Introduction

The complete set of rheological properties of a complex fluid in shearing flows includes not only the shear stress τ 21 but also the normal stress differences N 1 and N 2. The values of the normal stress differences N 1 and N 2 are very important to the stability of shearing flows present in commercial polymer manufacturing processes (Bird et al. 1987; Baek and Magda 2003). For example, it is observed in the literature that vortices occur in axisymmetric converging flows with a size that depends on the ratio β = − N 2/N 1 (Feigl and Ottinger 1994) and that secondary flows occur during profile coextrusion with a magnitude that depends on N 2 (Debbaut et al. 1997). Many methods have been proposed to measure both normal stress differences of a viscoelastic fluid, including shear-optical methods (Kannan and Kornfield 1992; Brown et al. 1995; Takahashi et al. 2002), rotating-rod method (Beavers and Joseph 1975; Magda et al. 1991a), tilted-trough method (Keentok and Tanner 1982), truncated cone–plate method (Alvarez et al. 1985), raised cone–plate method (Ohl and Gleissle 1992), hole-pressure method (Alvarez et al. 1985), segmented cone–plate method (Meissner et al. 1989; Schweizer 2002), strain potential total force method (Sui and McKenna 2007a), total force cone–plate parallel-plate method (TFCPPP; Ginn and Metzner 1969), pressure distribution cone–plate method (PDCP; Miller and Christiansen 1972; Gao et al. 1981; Magda et al. 1991a, b, 1993; Baek and Magda 2003), and pressure distribution parallel-plate method (PDPP; Adams and Lodge 1964). Here, we focus on the last three methods, which we denote for convenience as the TFCPPP, PDCP, and PDPP methods, respectively. In order to measure the pressure distribution during torsional shear flows, we use a novel rheometer plate, the normal stress sensor (NSS, RheoSense, Inc., San Ramon, CA), which has previously only been applied to N 2 measurements in cone–plate shearing flows (Baek and Magda 2003). The primary objective of the current work is to evaluate the accuracy of the PDPP method for measuring N 1 and N 2. This is important because the cone–plate geometry may be unsuitable for some complex fluids, for example suspensions containing non-Brownian particles comparable in size to the rheometer gap. As far as we know, the PDPP method has only been attempted once before, by Adams and Lodge (1964) in the era before the significance of hole-pressure errors (Broadbent et al. 1968) was fully appreciated. A secondary objective of this work is to evaluate the accuracy of the TFCPPP method (Ginn and Metzner 1969) by comparing its results for N 1 and N 2 vs. shear rate to those obtained by the cone–plate and parallel-plate pressure distribution methods. The TFCPPP method has the advantage that it does not require any special rheometer attachments such as the NSS, but it does require use of two sample geometries.Footnote 1 Furthermore, as pointed out by Ohl and Gleissle (1992), when β is small, then N 2 must be calculated in the TFCPPP method by finding the small difference between two large experimental quantities, each of which is measured with a different sample geometry. Hence, small measurement errors are amplified by the evaluation procedure. Walters (1983) found the TFCPPP method to be of low accuracy for measurement of N 2 when comparing to results measured by the cone–plate pressure distribution method. However, Keentok and Tanner (1982) and Brown et al. (1995) obtained good agreement between results obtained by the TFCPPP method and results obtained by the tilted-trough method and a shear-optical method, respectively.

Equations for rheometry calculations

Normal stress differences N 1 and N 2 in torsional shear flow

  1. 1.

    The TFCPPP method

In this method, the normal stress differences N 1 and N 2 are determined by measuring the total vertical thrust F in two shearing flow geometries: cone and plate and parallel plate. In steady cone-and-plate shear flow, N 1 can be calculated from the total axial force or the thrust, F CP, as follows (Bird et al. 1987):

$$ \label{eq1} N_1 =\frac{2F_{\mathrm{CP}} }{\pi R^2} $$
(1)

where R is the rheometer plate radius. In cone–plate steady shear flow experiments, small cone angles are used and therefore the flow is considered to be homogeneous (i.e., the shear rate is constant between the center and the rim of the plate). However, small cone angles may only be suitable for steady-state measurements, due to potential problems with axial compliance during transient tests.

