Mooney plot

In rheology and capillary rheometry, the Mooney plot (also called the Mooney diagram or Mooney analysis) is a graphical method used to detect and quantify Wall Slip during the pressure-driven flow of non-Newtonian fluids through capillary or slit dies.^[1] The technique was introduced by Melvin Mooney in 1931 and has since become one of three standard corrections routinely applied in capillary rheometry, alongside the Bagley correction and the Weissenberg-Rabinowitsch correction.^[2]^[3]^[4] The Mooney plot is described in all major rheology textbooks^[5]^[6]^[7] and is implemented as standard functionality in commercial capillary rheometer software.^[8]^[9]

The Mooney plot is constructed by plotting the apparent shear rate ${\dot {\gamma }}_{a}$ against the reciprocal capillary radius $1/R$ (or reciprocal diameter $1/D$ ) at constant values of the wall shear stress $\tau _{w}$ . In the presence of wall slip, the resulting data fall on straight lines whose slopes are directly proportional to the slip velocity $v_{s}$ .^[1]^[10]

Background

No-slip boundary condition

A fundamental assumption in fluid dynamics is the no-slip condition, which states that the velocity of a fluid at a solid boundary equals the velocity of that boundary.^[11] For Newtonian fluids this assumption holds under virtually all practical conditions.^[12] However, for complex fluids, including polymer melts, concentrated suspensions, emulsions, foams, and viscoplastic materials, the fluid may exhibit a finite velocity at the wall when the wall shear stress exceeds a critical value $\tau _{w,c}$ . This phenomenon is known as wall slip and leads to systematic errors in rheological measurements if not properly accounted for.^[13]^[10] The critical shear stress for the onset of slip in linear polyethylenes has been reported to be approximately 0.09 to 0.14 MPa.^[14]^[15]

Origin of apparent wall slip

In concentrated suspensions, particles cannot occupy the space adjacent to a rigid surface as efficiently as in the bulk, producing a thin particle-depleted layer (or apparent slip layer) of thickness $\delta$ that consists predominantly of the continuous phase.^[16]^[17] This layer has a viscosity much lower than the bulk suspension and acts as a lubricant, producing an apparent velocity jump at the wall. The same mechanism has been documented in emulsions, foams, colloidal gels, and food pastes.^[18]^[19]^[20]

In polymer melts, wall slip has been attributed to chain disentanglement at the polymer-wall interface.^[12] Migler, Hervet, and Leger performed direct optical measurements using evanescent waves that confirmed a sharp transition between weak and strong slip near smooth surfaces.^[21]

Derivation

Flow in a circular capillary

Consider the steady, fully developed, isothermal, creeping flow of an incompressible fluid through a circular capillary of radius $R$ and length $L$ . The wall shear stress is obtained from the pressure drop $\Delta P$ across the capillary (after applying the Bagley correction^[22]) as

\tau _{w}={\frac {R\,\Delta P}{2L}}\,.

The apparent shear rate (assuming a Newtonian velocity profile with no slip) is defined as^[3]^[5]

{\dot {\gamma }}_{a}={\frac {4Q}{\pi R^{3}}}={\frac {4{\bar {v}}}{R}}

where $Q$ is the volumetric flow rate and ${\bar {v}}=Q/(\pi R^{2})$ is the mean velocity in the capillary.

Velocity decomposition

When wall slip occurs, the total mean velocity is the sum of a bulk deformation contribution and a slip contribution. At the wall ( $r=R$ ), the fluid has a finite tangential velocity $v_{s}$ rather than zero. The volumetric flow rate then decomposes as^[1]

Q=Q_{\text{bulk}}+Q_{\text{slip}}=Q_{\text{bulk}}+\pi R^{2}v_{s}\,.

Dividing through by $\pi R^{3}/4$ :

{\frac {4Q}{\pi R^{3}}}={\frac {4Q_{\text{bulk}}}{\pi R^{3}}}+{\frac {4v_{s}}{R}}

which may be written as

{\dot {\gamma }}_{a}={\dot {\gamma }}_{a,{\text{true}}}(\tau _{w})+{\frac {4v_{s}(\tau _{w})}{R}}\,.

