Thermally Conductive Polymer Matrix Composites

Abstract

Thermally conductive polymer matrix composites are used increasingly for thermal management of electronic packaging system. The polymer matrix includes both thermosetting and thermoplastic types. Different kinds of fillers or reinforcements have been developed to process composite materials with desired thermal, mechanical, and electrical properties. Reinforcement fillers have an important role to play in maximizing polymer performance and production efficiency. Cost reduction, density control, optical effects, thermal conductivity, magnetic properties, flame retardancy, and improved hardness and tear resistance have been increased the demand for high-performance fillers. Several types of reinforcements, especially nanoparticulate fillers, have been used in polymer matrix composites: vapor grown carbon fiber (VGCF), carbon foam, carbon nanotube (CNT), and other thermal conductive particles, such as ceramic, carbon, metal or metal-coated particles, as well as metal or carbon foams. Nanoparticles of carbides, nitrides, and carbonitrides can be used to reinforce polymer matrix nanocomposites with desirable thermal conductivity, mechanical strength, hardness, corrosion, and wear resistance. To achieve these of desirable properties, however, polymer matrix and layout or distribution of reinforcements including nanoparticles need to be optimized. This chapter will give a brief review on the advanced thermally conductive polymer matrix composites, including polymer matrix types, reinforcement selection and its effect on thermal conductivity, fabrication and manufacturing process, and typical applications for thermal management of electronic packaging system.

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Introduction

Thermally conductive polymer matrix composites have been increasingly used for thermal management of electronic packaging system. Composites with continuous reinforcements such as fibers, woven or cloths, are usually used as substrates, heat sinks, and enclosures. Composites with discontinuous reinforcements such as particles or whiskers, are generally used for die attach, electrically/thermally conductive adhesives, encapsulations, thermal interface materials (TIMs), and electrical interconnections (Chung 2003). The composite matrix can be thermal plastic or thermosetting polymer. Thermoplastic composites can be reworked by heating for the purpose of repair, whereas thermosetting composites do not allow reworking, but are attractive for their thermal stability and dielectric properties. The polymer matrices for thermally conductive composites shall preferably exhibit low dielectric constant, low dissipations factor, low coefficient of thermal expansion (CTE), and being compliable for electronic packaging applications. The composites can be electrically conducting or electrically insulating, or both electrically and thermally conducting, depending on the reinforcement materials used. With proper reinforcement display, the electrical or thermal performance of the composites can be isotropical or anisotropical (Chung 2003).

Typical composites for electronic packaging components exhibit tailored CTE by varying the architecture of reinforcements, such as carbon fibers, silicon carbide (SiC) particles, boron nitride particles, titanium nitride particles, and diamond particles. The resulting composite materials with these reinforcements have not shown substantial improvements in thermal conductivity. VGCFs that are grown through the pyrolysis of hydrocarbon gas in the presence of a metal catalyst, have a higher thermal conductivity than any other carbon fiber and may be produced at a lower cost. VGCFs also exhibit the highest degree of graphitic perfection of any known carbon fiber. Polymer matrix composites formed from the interwoven VGCF mats, exhibit high thermal conductivity (>400 W/m K), low density, and variable CTE (Ting 2002).

On the other hand, the through-thickness thermal conductivity in the polymer matrix composite becomes particularly important in applications such as composite space electronics enclosures where the heat dissipation is entirely dependent on thermal conduction to a heat sink. The spreading of heat at the composite surface and subsequent localized conduction in the through-thickness direction down to high thermal conductivity fiber is thought to be the key to designing a light weight, thermally efficient enclosure. Consequently, there exists a need for lightweight, thermally conductive through-thickness composite materials. For instance, low conductivity in the through-thickness direction of unidirectional fiber reinforced composites poses problems in any design that requires a thermal path to a high thermal conductivity fiber, particularly where large amounts of heat are input over small areas. As one approach, introducing metal particles into the matrix will increase the through-thickness conductivity throughout the composite; however, the weight density of the metallic reinforcements over a polymer matrix composite is greatly increased. Locally increasing the through-thickness thermal conductivity and then allowing the high thermal conductivity fibers to spread and orient the heat flow to a heat sink appears to be an optimal solution (Roberts 1999).

Polymer Matrix Types

The polymer matrices in advanced thermally conductive composites include both thermosetting and thermoplastic types. Table 5.1 gives general properties of some typical polymer matrices. The most widely used thermosetting resins are polyester, which provide a combination of low cost, versatility in processing, and relatively good property and performance. Polyester is generally made by reacting dibasic acids such as maleic anhydride or phthalic anhydride with dihydric alcohols such as ethylene glycol (antifreeze) in approximately equal amounts. The resulting polymer is a short chain polymer with a molecular weight of about 5,000 (about 35–40 repeat units or mers) and is a stable liquid. The liquid resin is set into an amorphous solid by cross linking the polyester chains to each other. The cross linking occurs by the addition of a small monomer molecule such as styrene. The monomer, like the active sites on the polyester chain has an unsaturated C=C bond and it is this that provides the bridge between the polyester chains (Pilling 2005).

Table 5.1 General properties of typical polymer matrices of polymer matrix composites (Alvino 1994)

Epoxy resins are of particular interest to structural and electronic applications because they offer a unique balance of chemical and mechanical properties combined with extreme processing versatility. The performance of epoxy resins is highly dependent on the formulation, including the base resin, curatives, and modifies (Nutt 2001). Epoxy resins are much more expensive than polyester resins because of the high cost of the precursor chemicals, most notably epichlorohydrin. However, the increased complexity of the epoxy polymer chain and the potential for a greater degree of control of the cross linking process gives a much improved matrix in terms of strength and ductility. Most epoxies require the resin and hardener to be mixed in equal proportions and for full strength require heating to complete the curing process. This can be advantageous as the resin can be applied directly to the fibers and curing need only take place at the time of manufacture, such as prepreg or preimpregnated fiber. Epoxy polymers are made by reacting epichlorohydrin with bisphenol-A in an alkaline solution which absorbs the HCl released during the condensation polymerization reaction. Each chain has a molecular weight between 900 and 3,000 (about 3–10 mers) with an epoxide grouping at each end of the chain but none within the polymer chain. The epoxy is cured by adding a hardener in equal amounts and being heated to about 120°C. The hardeners are usually short chain diamines such as ethylene diamine. Heat is usually required because the cross linking involves the condensation of water which must be removed in the vapor phase (Pilling 2005).

Silicones are used mainly for TIMs in electronic packaging. Silicone polymers composed of alternating atoms of silicon and oxygen with organic substituents attached to the silicon atoms, which may exist as liquids, greases, resins, or rubbers. They have good resistance to water and oxidation, stability at high and low temperatures, and lubricity. Silicones are obtained by the condensation of hydroxy organosilicon compounds formed by the hydrolysis of organosilicon halides. The first products are usually low in molecular weight (n = 2–7), and usually consist of a mixture of linear and cyclic species, especially the tetramer. Fluids having a wide range of viscosity are prepared by polymerizing further, using a monofunctional trichlorosilane to limit molecular weights to the value desired. Elastomers are made by polymerization of the purified tetramer using an alkaline catalyst at 100–150°C (212–302°F). Properties can be varied by partial replacement of some of the methyl groups by other substituents and by the use of reinforcing fillers. The wide range of structural variations makes it possible to tailor compositions for many kinds of applications. Low molecular weight silanes containing amino or other functional groups are used as treating or coupling agents for glass fiber and other reinforcements in order to cause unsaturated polyesters and other resins to adhere better. The liquids, generally dimethyl silicones of relatively low molecular weight, have low surface tension, great wetting power and lubricity for metals, and very small change in viscosity with temperature. They are used as hydraulic fluids, as antifoaming agents, as treating and waterproofing agents for leather, textiles, and masonry, and in cosmetic preparations. The greases are particularly desired for applications requiring effective lubrication at very high and at very low temperatures. Silicone resins are used for coating applications in which thermal stability in the range 300–500°C (570–930°F) is required. The dielectric properties of the polymers make them suitable for many electrical applications, particularly in electrical insulation that is exposed to high temperatures and as encapsulating materials for electronic devices. Silicone rubbers are compositions containing high molecular weight dimethyl silicone linear polymer, finely divided silicon dioxide as the filler, and a peroxidic curing agent. The silicone rubbers have the remarkable ability of remaining flexible at very low temperatures and stable at high temperatures.

There are also many other polymers that can be chosen as the matrix. Basically, the matrix should provide (Trostyanskaya 1995): (1) good wettability with reinforcements to couple over their surfaces as a result of chemical and polar interactions preserving a clear phase boundary; (2) controllable or limited chemical reaction with reinforcements to be able to modify bulk properties of the reinforcement; (3) monolithization and configuration forming or shaping in regimes preventing thermodegradation and mechanical fracture or destruction of reinforcements and disturbing their distribution; (4) continuity of an uniform matrix distribution over the whole interstitial space at a required filling volume of the reinforcement.

