear, bulk, and elastic relaxations while strains are small. The advantage of such an approach is that while modeling the behavior of viscoelastic materials, only one modulus needs to be measured (e.g. the shear relaxation modulus) and the other can be calculated from Equation 2.15.

The effect of temperature in Equation 2.12 is usually not included as G(t,T), but as G(t_red[T]), where t_red is the so-called reduced time scale and G(t_red) is referred to as the master curve. The reduced time scale is defined as

[2.16]

where α_T is the temperature-dependent shift factor, which can be described by the WLF (Williams, Landel, Ferry) equation

[2.17]

where T_ref is the reference temperature, C₁ and C₂ are material parameters. The reduced time scale t_red can be replaced by reduced frequency scale f_red = α_Tf.

The standard way of measuring dynamic changes of the mechanical parameters of shear modulus for polymeric materials is by a continuous monitoring of the thermo-mechanical properties from the liquid to the fully cured state by applying the liquid compound (e.g. resin + hardener) in small gaps between parallel plates of the shear clamps of a dynamic mechanical analyzer (DMA). The samples are subject to a series of sinusoidal strains or stresses at different frequencies (a frequency sweep). The temperature is then increased by 5–10 °C and another frequency sweep is applied. This procedure is repeated from about 80 °C below the glass transition temperature (T_g) to about 80 °C above it. Far below the glass transition temperature the modulus data (stress amplitude divided by strain amplitude) is frequency independent. This modulus is called the glassy modulus. Far above the glass transition temperature, the material either melts, as is typical of thermoplasts, or displays the non-zero, frequency-independent rubbery modulus of thermosets. In between, in the so-called viscoelastic region, the modulus is frequency-dependent and lies between the glassy and the rubbery values. It is customary to shift the individual modulus vs frequency curves along the logarithmic frequency axis until they overlap and form a master curve. The shift (α_T) is different for each temperature. The master curve, together with this shift factor, completely describes the temperature and frequency-dependent modulus data and can even be used to predict the mechanical behavior at time scales and temperatures different from those of the test conditions.⁷²

When constructing the master curves, the shift factor a_T has to be evaluated (see Equation 2.17) by estimating the values of the WLF equation parameters. As an example, the parameters of the WLF equation for an epoxy with volume shrinkage of ~ 3∙10^− 2 (the linear reaction cure shrinkage assumed as 1 × 10^− 2) are collected in Table 2.2.⁶⁹

2.5.4 Influence of polymer base materials on contact pressure between filler particles

To study the contact pressure occurring between the filler particles due to the cure shrinkage of the polymer matrix (with parameters from Table 2.2) and its influence on thermal conductivity, the 2D model shown in Fig. 2.12 was implemented by ABAQUS FEM software.⁷³ The model represents two identical spherical silver particles surrounded by cylindrically shaped epoxy resin. The particles touch each other at one point (A) at the beginning of the simulation and the contact pressure and its change in time after curing at different temperatures were monitored at this point.

The simulation procedure was divided into two steps. In the first step, a steady state analysis was performed to simulate the shrinkage caused by curing the epoxy and to monitor the initial contact pressure between silver particles. In the second step, transient analysis was performed and various temperatures (T₀ changes from 40 °C to 70 °C) were applied to observe the temperature dependence of the contact pressure relaxation. The results of the simulations, i.e. the contact pressures between particles vs time for different temperatures, are shown in Fig. 2.14.⁶⁹

The contact pressure occurring between filler particles due to the cure shrinkage relaxes with time. The time needed to reach the fully relaxed state strongly depends on the working temperature of the system. In the considered case, the contact pressure is fully relaxed when it decreases from an initial value of 0.73 GPa to 0.03 GPa, as it is shown in Fig. 2.14. When the temperature is 70 °C, the contact pressure becomes fully relaxed after 10 seconds, but when it is lower (e.g. 40 °C), the full relaxation state is reached after about 10⁹ seconds (more than 30 years!).

The thermal conductivities of the tested structures have been calculated (using Finite Elements Method) before and after relaxation. After the relaxation of the contact pressure, the thermal conductivity drops to around 50 % (0.484) of its initial value for the unrelaxed structure.⁶⁹

The shrinkage phenomenon of the polymer base seems to be very important in heat transport. On the other hand, shrinkage means lowering of the formulation volume, which may lead to the occurance of some defects. Much more dangerous in effects may be the presence of solvents as polymer base additives.²⁸ The minimum void ratio was obtained when the solvent vaporized effectively before the resin had been cured.

2.6.1 Steady-state methods of adhesive thermal conductivity measurement

Steady-state methods are based on establishing a steady temperature gradient over a known thickness of a sample, and on controlling the heat flow from one side to the other. The determination of the thermal conductivity of the insulation follows from the basic law of heat flow (Equation 2.1).