Here ${\dot {\gamma }}_{a,{\text{true}}}$ is the apparent shear rate that would be observed in the absence of slip; it depends only on the wall shear stress $\tau _{w}$ for a given fluid.^[2]^[4]

The Mooney equation

Mooney's fundamental assumption is that the slip velocity is a function of the wall shear stress alone and is independent of the capillary geometry:^[1]

v_{s}=f(\tau _{w})\,.

Under this assumption, at a fixed value of $\tau _{w}$ , the quantity ${\dot {\gamma }}_{a,{\text{true}}}$ is a constant across capillaries of different radii, and the apparent shear rate becomes a linear function of $1/R$ :

{\dot {\gamma }}_{a}={\dot {\gamma }}_{a,{\text{true}}}(\tau _{w})+4\,v_{s}(\tau _{w})\cdot {\frac {1}{R}}\,.

This is the Mooney equation.^[1] When expressed in terms of the capillary diameter $D=2R$ :

{\dot {\gamma }}_{a}={\dot {\gamma }}_{a,{\text{true}}}(\tau _{w})+8\,v_{s}(\tau _{w})\cdot {\frac {1}{D}}

the slope of ${\dot {\gamma }}_{a}$ versus $1/D$ at constant $\tau _{w}$ equals $8v_{s}$ , and the $y$ -intercept yields the slip-corrected apparent shear rate ${\dot {\gamma }}_{a,{\text{true}}}$ .^[23]^[2]

Procedure

The experimental procedure for constructing a Mooney plot is described in standard references^[2]^[3]^[10]^[8] and comprises the following steps:

Obtain flow curves (plots of $\tau _{w}$ vs. ${\dot {\gamma }}_{a}$ ) using a set of capillary dies with at least three different diameters $D_{1},D_{2},D_{3}$ but the same $L/D$ ratio. A practical die combination is L/D = 40/2 mm and L/D = 20/1 mm.^[8]
Apply the Bagley correction to each die to obtain the true wall shear stress, free of entrance and exit pressure losses.^[22]
At each value of constant $\tau _{w}$ , read off the apparent shear rate from each flow curve.
Construct the Mooney plot: plot ${\dot {\gamma }}_{a}$ (ordinate) against $1/R$ (abscissa) at each constant $\tau _{w}$ .
If the data at each $\tau _{w}$ fall on a straight line, the Mooney assumption $v_{s}=f(\tau _{w})$ is validated. The slip velocity is obtained from the slope: $v_{s}={\tfrac {1}{4}}\times {\text{slope}}$ . The slip-corrected apparent shear rate is the $y$ -intercept.
Repeat at other values of $\tau _{w}$ to determine $v_{s}(\tau _{w})$ .
Apply the Weissenberg-Rabinowitsch correction to the slip-corrected data to obtain the true wall shear rate and hence the true viscosity.^[24]

Kalika and Denn described an alternative approach in which a power-law model is assumed for the bulk flow curve and slip velocities are computed by subtraction at each data point, yielding a master plot of slip velocity independent of capillary diameter.^[25]

Detection of wall slip

Wall slip is detected by observing divergence of the flow curves obtained from capillaries of different diameters.^[14]^[15]^[12] In the absence of slip, all flow curves collapse onto a single master curve, since for a given fluid

{\dot {\gamma }}_{a}=f(\tau _{w})\quad {\text{(independent of }}R{\text{)}}

When slip is present, the flow curves shift apart. At a given $\tau _{w}$ , smaller-diameter capillaries yield higher apparent shear rates, because the surface-to-volume ratio $2/R$ is larger and the relative contribution of slip is greater:^[10]

{\dot {\gamma }}_{a}{\bigg |}_{R_{1}}>{\dot {\gamma }}_{a}{\bigg |}_{R_{2}}\quad {\text{for}}\quad R_{1}<R_{2}\quad {\text{at fixed }}\tau _{w}\,.