One typical example is to use polymer as the matrix of the conductive adhesives that have been used for die mounting and terminal bonding of components in some types of hybrid circuits, especially for TIM applications. Polymers are long-chain molecules, such as epoxies, acrylics, and urethanes that are widely used to produce structural products such as films, coatings, and adhesives. Although polymers occur naturally, most are now synthesized. Their properties can be tailored to meet thousands of different applications. Polymer-based adhesives are used in virtually every industry because of this capability to customize performance. Polymers have excellent dielectric properties and, for this reason, are used extensively as electrical insulators. Although a narrow class of conductive polymers, called intrinsically conductive polymers, does exist, their other properties do not lend themselves for use as conductive adhesives. Therefore, adding conductive fillers to nonconductive polymer binders makes virtually all-conductive adhesives. The most common conductive adhesives are silver-filled thermosetting epoxies that are typically provided as thixotropic pastes. They are used to electrically interconnect and mechanically bond components to circuits. Heat is most often used to activate a catalyst or coreactant hardener that converts the paste to a strong, conductive solid. The products, which conduct equally in all directions, are referred to as isotropic conductive adhesives. These metal-filled thermosetting conductive adhesives have been used as die attach materials for many decades and are still the most popular products for bonding integrated circuits to lead frames. More recently, metal-filled thermoses have been formulated as component assembly materials. Novel polymer-based materials are now being used to replace metallurgical solders, especially for surface mount assembly (Nordic 2010).

Two-type polymers are usually used for the conductive adhesives: thermoplastics and thermosetting polymers. Thermoplastic binders, those polymers that are already polymerized, have not found widespread use for component assembly, as have the thermoses isotropic type conductive adhesives. One problem is that useful thermoplastics are solids that must either be melted or dissolved in solvent to be used. Very limited use of isotropic thermoplastics has occurred in calculators, but the difficulty in application has severely limited them. Anisotropy conductive adhesives do use thermoplastic binders because they do not require selective application. Most anisotropy products are provided in film form. The material is used by first applying it to the circuit (and sometimes to the component). Because electrical conductivity only occurs in the z-axis where opposing conductors are forced together, the film can be applied to the entire circuit. Assembly is accomplished by forcing components against the adhesive-coated circuit conductors while adding heat. A thermoplastic binder will soften and bond to the adherents. Thermoplastic materials can also be mixed with thermosetting polymers to allow lower temperature assembly by higher temperature performance. It should be emphasized that thermoplastics can be remelted. They are not altered during the assembly heating process like thermoses. They also have excellent storage characteristics and do not require refrigeration like the one-part thermoses (Nordic 2010).

Most thermosetting polymers, especially the pastes used for isotropically conductive adhesives, are polymer precursors (ingredients that will polymerize). Epoxies typically consist of a low weight liquid with reactive epoxy groups and a coreacting hardener. The addition of heat causes the two ingredients to chemically react forming very high weight, cross-linked polymers. Cross links, or chemical bonds between adjacent chains, produce the thermoses characteristic of shape retention on heating. The thermoplastics are made up of polymer chains that are independent (not linked). Heat allows the individual chains to move past one another and to be reshaped. The reapplication of heat again softens the thermoplastic making them somewhat analogous to solder. The three-dimensional (3-D) network of cross links in the thermoses prevents chain movement. Thermoses adhesives are provided as both single component (one-part) and as unmixed two-part is stable almost indefinitely. Prior to use, the ingredients must be mixed, however. This often introduces air, a serious problem, and the user may not have the equipment to produce good mixing. Both users and manufacturers prefer one-part systems. The one-part system’s major disadvantage is that the mixed system has a limited storage life unless refrigerated. Recalling that a chemical reaction rate increases for every 10°C, the converse is also true. Refrigeration slows the polymerization rate to a level where storage can be extended to 6 months or more. While it is possible to make up systems that are stable at room temperature, higher temperatures over extended times are required for curing. However, improvements in catalysts and hardeners continue and pot life, now up to 6 days at room temperature for some of the products that cure quickly (3–6 min) even as low as 130–150°C, can be expected to continue to improve (Nordic 2010).

Reinforcements of Conductive Polymer Composites

Different kinds of fillers or reinforcements have been developed to process composite materials with desired thermal, mechanical, and electrical properties. Fillers may be in the form of fibers or in the form of particles uniformly distributed in the polymer matrix material. The properties of the polymer composite materials are strongly dependent on the filler properties as well as on microstructural parameters such as filler diameter, length, distribution, volume fraction, and the alignment and packing arrangement of fillers. It is evident that thermophysical properties of fiber-filled composites are anisotropic, except for the very short, randomly distributed fibers, whereas, thermophysical properties of particle-filled polymers are isotropic (Tavman 2004).

For instance, silver is by far the most popular conductive filler for conductive polymer adhesives, although gold, nickel, copper, and carbon are also used. Metal-plated con-conductors are also used, especially for anisotropy adhesives. Silver has high thermal and electrical conductivity. The metal forms a silver oxide layer at the surface. This layer, unlike many other oxide layers, has good electrical conductivity. Due to this special characteristic the aging of the silver flakes will not result in significantly reduced conductivity in a composition including flakes. A disadvantage of silver is its tendency to migrate. Furthermore, it is relatively expensive. Even so, silver is unique among all of the cost-effective metals by nature of its conductive oxide. Oxides of most common metals are good electrical insulators and copper powder, for example, becomes a poor conductor after aging. Nickel and copper-based adhesives do not have good stability. Copper has good conductivity and is not very expensive but in the presence of oxygen it will form a continually growing oxide layer, which reduces the conductivity of the composition. This means that copper cannot offer stability with regard to thermal and electrical conductivity. Nickel forms well-defined oxides layers. Both for nickel and copper the relatively low thermal and electrical conductivity limits the use of this material to low-cost applications that do not require high thermal and electrical conductivity. Aluminum has the same tendency of forming an oxide layer resulting in reduced electrical conductivity as with copper, but it is usually used as low-cost thermally conductive filler for TIMs. Even with antioxidants, copper-based materials will show an increase in volume resistivity on aging, especially under high humidity conditions. Silver-plated copper has found commercial application in conductive inks and this type of filler should work in adhesives as well. While composites made with pure silver particles often show improved conductivity when heat aged, exposed to heat and humidity or thermal cycled, this is not always the case for silver-plated metals, such as copper flake. Presumably, the application of heat and mechanical energy allows the particles to make more intimate contact, but the silver-plated copper may have coating discontinuities that allow oxidation of the copper. Therefore, silver is found to be the most frequently used filler particle material in conductive adhesives. Gold has lower conductivity than silver and is much more expensive but it has one great advantage compared with silver, and that is a very low ability to migrate. As silver, gold forms a thin and relatively conductive oxide layer. Gold is often chosen for use in military and space electronics (hybrid circuits) to avoid the risk of silver migration. In addition to the pure metals, attempts have been made to develop composites for use as filler material. Examples are copper and glass plated with silver, and coated reformative (Nordic 2010).

Fillers for anisotropy conductive adhesives are often very different from those for isotropic adhesives. There is usually only one layer of conductive particles between the two adherents in anisotropy configurations. The conductive particle makes a mechanical contact between the conductors with the polymer binder supplying the tensional force. Although both hard and soft conductors are being used, most systems have moved toward resilient particles that deform and act like small springs. The most common type has a plastic core that is overcoated with a good conductor like gold or silver. The most popular conductive particle is a polymer sphere that has been first plated with nickel and then pure gold (Nordic 2010).

It is not only the choice of filler material but also its size and shapes that influence the properties of the final adhesive. Some of the properties that can be influenced by particle size and shape are electrical conductivity, thermal conductivity, tensile strength, viscosity, weight loss, and rheology. The production of silver flakes are often based on silver powder, which is mechanically worked using fluid energy milling or ball milling. The latter is the most frequently used. Fluid energy milling is a method in which a mixture of silver powder and surfactant is accelerated in a stream of compressed air or superheated steam. When the particles collide under the turbulent flow, the densification takes place (particle size decreases but density increases). In ball milling the silver powder mixed with solvent and surfactant is tumbled in a rotary mill loaded with balls. When collision occurs the powder is flaked. The chosen solvent must have a high flash point and low vapor pressure because of heat generation. The surfactant used must be soluble in the solvent to ensure even distribution. Often fatty acids are used as surfactant. A surfactant is used either to prevent agglomeration (fluid energy milling) or to prevent cold welding (ball milling) in both milling methods. A part of this surfactant is absorbed at the surface of the flakes produced resulting in a coating that can affect the properties of the final flake. It is very important for the resulting conductivity of the adhesive that the surfactant shows compatibility with the resin. The size of the resulting flakes, after cleaning and drying is reported to be 2–30 μm. Three other manufacturing processes for silver filler particles are chemical reduction of an alkaline silver nitrate solution–the resulting flake size being 0.5–10 μm; electrochemical cathodic precipitation in nitrate or sulphate solution–the resulting flake size being 1.2–50 μm; atomization-melted silver is induced into a high speed water- or gas-flow resulting in flakes that are 5–100 μm. The electrical conductivity of an isotropic adhesive is dependent on the amount of contact points between the filler particles. This means that the conductivity will increase with increased amounts of filler particles. However, as the amount of filler increases the amount of polymer must obviously decrease which means that there is a limit for the amount of filler that can be used without too much decrease in the properties of the polymer matrix (adhesion and tensile strength). Eighty weight percent of silver is found to be the limit in epoxy adhesives. The achieved particle size influences the amount of filler material that provides the optimum balance between electrical conductivity and tensile strength. It is found that the larger the flake particles, the greater the number of contacts. This means that larger particles, when formed as flakes, should be preferred to smaller ones (tap density, packing ability of flakes, increase with increased particle size). With regard to viscosity, as the flake particles become larger, the increase in frictional forces between particle surface results in higher viscosities. With regard to filler particle shape some influence on rheology, conductivity, and tensile strength is also observed. The use of spherical filler particles decreases the viscosity while flakes, needles, and fibers offer ideal overlapping conditions for isotropic adhesives. The manufacturer therefore can optimize the adhesive to a specific use by mixing filler particles with different shapes. In addition, the tensile strength of the final adhesive is found to be influenced by the properties of the surface of the filler particles/coatings. It is found that when nickel is used as filler, the strength increases until 40 wt% is added, whereas it only increases up to 10 wt% for silver. This is explained by the rough surface of nickel compared with a smoother surface of silver. This must only be taken as an example of the relationship between surface conditions and strength and not as a general rule in comparing nickel with silver (Nordic 2010).