In the guarded hot plate method, the specimen is placed between uniformly distributed heaters and a heat sink (or cold plate), both of which contain temperature sensors that measure the temperature drop across the sample after a steady-state heat flow has been established. The method’s error is diminished by modifying the base measurement configuration, e.g. by the line heat source applying,⁷⁹ more stable operation at liquid nitrogen temperatures⁸⁰ or measurement under vacuum conditions.⁸¹ In result, the temperature drop ΔT_T can be measured (Fig. 2.7) and by changing the thickness of the sample, it is possible to extract the thermal conductivity 1_TA of the tested material.^17,22 The measured sample can be prepared either as a disc and directly placed between hot and cold plates,^17,22 or additional blocks made from a material with a low value of λ can be joined to the sample.⁸² In such a case, the additional temperature drops (Fig. 2.7) have to be taken into consideration.

As an example, the measurement setup is shown in Fig. 2.15. The system consists of a heater, a heat sink, a pair of contact members (with a known thermal conductivity λ, e.g made from iron,^17,82 aluminum,⁸³ or copper,^84,85), as well as temperature sensors.²¹ The whole setup is placed in a high vacuum environment in order to minimize the effect of heat dissipation through convection. The radiation is out of the temperature work-range, and therefore is neglected, and the heat conduction is a dominating factor in transport heat energy. The limiting error of the measurement method described previously is known, and can be added to information about the measured value. Thus, in this setup, the thermal conductivity of the adhesive was found to be 1.26 W/m·K, with 0.44 W/m.K standard deviation.⁸⁶

2.6.2 Transient methods of adhesive thermal conductivity measurement

The transient hot-wire technique is a transient dynamic technique based on a linear heat source of infinite length and infinitesimal diameter and on the measurement of the temperature rise at a defined distance from the linear heat source embedded in the test material. As an electric current of a fixed intensity flows through the wire, the thermal conductivity can be derived from the resulting temperature change over a known time interval.⁸⁷ Nowadays, four variations of the method are in use:⁸⁸ hot wire standard technique, hot wire resistance technique, two-thermocouple technique, and hot wire parallel technique. The theoretical model is the same, and the basic difference among these variations lies in the temperature measurement procedure. This makes it possible to determine the thermal conductivity of a wide range of materials, including cured thermally conductive adhesives.

Modification of the transient hot-wire technique may also concern the shape of the heat source, which can be formed as a strip or a disc. For measuring the thermal properties of materials available as thin slabs, a hot disc can be used. It is assumed that the slabs are thermally insulated in such a way that the heat losses at the boundaries are negligible, compared with the total input of power.⁸⁹

There are also a number of presently existing transient methods of measuring thermal conductivity indirectly – by measurement of a material’s thermal diffusivity. The relation between the thermal conductivity λ (W/m·K) and the thermal diffusivity α (m²·s^− 1) is given by

[2.18]

where ρ (kgm^− 3) is the density and c_p (Jkg^− 1·K^− 1) is the specific heat of the tested composite.

Among the transient methods, the so-called flash method is the most popular in determining thermal diffusivity. The method was described in was developed over the next few years.^91–93 In this method, a pulse of energy is absorbed on the front face of a specimen and the subsequent temperature change at the rear face is recorded. The front surface of the sample must be uniformly irradiated for a short time compared with the rise time of the back surface temperature. A laser beam is the most popular heat source, but also a xenon lamp or more flexible energy sources can be used (e.g. an electron beam).^94,95 The measured sample usually has the form of a disc with a diameter of a dozen or so millimeters and a few millimeters in thickness. The shape of the time–temperature curve on the back side of the sample is used in analysis.^90,91,96,97 The method’s errors result from an assumption that the sample is perfectly insulated from the environment during the test (there is no heat exchange) and the whole energy is absorbed instantaneously (zero pulse width) in a very thin layer of the sample material.

The popular transient method described above does not exhaust all the possibilities of thermal properties measurement. In the photoacoustic technique, a heating source (normally a laser beam) is periodically irradiated on the sample surface. The acoustic response of the gas above the sample is measured and related to the thermal properties of the sample. The method can be used for a single layer on a substrate⁹⁸ as well as multilayered materials measurement.⁹⁹ The 3ω method employs a metallic strip¹⁰⁰ or wire¹⁰¹ in intimate contact with the specimen surface. An AC electrical current modulated at a regular frequency ω generates thermal waves in the specimen. Because the electrical resistance of the strip depends on the temperature, the resistance is modulated and it is possible to extract λ from an electrical measurements.¹⁰²

2.6.3 Thermal conductivity of adhesives

Thermal conductivity (W/m·K) of selected adhesives measured by various methods is listed in Table 2.3. The data are taken from papers, not from manufacturers’ offers.