The critical wall shear stress $\tau _{w,c}$ at which the flow curves begin to diverge marks the onset of wall slip.^[14]^[25] Ramamurthy demonstrated this divergence for several LLDPE and HDPE resins and showed that the critical stress is relatively insensitive to molecular weight, molecular weight distribution, and temperature.^[14]

In rotational rheometry, the analogous diagnostic involves comparing flow curves measured at different gap heights in a parallel plate geometry; if the curves do not superimpose, wall slip is indicated.^[18]^[26]

Slip velocity models

Navier linear slip

The simplest model, due to Navier (1827),^[27] assumes a linear relationship:

v_{s}=\beta \,\tau _{w}

where $\beta$ is the Navier slip coefficient. For this model, the Mooney equation becomes

{\dot {\gamma }}_{a}={\dot {\gamma }}_{a,{\text{true}}}(\tau _{w})+{\frac {4\beta \,\tau _{w}}{R}}\,.

Nonlinear (power-law) slip

For many practical systems, particularly those with shear-thinning phases, the slip velocity follows a power-law dependence:^[10]^[16]

v_{s}=\beta '\,\tau _{w}^{s_{\beta }}

where $\beta '$ is a slip coefficient and $s_{\beta }$ is the slip exponent. Hatzikiriakos and Dealy found experimentally that the slip velocity of HDPE melts follows an approximate power-law relationship with wall shear stress above the critical stress.^[23]^[15] A Newtonian binder ( $s_{\beta }=1$ ) recovers the linear Navier model.

Corrections in capillary rheometry

The Mooney slip analysis is one of three standard corrections applied in capillary rheometry. The recommended order is:^[2]^[3]^[28]^[29]

Bagley correction for entrance and exit pressure losses:^[22]
$\tau _{w}={\frac {R(\Delta P-\Delta P_{\text{end}})}{2L}}={\frac {R\,\Delta P}{2(L+eR)}}$

where $e$ is the Bagley correction factor (end correction).
Mooney correction for wall slip (as derived above).
Weissenberg-Rabinowitsch correction for non-Newtonian velocity profiles:^[24]
${\dot {\gamma }}_{\text{true}}={\frac {3n'+1}{4n'}}\,{\dot {\gamma }}_{a,{\text{true}}}$

The true shear viscosity after all corrections is^[2]

\eta ({\dot {\gamma }}_{\text{true}})={\frac {\tau _{w}}{{\dot {\gamma }}_{\text{true}}}}\,.

Extension to other geometries

Kalyon (2005) extended the Mooney analysis to viscoplastic (Herschel-Bulkley) fluids and showed that the same slip coefficient is obtained from Couette, capillary, and slit flow geometries, validating the physical basis of the method.^[16]

For flow through a rectangular slit die of height $2H$ and width $W\gg 2H$ :^[16]^[30]

{\dot {\gamma }}_{a,{\text{slit}}}={\dot {\gamma }}_{a,{\text{true}}}(\tau _{w})+{\frac {6v_{s}(\tau _{w})}{2H}}

so that plotting ${\dot {\gamma }}_{a,{\text{slit}}}$ versus $1/H$ at constant $\tau _{w}$ yields a straight line with slope $3v_{s}$ .

Yoshimura and Prud'homme developed the analogous correction for parallel plate and Couette geometries in rotational rheometers, extending the applicability of the Mooney approach beyond capillary flow.^[18]

Limitations and modifications

Limitations of the classical Mooney analysis

Despite its status as a standard method, the original Mooney analysis has several known limitations that have been discussed extensively in the literature:

Geometry-independent slip assumption: The fundamental assumption $v_{s}=f(\tau _{w})$ can fail when particle migration, pressure-dependent effects, or gap-dependent mechanisms are significant.^[31]^[32]

Negative slip velocities: For some filled compounds and complex polymer systems, the Mooney analysis yields physically impossible (negative) values of $v_{s}$ , indicating breakdown of the underlying assumptions.^[33]^[34]

Pressure-dependent slip: Hatzikiriakos and Dealy demonstrated experimentally that the slip velocity of HDPE depends on the wall normal stress (and hence on pressure), producing an apparent $L/D$ dependence that violates the Mooney assumption. They developed a modified Mooney technique to account for this effect.^[23]