Consequently, reinforcement fillers have an important role to play in maximizing polymer performance and production efficiency. Cost reduction, density control, optical effects, thermal conductivity, magnetic properties, flame retardancy and improved hardness and tear resistance have increased the demand for high performance fillers. Several types of reinforcements, especially nanoparticulate fillers, have been used in polymer matrix composites: VCGF, carbon foam, CNT, and other thermal conductive particles, such as ceramic, carbon, metal or metal-coated particles, as well as metal or carbon foams. Nanoparticles of carbides, nitrides, and carbonitrides can be used to reinforce polymer matrix nanocomposites with desirable thermal conductivity, mechanical strength, hardness, corrosion, and wear resistance. To achieve these of desirable properties polymer matrix and layout or distribution of nanoparticles need to be optimized. For example, TiC nanoparticles as reinforcement were produced by the sol-gel method. The nanocomposites were obtained in situ, in the reaction mixture and synthesizing copolymers (Meija 2007).

Design and Modeling of Conductive Polymer Composites

Different types of fillers with high thermal conductivities, such as graphite, metals and ceramics, have been added to polymers to create composite materials with improved mechanical or electrical properties. A great many modeling efforts have been made and found that it is relatively difficult to predict the effective thermal conductivity of a polymer composite material when incorporated with large volume content of filler. For small volume content of filler, the effective thermal conductivity of the composite material generally increases linearly with an increasing volumetric fraction. The Maxwell-Eucken model can predict the effective thermal conductivity of this composite material. However, for large volume content of filler, the thermal conductivity behaves nonlinearly with the volumetric fraction. Many analytical and numerical models have been developed for dispersed phase, at small volume content, the filler is likely evenly dispersed in the homogeneous phase of the matrix. Hence, they are hardly touching each other. At high volume content, it is likely that some filler is touching others and forming conductive chains of filler. These conductive chains, if aligned along the direction of the temperature gradient, will create a preferable path for heat transfer. These kinds of conductive chains may behave differently with percolation of electrically conductive connection, however, this arrangement of thermally conductive filler, in the attached phase, subsequently leads to a drastic increase in the effective thermal conductivity of the composite material.

Many experiments have been conducted to determine the impact of incorporating filler to enhance the thermal conductivity of thermally conductive elastomers. Among them, Agari is most representative. Based on the thermal conductivity measurement of three composites: polyethylene with carbon black, polyethylene with graphite filler, and polyvinyl chloride with graphite filler (maximum volume content of graphite was about 30%), Agari stated that the data deviated from the Maxwell-Eucken equation at about 10% volumetric fraction (Agari and Uno 1986). At large volume content of filler, the measured value was much higher than the prediction using existing correlations. The thermal conductivity was measured for polyethylene and polystyrene composites filled with high volume content of quartz or alumina particles. At above 30% volume content, these fillers start to interact with each other and form an “attached” system. Semiempirical correlation has been developed to fit the data obtained from experiments. Using the experimental data, many analytical and numerical models have been developed and work well for the disperse phase; these model predictions are reliable as long as the filler is homogeneously dispersed. However, if the filler is in the attached phase, only a few models show promising results. Maxwell and Rayleigh developed the effective medium approximation model. This model assumes an isotropic distribution of fillers. Since then, many modifications have been added to this model, making it increasingly sophisticated. Recent versions include shape, size effects, and interfacial resistance. However, these models are unable to predict the effective thermal conductivity accurately if touching between filler particles exists. Agari and Uno (1986) used a generalization of parallel and series conduction models to estimate the thermal conductivities of filled polymers. His modified model considered the fillers being randomly dispersed and predicted an isotropic thermal conductivity with moderate success at low volume content. An assessment was also performed on model predictions and their comparison with experimental data at high volume fraction. Discrepancy existed, especially at high volume content of filler. The general observation was that empirical correlations did not capture the structural complexity of the composite materials well. Hatta and Taya (1986) solved the heat conduction equation of a coated-fiber embedded in an infinite matrix subjected to a far-field heat flux using an analytical model. Analytical solutions for both single and multiple fibers were developed. No clear closed-form solution could be generated for high volume fraction with touching fibers. Davis and Artz (1995) used finite element analysis to investigate heat conduction of composite with bimodal distributions of particle size. Prediction is reliable as long as filler particles are not touching others. Contact resistance between matrix and particles (Kapitza resistance) was investigated by Hasselman and Johnson (1987). They developed formulas for spherical, cylindrical, and flat-plate geometry to predict an effective thermal conductivity of composites with interfacial thermal barrier resistance. The model worked well for a much-diluted system. If the interfacial resistance was zero or infinite, particle size effect on the effective thermal conductivity was minimal. For a finite resistance, decreasing the particle size usually decreased the effective thermal conductivity for a given volume content. Hence, just incorporating nanoparticles of the same size in a composite system might not work, but adding nanoparticles only in the depletion zone of a composite with filler of microns in diameter should be analyzed (Rightley et al. 2007).

Theoretical Modeling

Many theoretical, numerical, and experimental models have been proposed about reinforcement selection and their layout design for achieving high thermal conductivity of polymer composites. Figure 5.1 shows several kinds of filler layouts. The upper or lower bounds of effective thermal conductivity are given when materials are arranged in either parallel or series with respect to heat flow. For the parallel conduction model (Tavman 2004):

$${k_{\rm{c}}} = \varphi {k_{\rm{f}}} + (1 - \varphi ){k_{\rm{m}}},$$

(5.1)

Fig. 5.1

Typical layout of reinforcements

and for series conduction model:

$$\frac{1}{{{k_{\rm{c}}}}} = \frac{\varphi }{{{k_{\rm{f}}}}} + \frac{{1 - \varphi }}{{{k_{\rm{m}}}}},$$

(5.2)

where, k _c, k _m and k _f are thermal conductivities of composite, matrix and filler, respectively, and ϕ is the volume fraction of filler. In the case of geometric mean model, the effective thermal conductivity is given by:

$${k_{\rm{c}}} = k_{\rm{f}}^\varphi k_{\rm{m}}^{(1 - \varphi )}.$$

(5.3)

Using potential theory, Maxwell obtained a simple relationship for the conductivity of randomly distributed and noninteracting homogeneous spheres in a homogeneous medium (Maxwell 1954):

$$\frac{{{k_{\rm{c}}}}}{{{k_{\rm{f}}}}} = 1 + \frac{{3\varphi }}{{\left( \displaystyle{\frac{{{k_{\rm{m}}} + 2{k_{\rm{f}}}}}{{{k_{\rm{m}}} - {k_{\rm{f}}}}}} \right) - \varphi }}.$$

(5.4)

Rayleigh models heterogeneous medium for large volume fraction ϕ as homogeneous spheres embedded in the intersections of a cubic lattice (Tavman 2004):

$$\frac{{{k_{\rm{c}}}}}{{{k_{\rm{f}}}}} = 1 + \frac{{3\varphi }}{{\left( {\displaystyle\frac{{{k_{\rm{m}}} + 2{k_{\rm{f}}}}}{{{k_{\rm{m}}} - {k_{\rm{f}}}}}} \right) - \varphi + 1.569\left( {\displaystyle\frac{{{k_{\rm{m}}} - {k_{\rm{f}}}}}{{3{k_{\rm{m}}} - 4{k_{\rm{f}}}}}} \right){\varphi ^{10/3}}}}.$$

(5.5)

The interaction between spheres is found to be small even when the Rayleigh derivation is used. Thus, the simpler derivation from Maxwell is generally used to simplify the calculations.