In the parallel-plate steady shear flow experiments, the relationship between the total thrust F PP and the difference between the first and second normal stresses is given as (Kotaka et al. 1959; Bird et al. 1987):

$$ \label{eq2} N_1 -N_2 =\frac{F_{\mathrm{PP}} }{\pi R^2}\left[ {2+\frac{d\ln \left( {\frac{F_{\mathrm{PP}} }{\pi R^2}} \right)}{d\ln \dot {\gamma }}} \right] $$
(2)

where \(\dot {\gamma }\) is the shear rate at the rim, which is adjusted so that it equals that of the cone–plate experiment on the same viscoelastic material. In Eq. 2, N 1 and N 2 are shear rate dependent and are evaluated at the rim shear rate. Hence, by performing vertical thrust measurements on the same viscoelastic fluid in both geometries, Eqs. 1 and 2 can be used in principle to calculate both N 1 and N 2. However, Eq. 2 contains a Rabbinowitch-type shear rate correction, the second term within the brackets, which must be evaluated by measuring F PP at various rim shear rates and differentiating the results. This is a potential source of error. In addition, the measured normal thrust can be significantly affected by the curvature of the free surface (Schweizer and Stockli 2008; Venerus 2007; Shipman et al. 1991), by sample edge instabilities (Sui and McKenna 2007b; Lee et al. 1992), or by transducer axial compliance (Dutcher and Venerus 2008; Schweizer and Bardow 2006; Zapas et al. 1989; Niemiec et al. 1996). The curvature of the free surface and the onset of edge instabilities may differ between cone–plate and parallel-plate measurements on the same viscoelastic material. Furthermore, the effect of axial compliance on the apparent rheological properties may be greater with cone–plate measurements than with parallel-plate measurements because the gap is less. Hence, there are multiple potential sources of error associated with the use of two sample geometries in the TFCPPP method.

  1. 2.

    PDCP method

In a cone-and-plate shearing flow, there exists a relationship between the vertical diagonal component of the total stress tensor (Π22) and the radial position r which is given as follows (Bird et al. 1987):

$$ \label{eq3} -\Pi _{22} -P_0 =-\big(N_1 +2N_2 \big)\ln \left( {\frac{r}{R}} \right)-N_2 $$
(3)

where P 0 is the atmospheric pressure and r is the radial position. Here and throughout this paper, we employ the sign convention that Πii is positive for a tensile stress. N 1 and N 2 can be obtained from plotting − Π22 − P 0 vs. ln(r/R) which should be a linear plot. Therefore, the slope and the intercept at the rim (i.e., r = R) give the values of (N 1 + 2N 2) and N 2, respectively (Miller and Christiansen 1972; Gao et al. 1981; Magda et al. 1991a, b; Baek and Magda 2003). Equation 3 is derived assuming that the cone-and-plate flow is viscometric and of uniform shear rate, that the sample–air interface is spherical, and that there is no jump in the value of the total stress component Π33 at the sample–air interface (Bird et al. 1987). If the free surface is not spherical, this can significantly affect the normal stresses components near the rim (Venerus 2007). However, when this occurs, it should be obvious because the measured pressure profile will likely depart from that predicted by Eq. 3, with a change in slope observed near the rheometer rim. This method has widely been used to measure N 1 and N 2 for different fluids such as a standard viscoelastic fluid (SRM-1490) from the National Institute of Standards and Technology (NIST; Baek and Magda 2003), liquid crystal polymers (Magda et al. 1991b), colloidal suspensions (Lee et al. 2006), and polymer solutions (Magda et al. 1993).

  1. 3.

    PDPP method

In parallel-plate shearing flow, the radial pressure profile (i.e., Π22 vs. r) was derived assuming a viscometric flow field in which the shear rate \(\dot {\gamma }\) increases linearly with radial position r (Adams and Lodge 1964):

$$ \label{eq4} r\left( {\frac{d\Pi _{22} }{dr}} \right)=\frac{d\Pi _{22} }{d\ln r}=N_1 +N_2 +\dot {\gamma }\frac{dN_2 }{d\dot {\gamma }} $$
(4)

We note here that for an ideal cone-and-plate flow (Eq. 3), the slope of the radial pressure profile is constant in a logarithmic plot (dΠ22/d(ln r) = constant). However, according to Eq. 4, this is not the case for a parallel-plate shearing flow because of the dependence of normal stresses N 1 and N 2 on shear rate \(\dot {\gamma }\), which itself depends on the radial position r. Hence, it is difficult to determine N 1 and N 2 from the measured pressure profile in parallel-plate shearing experiments. However, the gap in parallel-plate fixture is independent of r, which may be advantageous for some fluids compared to that in cone-and-plate shear flow experiments where the rheometer gap varies with the radial position r. Bird et al. (1987) proposed the following relationships for determining N 1 and N 2 from the pressure profile in parallel-plate shearing flow