As was mentioned previously, not all thermal conductivity measurement methods are characterized by high accuracy and some thermal conductivity values from Table 2.3 may be questionable. Nevertheless, the data cited in the table point out that

Theoretically, according to Equations 2.6 and 2.7, improvement of thermal conductivity of adhesives saturated by filler with low thermal resistance can be achieved by:

The first step of conductivity improvement can be done by filler particle phase or shape changing during the base material polymerization (see Fig. 2.12) or additional thermal process. There are investigations of new formulations which incorporate novel fluxing polymeric resins in combination with a blend of metal alloy filler particles. During the curing process, the metal filler particles sinter together and are capable of forming ‘ solder-like’ metallurgical connections to a variety of metal surfaces, resulting in a lowering of thermal contact resistance values. These materials have both high electrical and thermal conductivity – closer to solder materials than most polymer- based materials but with the processing advantages of the best polymeric adhesives.¹⁰⁷

The idea of additional thermal processing has come from experiences with electrically conductive adhesives. An adhesive based on epoxy resin filled with a mix of silver flakes (Fig. 2.8a) and powder (Fig. 2.8b) was tested. The content of filler in the adhesive was 65.4 wt%. After ‘standard’ curing (30 min at 180 °C), the adhesive was additionally annealed for 2 hours at 180 °C. After this process, scanning electron microscopy inspection revealed an increase in size of the particles and a decrease of electrical resistance of about two orders of magnitude.²⁵ Probably the observed effect was caused by recrystallization of the silver. In general, recrystallization begins when the temperature exceeds the so-called recrystallization threshold T_r, which is expressed by the following equation

[2.19]

where T_m is the melting point (K). The applied additional post-curing temperature was about 37 % of the silver melting point temperature and a process of changing the shape of particles may occur, as the value of T_r strongly depends on material purity and for extremely pure metals, recrystallization can be observed at a much lower temperature. A similar thermal effect was observed in the case of adhesives with nanosilver annealed at 180 °C.¹⁰⁸ The particles were fused through their surface and many dumbbell-type particles could be found. The morphology was similar to the typical morphology of an initial stage in the sintering process of ceramic, metal and polymer powders.

Both the electrical and thermal conductivity increase has been stated for a typical isotropic conductive adhesive composed of an epoxy base material and silver filler (spherical and flake-shaped particles) content of 85 wt%.¹⁰⁹ As recommended by the manufacturer, a suitable curing condition was 150 °C for 0.5 hours. After this process, the adhesive was additionally annealed in 200 °C during 30 hours. The sample under investigation exhibited anisotropy in the thermal conductivity, due to an oriented dispersion of flake- shaped filler particles. In a direction parallel to the squeegee direction, the thermal conductivity, increased by about 85 % due to the additional thermal process.

When carbon nanotubes are dispersed in the polymer, decreasing Θ_TC can be achieved by functionalization of their surfaces. Experimental results have suggested⁴⁵ that such treatment can significantly reduce interfacial thermal resistance between filler and epoxy resin by forming stronger chemical bonding across the interface. Surface treatment improved the interaction between filler and polymer matrix, which can significantly reduce phonon scattering, and also increase thermal conductivity.

Increasing the number of parallel contact paths in the direction of the temperature gradient can be achieved by applying hybrid fillers. The idea is presented in Fig. 2.9, in which mixed fillers (micro-sized silver flakes and carbon nanotubes) are presented. However, the low content of nanotubes (about of 0.5 wt%) weakly influences the total thermal conductivity of the composite, and efforts of research labs are aimed at achieving a significant increase of the CNT content.³¹ A similar role may be played by nanowires, doped by the composite containing micro-sized silver particles^29,30 and a suitable ratio of filler with various particle sizes.⁵⁷

It seems to be necessary to develop a systematic method of producing various multiple junctions of ‘long’ nano fillers to establish a 3D network instead of a set of singular structures. There is information about experiments aimed at establishing a junction between crossed nanotubes, a first attempt at producing such web-like CNTs having been achieved with electron beam, ion beam, laser or other techniques of nano-welding.¹¹⁰ It is reported that under a high-voltage of 1.25 MeV at specimen temperatures of 800 °C from the random criss-crossing distribution of individual nanotubes and nanotube bundles on the specimen grid, several contact points could be identified where tubes were crossing and touching each other. After a few minutes of irradiating and annealing, different shapes of junctions were established.¹¹¹ It was demonstrated that also ion irradiation should result in the welding of crossed nanotubes and junctions were obtained.¹¹²

• higher conductivity is obtained when micro-sized, not nano-sized, particles of filler are used,

• the bulk thermal conductivity of the filler when particles are of micrometer size does not influence the thermal conductivity significantly,

• the higher the filler content, the better the thermal conductivity of the adhesive.

• lowering the numbers of and decreasing the thermal contact resistance between particles making chains for heat transport in the direction of the temperature gradient,

• increasing the number of parallel contact paths in the direction of the temperature gradient.

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