Viscous heating: At high shear rates, viscous dissipation can raise the temperature within the capillary, altering both the bulk viscosity and the slip behavior. Rosenbaum and Hatzikiriakos developed a nonisothermal model that corrects the slip velocity for this effect.^[35]

Shear-induced particle migration: In concentrated suspensions, particles migrate radially under shear gradients, altering the near-wall composition along the die length. This transient process introduces length-dependent apparent slip that the Mooney analysis cannot capture.^[36]^[31]

Die material and surface effects: The material of construction of the capillary die and its surface finish can influence the onset and magnitude of wall slip. Ramamurthy showed that using brass dies or adhesion promoters can substantially reduce slip in polyethylene extrusion.^[14] Chen and coworkers further investigated the role of surface roughness and chemical composition.^[37]

Jastrzebski modification

Jastrzebski (1967)^[38] proposed that the slip velocity depends inversely on the capillary radius:

v_{s}={\frac {\beta _{J}\,\tau _{w}}{R}}

so that plotting ${\dot {\gamma }}_{a}$ versus $1/R^{2}$ at constant $\tau _{w}$ yields straight lines (the Jastrzebski plot). Martin and Wilson argued that this condition lacks physical grounding and recommended against its use without further justification.^[32]

Tikhonov regularization

Owens, Corfield, and coworkers developed a Tikhonov regularization approach to the Mooney analysis that treats the extraction of slip velocity and bulk flow curves as an inverse problem, yielding more robust results when the standard graphical method produces noisy or inconsistent data.^[39]^[40]

Applications

Polymer melts

The Mooney analysis is routinely applied to polymer melts, particularly high molecular weight linear polymers such as LLDPE, HDPE, polypropylene, and polystyrene, which exhibit wall slip above a critical wall shear stress.^[14]^[25]^[15]^[10] In such systems, the slip velocity typically follows a power law in $\tau _{w}$ and the transition from weak slip (partial disentanglement at the wall) to strong slip (nearly plug flow) can be tracked on the Mooney plot.^[12]^[10] Hatzikiriakos and coworkers demonstrated good agreement between slip velocities obtained from the Mooney technique on HDPE and independent sliding plate rheometer measurements.^[23] Studies on metallocene-catalyzed LLDPE resins have used the Mooney plot to show that nearly complete plug flow occurs at high wall shear stresses.^[41]

Rubber compounds

Rubber compounds (SBR, EPDM) are known to exhibit wall slip, and the Mooney method is commonly attempted. However, Mourniac, Agassant, and Vergnes found that the standard Mooney analysis failed to produce physically acceptable results for SBR compounds and developed alternative characterizations based on gap-dependent slip.^[34] Leblanc reported similar difficulties with filled rubber compounds, observing negative slip velocities from the Mooney analysis.^[33]

Concentrated suspensions

Wall slip is particularly important in concentrated suspensions of rigid particles (ceramic pastes, pharmaceutical formulations, solid propellant simulants). Yilmazer and Kalyon measured slip layer thicknesses on the order of a few percent of the particle diameter using the Mooney technique in highly filled suspensions.^[42] Wilms and coworkers provided a systematic comparison of the accuracy of the original Mooney analysis and various modifications for concentrated suspensions, and proposed a best-practice approach.^[31]

Food and consumer products

The Mooney analysis has been applied to food pastes and consumer products. Corfield and coworkers critically examined capillary rheometry of starch-based pastes exhibiting wall slip, and assessed the robustness of the Mooney method for such materials.^[40]

Other materials

Further applications of the Mooney method have been reported for solder pastes,^[43] wood-polymer composites,^[44] powder injection moulding compounds,^[45] abrasive flow machining media,^[46] and natural fluid muds.^[47]

Distinction from the Mooney-Rivlin plot

The Mooney plot for wall slip should not be confused with the Mooney-Rivlin plot used in rubber elasticity. In the latter, the reduced stress is plotted against $1/\alpha$ (where $\alpha$ is the stretch ratio) for uniaxial tension data of a Mooney-Rivlin solid, and a linear fit yields the material constants $C_{1}$ and $C_{2}$ . Both plots are named after the same Melvin Mooney, who contributed to both rubber mechanics and capillary rheometry.^[1]

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