$$ \label{eq5} -N_2 \left( {\dot {\gamma }} \right)=-\Pi _{22} -P_0 \;\;\;\;\;\mbox{at}\;\;r=R $$
(5a)
$$ \label{eq6} N_1 \left( {\dot {\gamma }} \right)+N_2 \left( {\dot {\gamma }} \right)=-\frac{d\Pi _{22} \left( 0 \right)}{d\ln \dot {\gamma }_R } $$
(5b)

where Π22(0) and Π22(R) are the pressures measured at the center and rim of the plate, respectively, and \(\dot {\gamma }_R\) is the rim shear rate. Therefore, measurements of the pressure distribution in parallel-plate shearing flow can be used to determine N 1 and N 2 despite the fact that the flow is inhomogeneous. However, numerical differentiation of data is required, in contrast to the cone–plate method. This is discussed below in the “Results” section.

Experiments

Material

The material used in this work is a reference polymer fluid, SRM-1490, containing 10 wt.% of polyisobutylene (Exxon Vistanex L-100) in n-hexadecane that was produced by NIST, Gaithersburg, MD, USA. The NIST-certified values of the steady shear properties at 25°C, if extrapolated to zero shear rate, yield zero-shear-rate-limiting values of about 45 Pa s and 42 Pa s2, for the viscosity η and the first normal stress difference coefficient Ψ 1, respectively. This gives a value for the sample mean relaxation time of 0.5 s, estimated as Ψ 1/(2η).

Rheological equipment

The cone-and-plate and parallel-plate rheology measurements were performed at Texas Tech University in a controlled-strain Advanced Rheometrics Expansion System (ARES) rheometer (TA Instruments) with temperature control at 25°C. This rheometer has been modified in a previous study in order to reduce transducer compliance errors (Zapas et al. 1989). The torque and normal force rebalance transducers from the rheometer manufacturer (FRT2000) were replaced with a torque and normal force transducer that uses semiconductor strain gages as the transducing elements (Hutcheson et al. 2004). The semiconductor strain gage transducer was custom-manufactured for the rheometer by Sensotec, Inc. (now a division of Honeywell, Inc.). After this modification, the axial compliance of the rheometer was measured to be 0.7 μm/N, and the torsional compliance was measured to be 0.007 rad/Nm. Using these compliance values, the axial and torsional response times are calculated to be 0.7 and 0.3 ms, respectively (Vrentas and Graessley 1981), and the maximum value of the compliance-induced change in rheometer gap is calculated to be 0.3 μm. In addition, the time required for N 1 to reach steady state upon startup of cone–plate shear flow was observed to be approximately 0.5 s (Huang 2007), which is similar to the mean relaxation time of the NIST fluid. This result also indicates that the rheometer axial compliance time is small. The semiconductor strain gage transducers (5,000 g full-scale) were used to measure the total force and total torque in cone–plate and parallel-plate experiments, and these quantities were measured simultaneously with the measurements of the radial pressure distribution made with the pressure sensor plate (see below). The radius of the plate fixture used was R = 1.25 cm, and the gap was varied between 0.8 and 1.1 mm in the parallel-plate tests. The parallel-plate measurements were made on the ARES rheometer in shear rate sweep experiments approximately 3 min after loading the sample into the rheometer. First, the sample was sheared in the clockwise (CW) direction with the rim shear rate increasing from 10 to 100 s\(^{-^1}\) in increments of 10 s\(^{-^1}\), with a shearing time of 60 s at each value of the shear rate. After completing the clockwise shear rate sweep test, a counterclockwise (CCW) shear rate sweep test was performed. The difference between the N 1 and N 2 values calculated using data from clockwise and counterclockwise experiments at each shear rate is used in the following section to estimate the reproducibility of each method. The cone-and-plate measurements were also performed in shear sweep experiments, but in this case the shear rate values were 1, 5, 10, 15, 20, 40, 50, 80, 100, and 120 s\(^{-^1}\). Compliance effects may be more significant with the cone–plate measurements, given that the effective gap is less than in the parallel-plate measurements. The cone radius was 1.25 cm with a cone angle of 0.1 rad. A novel rheometer plate, the NSS (RheoSense, Inc., San Ramon, CA, USA), was used to measure the radial pressure distribution in both parallel-plate and cone–plate shearing flows. The NSS rheometer plate has a radius of 1.25 cm and contains eight miniature pressure sensors, four on each side of the plate at locations symmetric about the plate center point (Baek and Magda 2003).