For nonspherical fillers in a continuous matrix, when square arrays of long cylinders parallel to z axis, Raleigh’s derivation is described as (Tavman 2004):

$$\frac{{{k_{{\rm{c}},zz}}}}{{{k_{\rm{f}}}}} = 1 + \left( {\frac{{{k_{\rm{m}}} - {k_{\rm{f}}}}}{{{k_{\rm{f}}}}}} \right)\varphi.$$

(5.6)

When the composite is anisotropic (the thermal conductivity is not the same in all directions) (Tavman 2004):

$$\frac{{{k_{{\rm{c}},zz}}}}{{{k_{\rm{f}}}}} = \frac{{{k_{{\rm{c}},yy}}}}{{{k_{\rm{f}}}}} = 1 + \frac{{2\varphi }}{{\left( {\displaystyle\frac{{{k_{\rm{m}}} + {k_{\rm{f}}}}}{{{k_{\rm{m}}} - {k_{\rm{f}}}}}} \right) - \varphi + \left( {\displaystyle\frac{{{k_{\rm{m}}} - {k_{\rm{f}}}}}{{{k_{\rm{m}}} + {k_{\rm{f}}}}}} \right)(0.30584\,{\varphi ^4} + 0.013363\,{\varphi ^8} + \cdot \cdot \cdot )}}.$$

(5.7)

Complex nonspherical approximation models simple unconsolidated granular beds as complex nonspherical fillers in a continuous solid matrix (Tavman 2004):

$$\frac{{{k_{\rm{c}}}}}{{{k_{\rm{f}}}}} = \frac{{(1 - \varphi ) + \alpha \varphi ({k_{\rm{m}}}/{k_{\rm{f}}})}}{{(1 - \varphi ) + \alpha \varphi }},$$

(5.8)

where $\alpha = 1{\rm{/}}3\sum\nolimits_{k = 1}^3 {{{\left[ {1 + (({k_{\rm{m}}}{\rm{/}}{k_{\rm{f}}}) - 1){g_k}} \right]}^{ - 1}}}$, g _k are “shape factors” for granules.

When the solid matrix contains gas pockets, assumes thermal radiation important and parallel planar fissures perpendicular to the direction of heat conduction (Tavman 2004):

$$\frac{{{k_{\rm{c}}}}}{{{k_{\rm{f}}}}} = \frac{1}{{1 - \varphi + {{\left( {\displaystyle\frac{{{k_{\rm{m}}}}}{{{k_{\rm{f}}}\varphi }} + \frac{{4\sigma {T^3}L}}{{{k_{\rm{f}}}}}} \right)}^{ - 1}}}},$$

(5.9)

where σ is the Stefan–Boltzmann constant, and L is the total thickness of the material in direction of heat conduction.

For a parabolic distribution of the discontinuous phase reinforcement in a solid matrix, assuming that the constants of the parabolic distribution is a function of the discontinuous phase volume fraction, the effective thermal conductivity of the composite is given for the case k _f > k _m (Cheng and Vachon 1969; Tavman 2004):

$$\frac{1}{{{k_{\rm{c}}}}} = \frac{1}{{\sqrt {C({k_{\rm{f}}} - {k_{\rm{m}}})\,({k_{\rm{m}}} + B({k_{\rm{f}}} - {k_{\rm{m}}}))} }}\ln \frac{{\sqrt {{k_{\rm{m}}} + B({k_{\rm{f}}} - {k_{\rm{m}}})} + \frac{B}{2}\sqrt {C({k_{\rm{f}}} - {k_{\rm{m}}})} }}{{\sqrt {{k_{\rm{m}}} + B({k_{\rm{f}}} - {k_{\rm{m}}})} - \frac{B}{2}\sqrt {C({k_{\rm{f}}} - {k_{\rm{m}}})} }} + \frac{{1 - B}}{{{k_{\rm{m}}}}},$$

(5.10)

where $B = \sqrt {3\varphi /2}$, $C = - 4\sqrt {2/3\varphi }$.

Considering the effect of the shape of the particles and the orientation or type of packing for a two-phase system or single phase reinforcement composite, the effective thermal conductivity can be expressed as (Lewis and Nielsen 1970):

$${k_{\rm{c}}} = {k_{\rm{m}}}\frac{{1 + A\beta \varphi }}{{1 - \beta \varphi \psi }},$$

(5.11)

where $\beta = (({k_{\rm{f}}}{\rm{/}}{k_{\rm{m}}}) - 1){\rm{/}}(({k_{\rm{f}}}{\rm{/}}{k_{\rm{m}}}) + A)$, and $\psi = 1 + ((1 - {\varphi _{\rm{m}}}){\rm{/}}{\varphi _{\rm{m}}}^2)\varphi$, A is constant and depends upon the shape and orientation of the dispersed particles. ϕ _m is the maximum packing fraction of the dispersed particles. For randomly packed spherical particles A = 1.5 and ϕ _m = 0.637, whereas for randomly packed aggregates of spheres or for randomly packed, irregularly shaped particles A = 3 and ϕ _m = 0.637.

Based on the generalization of models for parallel and series conduction in composites as (5.1) and (5.2), the effective thermal conductivity of the polymer composites is given (Agari and Uno 1986):

$$\log \,{k_{\rm{c}}} = (1 - \varphi )\log ({C_1}{k_{\rm{m}}}) + \varphi \log ({C_2}{k_{\rm{f}}}),$$

(5.12)

where, C ₁, C ₂ are experimentally determined constants of order unity. C₁ is a measure of the effect of the particles on the secondary structure of the polymer, like crystallinity. C ₂ measures the ease of the particles to form conductive chains, the more easily particles are gathered to form conductive chains, C ₂ becomes closer to 1. However, experimental data are needed for each type of composite in order to determine the necessary constants (Tavman 2004).

For a specific situation, such as the polymer composite reinforced with pitch-based carbon fibers, the in-plane thermal conductivity for a ply of carbon fiber is determined by the rule of mixtures (Bootle 2001):

$${k_x} = {k_{\rm{f}}}\varphi {\sin ^2}\alpha,$$

(5.13)

$${k_y} = {k_{\rm{f}}}\varphi {\cos ^2}\alpha,$$

(5.14)

where ϕ is fiber volume percentage, k _f is the longitudinal fiber thermal conductivity, and α is the angle of fibers in a particular ply. The thermal conductivity through the thickness of the laminate k _z , typically in the region of 1.5 W/m K, is dominated by the resin thermal conductivity and radial thermal conductivity of the fiber, k _f. However, the k _f usually various depending on the plane through the fiber, therefore, a statistical term must be included for k _f when calculating it. For practical purposes, it is generally better to work from measured data. For instance, it is possible to increase k _z up to about 4–6 W/m K by adding boron nitride powder to the laminate.

Thus far the effective thermal conductivity of composites and mixtures as discussed above is derived from continuum-level phenomenological formulations that typically incorporate only the particle shape and volume fraction as variables and assume diffusive heat transport in both phases, providing a good description of systems with micrometer or larger-size particles. However, this approach fails to describe thermal transport in nanocomposites, as interfaces between materials become increasingly important on small length scales. For sufficiently small particles, the properties of the polymer/nanoparticle interface also control thermal transport in the composite. Interface effects in thermal transport can be captured by effective medium models if the introduction of particle fillers does not significantly alter the thermal conductivity of the matrix material. Because local vibrations of the atoms and molecules dominate heat transport in an amorphous polymer, this assumption should be well satisfied in a polymer matrix composite. In the case of a low volume fraction ϕ of high thermal conductivity spherical particles, the thermal conductivity of the composite k _c can be expressed as (Tavman 2004):

$${k_{\rm{c}}} = {k_{\rm{m}}}(1 + 3\varphi )\frac{{\gamma - 1}}{{\gamma - 2}},$$

(5.15)

where, k _m is the thermal conductivity of the matrix and γ = rG/k _m; r is the radius of the particle and G is the thermal conductance per unit area of the particle/matrix interface. A critical particle radius is defined as r _c = k _m/G; the thermal conductivity of the composite is increased by particles of size r > r _c and decreased for r < r _c. The changes in conductivity are a relatively weak function of (rG), but the thermal conductivity is sensitive to G over a wide range of particle sizes 0.3 < r/r _c < 8 (Tavman 2004). Classical molecular dynamics simulations are emerging as a powerful tool for calculations of thermal conductance and phonon scattering, and may provide for a lively interplay of experiment and theory to understand the conductivity of nanocomposite materials in the near future.

Computational Modeling

A set of computational tools have been applied to model the heat conduction across the conductive polymer composites, such as TIMs and calculate the effective thermal conductivity of the composite for different volume contents of filler. The analysis procedure involves (Rightley et al. 2007): (1) generating a computational geometric model representing the structure of the composite; (2) simulating the heat transfer across the composite for a set of boundary conditions using a finite element thermal analysis tool; and (3) calculating and evaluating the results.

CUBIT: A Computational Mesh Generator

CUBIT is a 2-D or 3-D finite element mesh generation toolkit for solid models (Rightley et al. 2007). Usually a geometry file can be ported into CUBIT to generate a computation mesh system for analysis. It can produce quadrilateral and hexahedral meshes, as well as triangle, tetrahedral, and hex-dominant meshes. CUBIT follows a toolkit approach, offering a variety of meshing techniques: 2-D and 3-D mapping, multiple sweeping, paving, and the latest meshing techniques, such as plastering, whisker weaving, and grafting. It runs on both UNIX and PC platforms. In addition to mesh generation, CUBIT has some basic features to construct a 3-D solid model. Other complex solid modeling capabilities include advanced geometry creation and modification, Boolean operations, web-cutting, healing, autodecomposition, defeaturing, and nonmanifold topology (Rightley et al. 2007).

Calore: A Finite Element Thermal Analysis Program

Calore is a computational program for transient 3-D heat transfer analysis. It is built upon the SIERRA finite element framework to run on both desktop and parallel computers (Rightley et al. 2007). Advanced thermal analysis capabilities include anisotropic conduction, enclosure radiation, thermal contact, and chemical reaction. The governing energy conservation equation in Calore is expressed as (Rightley et al. 2007):

$$\frac{\partial }{{\partial t}}(\rho \cdot c \cdot T) + \sum\limits_{i = x,y,z} {\frac{\partial }{{\partial {x_i}}}} (\rho \cdot c \cdot {u_i} \cdot T) = \sum\limits_{i = x,y,z} {\frac{\partial }{{\partial {x_i}}}} ({q_i}) + S,$$

(5.16)

where ${q_i} = \sum\nolimits_{i = x,y,z} {{k_{ij}}(\partial T/\partial {x_i})}$, ρ is density, c is heat capacity, T is temperature, u _i is convective velocity, q is heat flux, and S is volumetric heat generation rate. Boundary conditions available include specified temperature, heat flux, force or free convection, and surface radiation.

Calore also has many state-of-the-art computational features, such as element death, automatic selection of the time step size, mesh adaptivity, and dynamic load balancing for massively parallel computing (Rightley et al. 2007).

Code Validation and Modeling Strategy

For a given volume content of filler, the measured thermal conductivity of the composite is reasonably repeatable, even though the filler is randomly distributed in the adhesive polymers. These measured values generally follow a normal distribution with a very small standard deviation, provided that the samples are produced from a well-controlled and consistent manufacturing process. The computational model can be assessed by comparing its predictions against measurement. The computational model in the initial analysis has the filler randomly distributed. For a low volume content of filler, the numerical algorithm that searches and places filler particle into an open space within a defined volume works very well. However, when the volume content of filler gets too large, the “search-and-place” algorithm becomes inefficient and time consuming. To overcome this deficiency, a new numerical algorithm was developed that has filler initially distributed in a crystallographic structure, such as face-centered cubic (fcc) or body-centered cubic (bcc) crystal structure. Then for the next 100 iterations, it allows the filler to move randomly. This approach works well to build a computational model and calculate an effective thermal conductivity of a composite. Nevertheless, randomly distributed filler particles have a drawback. It generates significant variations in the calculated thermal conductivity as a result of the randomness of filler distribution. A large number of simulations are needed to suppress the noises and calculate an average value. Obtaining an average is important to recognize the pattern of thermal conductivity behavior as a function of volume content of filler (Rightley et al. 2007).

Modeling Filler in the Attached Phase

When a group of filler particles are in contact with one another, forming a chain of attached filler particles in the direction of temperature gradient, the effective thermal conductivity can drastically increase. This effect is predicted especially for those cases when the volume content of filler is very high. For the composite with filler in the bcc lattice structure, the drastic increase of conductivity occurs at the volumetric fraction of about 60%. Similarly, for the composite with filler in the fcc lattice structure, the drastic increase occurs at the volumetric fraction of about 65%. When chains of attached filler are created in the direction of temperature gradient, a preferable path for heat conduction will be generated, leading to a significant reduction of temperature at the hotspot. Hence, the effective thermal conductivity of the composite is much higher (Rightley et al. 2007).

Effect of Contact Resistance

For the modeling results presented thus far, the contact resistance between adhesive polymer and filler particle is assumed to be zero. The effect of nonzero contact resistance between two dissimilar materials has been analyzed; it would be interesting to evaluate the contact resistance between adhesive polymer and filler on the overall effective thermal conductivity of the composite. The cases analyzed have the following conditions: volume content = 40%; contact conductance between adhesive polymer and filler particles, h = 40 W/cm² K. For those cases in which the filler particles are in contact with each other to form a string of particles, the effective thermal conductivity of the composite does not change drastically if the contact resistance between polymer and filler increases significantly. For example, for a string of 12 contact filler particles, the calculated effective thermal conductivity of the composite decreases by less 4%, from 0.0203 to 0.0195 W/cm K if the contact conductance between polymer and filler decreases to 40 W/cm² K. However, for those cases in which the filler particles are dispersed evenly without any contact with any other particle, the effective thermal conductivity of the composite does change drastically (Rightley et al. 2007).

Percolation Theory

Consider a large square lattice in which each site is either occupied with a probability p or empty with a probability 1−p to simulate the structure of the polymer matrix composite reinforced with thermally conductive reinforcements. An occupied site, i.e., reinforcement, is assigned a conductivity k _s and an unoccupied site, i.e., polymer matrix, a conductivity k _f.

The fundamental premise of the percolation theory is contained in the idea of a sharp increase in the effective conductivity of the disordered media, polymer matrix composite, at a critical volume fraction of the reinforcement known as the percolation threshold ϕ _c at which long-range connectivity of the system appears.

When k _s ≠ 0, k _f = 0, and ϕ _s < ϕ _c, no macroscopic conducting pathway exists and the composite remains in the insulating phase. When ϕ _s ≥ ϕ _c, however, the system becomes conducting as a cluster of bonds of conductivity k _s almost certainly forms a connected bridge between the two boundaries of the disordered media across which the potential is applied. In the vicinity of the transition volume fraction ϕ _c, one has (Stauffer 1992; Karayacoubian 2006):

$${k_{\rm{e}}} \approx 0,\quad {\rm{when}}\,{\varphi _{\rm{s}}} \,< \,{\varphi _{\rm{c}}}$$

(5.17)

$${k_{\rm{e}}} \approx {k_{\rm{s}}}{({\varphi _{\rm{s}}} - {\varphi _{\rm{c}}})^t},\quad {\rm{when}}\,{\varphi _{\rm{s}}} \ge {\varphi _{\rm{c}}}$$

(5.18)

where critical exponent t has a universal value of t = 2.0 in 3-D and t = 1.3 in 2-D problems.

The existence of a critical percolation threshold for electrical conductivity has since been demonstrated for a wide variety of fillers, all at concentrations below the maximum packing fraction. The percolation threshold in an actual granular aggregate is in general a function of the lattice structure of the phases, and ranges from ϕ _c = 0.2 for a fcc arrangement to ϕ _c = 0.7 for a honeycomb arrangement and can be calculated exactly for certain simple lattices. The critical point can be approached in different ways. For example, when k _s → ∞, and k _f  ≠ 0, the effective conductivity diverges as ϕ _c is approached from below and is written as (Karayacoubian 2006):

$${k_{\rm{e}}} \approx {k_{\rm{f}}}{({\varphi _{\rm{c}}} - {\varphi _{\rm{s}}})^{ - t}},\quad {\rm{when}}\,{\varphi _{\rm{s}}} \,< \,{\varphi _{\rm{c}}}$$

(5.19)

$${k_{\rm{e}}} \approx \infty,\quad {\rm{when}}\,{\varphi _s} \ge {\varphi _c}$$

(5.20)

and when both k _s and k _f ≠ 0 and k _f/k _s → 0, one has (Karayacoubian 2006):

$${k_{\rm{e}}} \approx k_{\rm{s}}^{1 - t}k_{\rm{f}}^t,\quad {\rm{when}}\,{\varphi _{\rm{s}}} = {\varphi _{\rm{c}}}$$

(5.21)

Because of the complexity of implementing percolation theory and the eventual necessity of numerical work, the method has not been very popular. In addition, it has been suggested that in most cases, the presence of a network of filler particles does not change the basic mechanism of thermal transport in composite systems. A thermal transport network may develop only at the maximum packing fraction. Moreover, all the previous discussions on the percolation model are for site percolation. There is another type of percolation known as bond percolation, which is applicable to fiber-type reinforcements. The bond percolation model has been used to a TIM made from CNT and Ni particles, showing the existence of a percolation threshold at a very small volume fraction of CNT. This indicates the potential of using CNTs in TIMs (Prasher 2006).

General Fabrication and Manufacturing Processes of Polymer Matrix Composites

The process to fabricate and manufacture polymer matrix composites (PMCs) have been pursued and ranged from hand lay-up with labor- and cost-intensive autoclave processing to the use of automated processes such as sheet molding, bulk molding, injection molding, extrusion, and pultrusion. Transport processes during fabrication and manufacturing of PMCs encompass the physics of mass, momentum, and energy transfer on all scales. There are simultaneous transfers of heat, mass, and momentum at micro-, meso-, and macroscales, which often occurs with chemical reactions in a multiphase system with time-dependent material properties and boundary conditions. Based on the dominant flow process, PMCs manufacturing processes can be categorized as short-fiber or particulate suspension methods; squeeze flow methods; and porous media methods (Advani 2001).

Short-fiber or particulate suspension manufacturing consists of manufacturing processes that involves the transport of fibers, whiskers or particulates, and resin as a suspension into a mold or through a die to form the composite. In such processes, the reinforcements in the molten deforming resin can travel large distances and are usually free to rotate and undergo breakage. The reinforcement distribution in the final product is linked with the processing method and the flow of the suspension in the mold. Typical processing methods include injection molding, compression molding, and extrusion processes, which are most commonly used for producing highly conductive PMCs reinforced with discontinuous fibers, whiskers, particulates, or nanoparticles. Squeeze flow manufacturing methods usually involve continuous or long, aligned, discontinuous fibers either partially or fully preimpregnated with thermoplastic resin, in which the fibers and the resin deform together to form the composite shape. These processes typically include thermoplastic sheet forming, thermoplastic pultrusion, and fiber tape-laying methods. Porous media manufacturing methods usually involve continuous and nearly stationary fiber networks into which the resin will impregnate and displace the air to form the composite product in an open or a closed mold. Low viscosity thermoset is almost always used in such processes, typically including liquid composite molding, thermoset pultration, filament winding, and autoclave processing.

The common feature to all polymeric composite processes is the combining of a resin, a curing agent, some type of reinforcing fiber, and in some cases, a solvent. Typically, heat and pressure are used to shape and cure the mixture into a finished product. Formulation is the process where the resin, curing agent, and any other component required are mixed together. This process may involve adding the components manually into a small mixing vessel or, in the case of larger processes, the components may be pumped into a mixing vessel. Prepregging is the process where the resin and curing agent mixture are impregnated into the reinforcing fiber. Somme common used processes are discussed below.

Sheet molding is usually used to make composite that contains unsaturated polyester resin, thermoplastic resin, styrene monomer, initiator, catalyst, thickener (earth oxide or earth hydroxide), filler, and a mold release agent. This resin paste is combined with fiber reinforcement via a continuous line process to form a prepreg, which is stored between thin plastic sheets at low temperature for later use in molding finished parts. During molding, the cure of the paste is advanced to produce the final rigid product. Bulk molding also uses precompounded reinforcements intended for fast processing by compression, transfer, and injection molding.

Most parts made by hand lay-up or automated tape lay-up must be cured by a combination of heat, pressure, vacuum, and inert atmosphere. To achieve proper cure, the part is placed into a plastic bag inside an autoclave. A vacuum is applied to the bag to remove air and volatile products. Heat and pressure are applied for curing. Usually an inert atmosphere is provided inside the autoclave through the introduction of nitrogen or carbon dioxide, as shown in Figure 5.2. In many circumstances, an autoclave provides enhanced processing flexibility compared with other common processing techniques such as ovens and presses. However, composite fabrication by autoclave is often costly in terms of labor consumption as well as capital investment. Furthermore, autoclave fabrication techniques typically limit the size of the parts which can be produced. One technique utilized to overcome disadvantages of autoclave fabrication is single-vacuum-bag (SVB) processing in an oven utilizing vacuum bag pressure. To date, this is believed to be one of the most -ffective out-of-autoclave fabrication techniques for fiber-reinforced resin matrix composites. However, this process and technique is often ineffective when a reactive resin matrix or solvent containing prepreg is present. A reactive resin (e.g., poly(amide acid)/N-methyl pyrrolidone [NMP]) typically generates reaction by-products (e.g., water) during curing at elevated temperatures. In order to produce a void-free quality laminate, it is often imperative to deplete these volatiles and solvents before commencing forced consolidation. The traditional SVB assembly inherently hinders and/or retards the volatiles depletion mechanisms during composite fabrication because a vacuum-generated compaction force is applied to the laminate during volatile depletion. In addition, the one atmospheric pressure associated with the SVB processing tends to create excessive resin flash out of the composite during the B-stage period. As a result, resin content and net shape of the consolidated laminate become difficult to control. Polymeric prepreg material is commonly impregnated with a solution of resin to provide tack and drape for handle ability. The SVB assembly and process are simply too primitive, too time consuming (i.e., costly), and ineffective in removing solvent and reaction by-product during composite fabrication (Hou and Jensen 2007).

Fig. 5.2

Schematic illustration of autoclave processing

Open molding processes are those where the part being manufactured is exposed to the atmosphere. The worker typically handles the part manually, and there is a higher potential for exposure. The resin mixture may be a liquid being formed onto a reinforcing material or it may be in the form of a prepreg material being formed for final cure. Closed molding processes are those in which all or part of the manufacture takes place in a closed vessel or chamber. The liquid resin mixture or prepreg material may be handled or formed manually into the container for the curing step. In the case of liquid resin mixtures, these may be pumped into the container, usually a mold of some type, for the curing step. Sequential or batch processes involve manufacture of a single part at a time, in sequence. This type of process is usually required where the part being made is small and complex in shape, when the curing phase is critical, when finishing work must be minimized, or where a small number of parts are involved. Continuous processes are typically automated to some degree and are used to produce larger numbers of identical parts relatively quickly. These processes are typified by pumping of the resin mixture into the mold, followed by closed curing. One of the older plastics processes, injection molding is also the most closed process, as shown in Figure 5.3. It is not normally used in PMC processes due to fiber damage in the plasticating barrel. Thermoplastic granules are fed via a hopper into a screw-like plasticating barrel where melting occurs. The melted plastic is injected into a heated mold where the part is formed. This process is often fully automated. As shown in Figure 5.4, resin transfer molding is used when parts with two smooth surfaces are required or when a low-pressure molding process is advantageous. Fiber reinforcement fabric or mat is laid by hand into a mold, and resin mixture is poured or injected into the mold cavity. The part is then cured under heat and pressure.

Fig. 5.3

Schematic injection molding/extrusion process

Fig. 5.4

Schematic resin transfer molding process

In the pultrusion process, as shown in Figure 5.5, continuous roving strands are pulled from a creel through a strand-tensioning device into a resin bath. The coated strands are then passed through a heated die where curing occurs. The continuous cured part, usually a rod or similar shape, is then cut to the desired length.

Fig. 5.5

Schematic pultrusion process

Most of the parts made in PMC processes require some machining and/or finishing work. This traditionally involves drilling, sanding, grinding, or other manual touch-up work. These processes vary widely, depending on the size of the finished part and the amount of finishing work required, and often requires complex drill and trims fixtures (OSHA 2010).

Typical Applications for Thermal Management

The development of carbon fibers, whiskers, and foams with extremely high thermal conductivities has resulted in thermally conductive polymer matrix composites, metal matrix composites and carbon/carbon composites that increasingly are being used in commercial and industrial equipment and microelectronic, optoelectronic and microelectromechanical system packaging.

Polymer–Carbon Composites

Carbon fibers derived from polyacrylonitrile (PAN) precursors have been the dominant reinforcements in high strength PMCs since their commercialization in the late 1960s, however they are limited to use in applications for which high thermal conductivity is not a requirement due to their relatively low thermal conductivities. The development of pitch-based carbon fibers, VGCBs, carbon foam, and CNTs having high thermal conductivities opens up significant new markets and potential applications. Thermally conductive carbon fibers which are made from petroleum pitch, coal tar pitch, and gaseous hydrocarbon precursor materials are the most highly developed and of greatest commercial interest.

An important consideration in evaluating competing thermal management materials is that convection often is the limiting factor, rather than material thermal conductivity in some applications. That is, in many cases, heat dissipation is not improved by increasing thermal conductivity beyond a certain value. This makes it possible, for example, to replace aluminum, and even copper, with carbon-fiber-reinforced polymers having much lower thermal conductivities in some designs.

Table 5.2 shows thermal conductivity of some typical polymer–carbon composites, sparked in their potential to improve the materials’ thermal, electrical, and mechanical performances. The thermal conductivities of carbon fiber-in-oil suspensions as carbon fiber-in-polymer composites could also be significantly enhanced by up to, for example, 150% with just 1% volume fraction loading of fibers.

Table 5.2 Thermal properties of some typical carbon reinforced polymer composites

Mesophase pitch-derived graphitic foam can be considered an interconnected network of graphitic ligaments and, exhibit isotropic material properties. More importantly, such a foam will exhibit extremely high thermal conductivities along the ligaments of the foam (up to five times better than copper) and, therefore, will exhibit high bulk thermal conductivities. The graphite foam reinforced composite structure can be made using equipment consisting of a flexible bag for holding a foam preform. A resin inlet valve and resin source are attached to one end of the bag. A vacuum port and vacuum source are attached to the other end of the bag. The bag containing the preform is thermally coupled to a heat source. A vacuum is created within the bag. The preform is then heated. Resin is introduced into the bag through the resin inlet valve. After the resin is amply impregnated into the preform, the inlet valve is closed and the preform is cured. Once the preform is cured, the resulting composite structure is removed from the bag. As the self-rigidizing carbon or graphite foam reinforcement is both isotropic and continuous, the resulting composites are more uniformly strong in all directions and have greater resistance to shear forces than composites formed from fibrous preforms.

The use of polymer carbon composites is growing considerably. They have been used in thermal applications as both heat conductors or insulators because of their thermal conductivity and thermal expansivity that can be varied and even oriented (as in the case of fibers) in a preferred direction. They can also be suitable for electronic applications especially because specific values of their electric conductivity can be attained, as well as provide new suitable materials in the battery industry. However, structural applications are by far the most studied and applied for this type of composites; the importance of understanding and prediction of their mechanical properties of polymer carbon composites are reflected in the many studies for a variety of structures of the carbon dispersed phase, from macrolaminates to the CNTs (single and multiwalled). Coupled to a carbon fiber reinforcement which can be used as a sensor or local heating source, some PMCs can be self-healing of cracks without significant opening. Polymer–carbon composites can also be the dispersed phase in other composites as in the case of concrete matrix composites.

Polymer–Metal Composites

Polymer composites with metal fillers possess intrinsic advantages including low cost and ease of processing, high electric conductivity, paramagnetism, high thermal conductivity, as well as good mechanical properties. In other words, the interest in polymer–metal composites arises from the fact that the electric and magnetic characteristics of such materials are close to those of metals; there can also be significant improvement in the thermal properties of the pure polymers and a mechanical reinforcement effect might also be achieved, whereas the processability is the same as for the neat polymer, a great advantage for speed of production and processing costs.

The thermal and/or electric conductivity of PMCs is one of the main properties of interest in electronic packaging. For example, poly(ethylene oxide) with added aluminum particles have found that only 1–2 wt% is needed in order to increase the conductivity of the pure polymer by over one order of magnitude, reaching a value of 5 × 10^–6 Scm⁻¹. There is a big advantage of using aluminum instead of a ceramic as the filler: in order to get the maximum conductivity with the ceramic, a concentration of around 40 wt% is needed; this of course is reflected in the cost of production of the material. For copper and nickel powders as fillers in an epoxy resin and in poly(vinyl chloride), the concentration dependence of the electric and thermal conductivity of the resulting composites: less than 1 vol% was sufficient in order to reach the percolation concentration and achieve a big jump in electric conductivity (around 12 orders of magnitude). On the other hand, the thermal conductivity increases linearly with the metal powders concentration. PMC with a soft Al–Fe–Si magnetic powder to a polymeric matrix to produce magnetic films to be used for shielding of electromagnetic waves was found that the level of shielding is mostly dependent of the film thickness and density and related to an increase of magnetic permeability and electric conductivity. In addition, metallic particles are also added to polymers to improve their mechanical properties. However, polypropylene composites with silver powder (several tens of microns) as filler have been found that the tensile modulus, strength, and elongation at break decrease initially when adding the Ag particles and then start increasing when adding more powder. Impact strength also decreased with Ag powder content. This negative effect on the composites properties is attributed to a discontinuity in the structure as well as poor interaction between the two phases. Nevertheless, flexural modulus and strength increase with filler content due to an increase in rigidity. A different way of improving mechanical properties of PMCs is irradiating the material. This improvement is attributed to a cross-linking effect of the polymer chains due to the irradiation making the material stronger. Interest in PMCs is increasing because of their potential to be used as sensors and actuators, especially for biomedical and biological applications. When a voltage is applied to the composite in a circuit, there is a mechanical response depending of the voltage value. This kind of materials can be used potentially as sensors for active noise damping and biosensors. Their large actuation strain and flexibility make them suitable for artificial muscles and other bioinspired devices, meanwhile keep a tailored thermal properties including appropriate thermal conductivity and thermal expansion coefficient (Mejia 2007).

Polymer–Ceramic Composites

Polymer–ceramic composites consist of an organic matrix, for instance polysiloxanes, and inorganic fillers such as oxides or silicates. The properties of the composites are mainly influenced by chemical cross linking between the organic components and between the organic components and functional groups on the surfaces of the filler particles. Polymer–ceramic composites combine typical properties of ceramics, including high thermal stability, thermal conductivity, and dimensional stability with typical properties of plastics, namely plastic formability and functional variability. The composite materials can be processed by plastic forming techniques, in particular injection molding. Polymer–ceramic composite materials have been used for power tool and automotive industries where the materials must endure high thermal loads. Examples include the encapsulation of high-performance resistors or metallic contact bars in heating registers as well as the use of polymer–ceramic composites for thermally loaded components such as brush holders in electric motors. Other applications include an adhesive for attaching electrical components to printed circuit boards, high power components such as transistors to heat sinks, laminates, hybrid substrates for electrical components, and encapsulating compositions for use in integrated circuits where dissipation of heat is a critical requirement of the encapsulant.

One major application of the polymer–ceramic composites is used for TIMs. There has been a heat-reducing grease of the type which uses silicone oil as a base material and a zinc oxide or alumina powder, as well as aluminum nitride as a thickener to achieve improved thermal conductivity. The silicone grease is prepared by mixing a silicone oil with a thickener usually having a low affinity for the silicone oil, so that they have a problem of separating the oil from the composition (in terms of the degree of oil separation) upon long-term standing at a high temperature or by a long-range repetition of cooling and heating cycles, and so on. This problem frequently arises in cases where the thickener used has a relatively large particle size and excellent thermal conductivity. To solve this problem, one silicone grease composite was made, which comprises 10–50 parts by weight of an organopolysiloxane modified by 2-phenylethyl, 2-phenylpropyl or 6–30C alkyl groups and 90–50 parts by weight of a metal oxide, such as silica, diatomaceous earth, zinc oxide, alumina, or titanium oxide (Yamada 2001), and another material is the thixotropic thermally conductive material which comprises an oily organosilicone carrier, a thermal conductivity-providing filler powder selected from a group consisting of thin-leaf aluminum nitride, dendrite-form zinc oxide, thin-leaf boron nitride, and a mixture of two or more thereof and a silica fiber acting as an exudation inhibitor (Aakalu 1981). Other materials are the thermally conductive silicone grease composition comprising an organopolysiloxane, SiC, and aerosol silica (Yamada 2001), and the thermally conductive silicone oil compound comprising a hydroxyl group-containing organopolysiloxane having a viscosity of 10–100,000 cs (centistokes) wherein the hydroxyl groups comprise 5–50 mol% of the total end groups and a powder of at least one metal compound selected from a group consisting of zinc white, alumina, aluminum nitride, and SiC (Yamada 2001). However, these grease-state silicone composites have a drawback of being insufficient in thermal conductivity. For further improvement, it has been found that the bleeding liability of a base oil can be controlled by combining a particular organopolysiloxane with a thickener, such as zinc oxide, alumina, aluminum nitride, boron nitride, or Sic, an organopolysiloxane having at least one hydroxyl group attached directly to a silicon atom and an alkylalkoxysilane, thereby forming a thermally conductive silicone composition having high reliability which can steadily display thermally conductive properties over a long period of time, brings no oil stain on the surroundings, and does not cause contact point disturbance. The typical thermally conductive silicone composition comprises (1) 5–30 wt% of a liquid silicone; (2) 50–94.98 wt% of at least one thickener selected from the group consisting of a zinc oxide powder, an alumina powder, an aluminum nitride powder, a boron nitride powder and a SiC powder; (3) 0.01–10 wt% of an organopoly-siloxane having at least one per molecule of hydroxyl group attached directly to a silicon atom; and (4) 0.01–10 wt% of an alkoxysilane. The thermally conductive silicone composition can control effectively the bleeding of the base oil and the thermal conductive properties thereof can persist stably for a long time, so that it is suitable for thermal conductive silicone grease. In addition, novel nonbleeding thixotropic thermally conductive dielectric materials which find particular application in the cooling of electronic components, especially in direct contact with integrated circuit chips. The composition of the composite comprises a liquid silicone carrier, a thermal filler powder selected from the group consisting of lamellar aluminum nitride, dendritic zinc oxide, lamellar boron nitride, or mixtures thereof in combination with silica fibers which function as a bleed inhibiting agent.

Polymer ceramic composites have also been applied in the energy storage, electric, electronic, biomedical, and structural fields. Alumina and zirconia are often selected additives to fabricate high performance electrolyte composites to be used in. Other electronic applications are piezoelectric and dielectric materials where CdO, borium strontium titanate, strontium cesium titanate are used as the dispersed phase. In some cases the dielectric constant of the composite equals that of a pure ceramic material, such is the case of CdO plus polymer composites which show a dielectric constant of 2,200 while for pure BaTiO₃ the value is around 2,000. Among thermal materials those composed of Al₂O₃, SiO₂ and TiO₃ compounds where low isobaric expansivities are desired in high performance chips, wiring and electronic packaging.

Polymer Matrix Nanocomposites

CNTs have typical diameters in the range of ~1–50 nm and lengths of many microns (even centimeters in special cases). They can consist of one or more concentric graphitic cylinders. In contrast, commercial (PAN and pitch) carbon fibers are typically in the 7–20 μm diameter range, while VGCFs have intermediate diameters ranging from a few hundred nanometers up to around a millimeter.

The graphite sheet may be rolled in different orientations along any 2-D lattice vector (m, n) which then maps onto the circumference of the resulting cylinder; the orientation of the graphite lattice relative to the axis defines the chirality or helicity of the nanotube. As-grown, each nanotube is closed at both ends by a hemispherical cap formed by the replacement of hexagons with pentagons in the graphite sheet which induces curvature. Single-walled carbon nanotubes (SWNTs) are usually obtained in the form of so-called ropes or bundles, containing between 20 and 100 individual tubes packed in a hexagonal array. Rope formation is energetically favorable due to the van der Waals attractions between isolated nanotubes. Multiwalled carbon nanotubes (MWNTs) provide an alternative route to stabilization. They consist of two or more coaxial cylinders, each rolled out of single sheets, separated by approximately the interlayer spacing in graphite. The outer diameter of such MWNTs can vary between 2 and a somewhat arbitrary upper limit of about 50 nm; the inner hollow core is often (though not necessarily) quite large with a diameter commonly about half of that of the whole tube. Carbon nanofibers (CNFs) are mainly differentiated from nanotubes by the orientation of the graphene planes; whereas the graphitic layers are parallel to the axis in nanotubes, nanofibers can show a wide range of orientations of the graphitic layers with respect to the fiber axis. They can be visualized as stacked graphitic discs or (truncated) cones, and are intrinsically less perfect as they have graphitic edge terminations on their surface. Nevertheless, these nanostructures can be in the form of hollow tubes with an outer diameter as small as ~5 nm, although 50–100 nm is more typical. The stacked cone geometry is often called a herringbone fiber due to the appearance of the longitudinal cross section. Slightly larger (100–200 nm) fibers are also often called CNFs, even if the graphitic orientation is approximately parallel to the axis (Shaffer and Sandler 2006).

As reinforcement in a polymer matrix, the formed functional nanocomposites have exhibited the unique physical properties, such as high thermal or electrical conductivity. Electrically or thermally conductive polymer composites, for example, are used in antistatic packaging applications, as well as in electromagnetic interference shielding or thermal managing components in the electronics, automotive, and aerospace sector. The incorporation of conductive filler particles into an insulating polymer host leads to bulk conductivities at least exceeding the antistatic limit of 10^–6 S/m. Compared with common conductive fillers such as metallic or graphitic particles or fibrous, the incorporation of CNTs allows for a low percolation threshold, a high quality surface finish, a robust network, and good mechanical properties–a combination not obtained with any other filler. The use of CNTs/CNFs as a conductive filler in thermoplastics is their biggest current application, and is widespread across the automotive and electronic sectors (Shaffer and Sandler 2006).

Figure 5.6 illustrates the electrical resistivity in a metal particle-filled functional polymer composite with increasing filler loading fraction. In general, the electrical conductivity of a particulate composite reveals a nonlinear increase with the filler concentration, passing through a percolation threshold. At low filler concentrations, the conductive particles are separated from each other and the electrical properties of the composite are dominated by the matrix. With increasing filler concentration local clusters of particles are formed. At the percolation threshold, ϕ _c, these clusters form a connected 3-D network through the component, resulting in a jump in the electrical conductivity. Close to the percolation threshold, the electrical conductivity follows a power-law of the form (Shaffer and Sandler 2006):

$${\sigma _0} \propto {({\phi _{\rm{v}}} - {\phi _{\rm{c}}})^t},$$

(5.22)

Fig. 5.6

Variation of electrical resistivity of polymer composite as a function of metal filler volume, with approaching the percolation threshold (Tong 2009)

where φ_v is the volume fraction of the filler. The exponent t in this equation was found to be surprisingly uniform for systems of the same dimensionality. For 3-D percolating systems t varies between 1.6 and 2. The percolation threshold is reduced on increasing the aspect ratio, but the maximum conductivity is limited by the contact resistance between neighboring particles. A related percolation behavior is observed in the rheological properties, at the point when the filler particles begin to interact. In many cases, the electrical percolation threshold of bulk composites corresponds to the rheological threshold. The electrical properties of nanofibere–thermoplastic composites exhibit characteristic percolation behavior. In the case of untreated CNFs, the critical volume fraction is between 5 and 10 vol%, but depends on the processing technique and resulting degree of CNF dispersion and alignment. At a higher filler content of 15 vol%, even drawing of nanofiber-filled polymer composite fibers does not destroy the conductive network. However, a comparative study of bulk injection-molded nanofiber and PAN-based short carbon fiber reinforced polymer composites showed a lower percolation threshold and higher maximum bulk conductivity for the macroscopic filler (Shaffer and Sandler 2006).

The electrical performance of the composite depends on the intrinsic conductivity as well as the dispersion and alignment of the filler. Oxidation encourages interaction with the polymer, increasing the contact resistance, whereas graphitization both reduces polymer interactions and improves the intrinsic conductivity. As an alternative to graphitization, electro-deposition of copper on nanofiber surfaces has been shown to improve the maximum bulk composite conductivity. Similarly, the electrical percolation threshold of thin MWNT thermoplastic films also depends on the type of nanotube and surface treatment. Threshold values from around 5 wt% for oxidized catalytic MWNTs in polyvinyl alcohol (PVA) to around 0.06 and 0.5 wt% for arc discharge MWNTs in PVA and polymethylmethacrylate,  respectively. Interestingly, the optical transparency of such conductive thin film composites for antistatic applications can be significantly improved by using SWNTs, even in bundled form. These very low percolation thresholds are far below the expected values for randomly distributed fibers, and are the result of active aggregation processes (Bal and Samal 2007).

A significant reduction in the critical nanotube volume fraction for electrical percolation can be achieved by exploiting the concept of double percolation through the formation of a cocontinuous morphology in nanotube-filled polymer blends. As in the case of the thermosetting systems, however, the thermal conductivity of nanofiber–thermoplastic composites does not show a percolation transition, even at higher filler volume fractions. A linear increase in thermal conductivity is observed, although the magnitude depends to some extent on the alignment of the filler, in agreement with data for short carbon fiber composites. The overall performance increased for nanofibers was similar to that observed for short carbon fibers in a similar system (Shaffer and Sandler 2006).

For instance, the thermal conductivity of aligned, vapor grown carbon nanoscale fiber reinforced polypropylene composite has been measured in the longitudinal and transverse directions for 9, 17, and 23% fiber reinforcements by volume. The values of thermal conductivity are 2.09, 2.75, 5.38 W/m K for the longitudinal directions and 2.42, 2.47, 2.49 W/m K for the transverse direction respectively, while the thermal conductivity of unfilled polypropylene is 0.24 W/m K (Tavman 2004).

The effective thermal conductivity of liquid with nanoparticle inclusions can be much higher than the normally used industrial heat transfer fluid, such a fluid has terminologized as nanofluid, and considered to be a novel enhanced heat transfer fluid. The possible mechanisms of enhanced thermal conductivity may mainly depend upon the size effect, the clustering of nanoparticles and the surface adsorption, while the Brownian motion of nanoparticles contributes much less than other factors. In particular, it was demonstrated that solid nanoparticle colloids (i.e., colloids in which the grains have dimensions of ≈10–40 nm) are extremely stable and exhibit no significant settling under static conditions, even after weeks or months. Furthermore, the enhancement of thermal-transport properties of such nanofluids was even greater than that of suspensions of coarse-grained materials. For example, the use of A1₂O₃ particles ≈13 nm in diameter at 4.3% volume fraction increased the thermal conductivity of water under stationary conditions by 30%.Use of somewhat larger particles (≈40 nm in diameter) only led to an increase of less than ≈10% at the same particle volume fraction; more in accord with theoretical predictions. An even greater enhancement has been reported for Cu nanofluids, where just a 0.3% volume fraction of 10 nm Cu nanoparticles led to an increase of up to 40% in thermal conductivity, a result that is more than an order of magnitude above the increase predicted by macroscopic theory (Tavman 2004).

Metals undergo considerable property changes by size reduction and their composites with polymers are very interesting for functional applications. Some new properties observed in nanosized metals are produced by quantum-size effects, such as electron confinement. These quantum effects arise from the fact that there are a big number of surface atoms compared to a normal bulk metal. These properties are size-dependent and can be tuned by changing the dimension; thus, the same material may show different sets of properties by changing its size. Particularly interesting are the dependence on the size of ferromagnetism and the superparamagnetism characterizing all metals; chromatism observed with silver, gold, and copper metals due to plasmon absorption; the photo and thermoluminescence; and the supercatalytic effect due to the very large superficial area of very fine particles. These materials are highly chemically reactive, highly absorbent, and show very different thermodynamic parameters. For example, they melt at much lower temperatures. Many of these unique chemical and physical characteristics of nanosized metals remain unmodified after embedding in polymers (i.e., optical, magnetic, dielectric, and thermal transport properties), and therefore they can be used to provide special functionalities to polymers (Bal and Samal 2007).

CNTs and nanofibers may not produce practical replacements for existing high-performance materials in the near future. However, there is a continuing market for electrically conducting polymer compounds, and immediate potential to develop the reinforcement of delicate composite structures such as thin films, fibers, and the matrices of conventional fiber composites. Although the full potential of nanotube composites remains to be realized, much progress has been made, and these nanocomposite systems have a bright future once the fundamental questions are resolved.

Summary

High thermal conductivity polymer matrix composites have been used increasingly for electronic packaging and thermal management. Composites with continuous reinforcements such as fibers, woven or cloths, are usually used as substrates, heat sinks, and enclosures. The polymer matrix in advanced thermally conductive composites includes both thermosetting and thermoplastic types. Different kinds of fillers or reinforcements have been developed to process composite materials with desired thermal, mechanical, and electrical properties. Fillers may be in the form of fibers or in the form of particles uniformly distributed in the polymer matrix material. The properties of the polymer composite materials are strongly dependent on the filler properties as well as on microstructural parameters such as filler diameter, length, distribution, volume fraction, and the alignment and packing arrangement of fillers. It is evident that thermophysical properties of fiber filled composites are anisotropic, except for the very short, randomly distributed fibers, whereas thermophysical properties of particle filled polymers are isotropic.

Reinforcement fillers have an important role to play in maximizing polymer performance and production efficiency. Cost reduction, density control, optical effects, thermal conductivity, magnetic properties, flame retardancy, and improved hardness and tear resistance have been increased the demand for high-performance fillers. Several types of reinforcements, especially nanoparticulate fillers, have been used in polymer matrix composites: VGCF, carbon foam, CNT, and other thermal conductive particles, such as ceramic, carbon, metal, or metal-coated particles, as well as metal or carbon foams. Nanoparticles of carbides, nitrides, and carbonitrides can be used to reinforce polymer matrix nanocomposites with desirable thermal conductivity, mechanical strength, hardness, corrosion, and wear resistance. To achieve these of desirable properties polymer matrix and layout or distribution of nanoparticles need to be optimized.