Results

  1. 1.

    PDCP method

Figure 1 shows the pressure profiles of the NIST fluid in cone-and-plate steady shearing flow for different shear rates between 1 and 120 s\(^{-^1}\). Each time-averaged value of the local pressure is an average of the signals obtained from two pressure transducers located on opposite sides of the rheometer plate but with the same radial coordinate, with error bars in Fig. 1 equated to the difference in values obtained from experiments with CW and CCW rotation. This figure confirms the existence of a linear relationship between the local pressure and ln(r/R), which is in agreement with Eq. 3. This linearity makes it easy to obtain N 2 by extrapolation of the local pressure to r = R. In accordance with Eq. 3, the slope observed from each curve in Fig. 1 can be used to obtain N 1. The values of N 1, N 2, and β = − N 2/N 1 ratio so obtained are plotted against shear rate in Fig. 2a, with error bars equated to the difference in values obtained from experiments with CW and CCW rotation. The corresponding values of the normal stress difference coefficients Ψ 1 and Ψ 2 are plotted in a logarithmic plot against shear rate in Fig. 2b, and the power law regime is evident. It can be seen in Fig. 2a that the values of N 1 and − N 2 increase with increasing shear rate. However, the ratio − N 2/N 1 is approximately constant at 0.1 with a slight degree of shear thinning as observed elsewhere (Magda et al. 1993; Sui and McKenna 2007a). The values of N 1 and N 2 shown in Fig. 2 are in reasonable agreement with those reported previously for a different batch of the same NIST standard fluid measured via the PDCP method (Baek and Magda 2003).

Fig. 1
figure 1

The local “pressure” ( − Π22) of the NIST fluid as a function of the logarithm of radial position at various shear rates as measured in cone-and-plate shearing flow experiments

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Fig. 2
figure 2

a The normal stress differences N 1, N 2, and the ratio − N2/N1 for the NIST fluid as a function of shear rate in cone-and-plate experiments measured using the pressure sensors (PDCP method), with error bars equal to the difference in values obtained from experiments with CW and CCW rotation. The values of N 1 and N 2 were determined from Fig. 1 using Eq. 3 in the text. b Logarithm of the normal stress differences coefficients ψ 1 and ψ 2, and the ratio β = − ψ 2/ψ 1 for the NIST fluid as a function of logarithm of shear rate in cone–plate experiments, as measured using the pressure sensors (PDCP method)

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  1. 2.

    PDPP method

The radial pressure profiles − Π22 − P 0 vs. ln(r) for the NIST fluid in parallel-plate steady shearing flows are shown in Fig. 3a. The pressure distribution was measured in a shear sweep experiment at 11 shear rates between 10 and 100 s − 1, but, for clarity purposes, results are shown for only four shear rates. The same averaging procedure was used as discussed previously for the cone–plate pressure profiles. It is observed in Fig. 3a that the measured values of − Π22 − P 0 are positive (i.e., Π22 is compressive) and decrease with increasing radial position. This is similar to what we observed in cone-and-plate shearing flow experiments (Fig. 1). However, the pressure profile in the parallel-plate shearing flow is concave down in a logarithmic plot (Fig. 3a), which is consistent with Eq. 4. Equation 4 predicts that at a given value of r, the slope in Fig. 3a (− dΠ22/dln r) equals \(-N_1 -N_2 -\mbox{ }\dot {\gamma }d\mbox{ }{N_2 } \mathord{\left/ {\vphantom {{N_2 } {d\dot {\gamma }}}} \right. \kern-\nulldelimiterspace} {d\dot {\gamma }}=N_1 \left( {-1+\beta } \right)-\dot {\gamma }d\mbox{ }{N_2 } \mathord{\left/ {\vphantom {{N_2 } d}} \right. \kern-\nulldelimiterspace} d\dot {\gamma }\). Since the magnitude of N 2 is only about one tenth of N 1, it may be reasonable to drop the last term in Eq. 4, in which case the slope is predicted to be negative as observed. Furthermore, the slope should become more negative as the local shear rate and therefore N 1 increases upon approaching the rheometer rim, as is also observed. The pressure profiles of Fig. 3a are somewhat similar to those observed previously in parallel-plate flow of a polymer solution by Adams and Lodge (1964). However, in contrast to the results of Adams and Lodge, the pressure profiles in Fig. 3a appear to extrapolate to a positive value as r → R, which implies a negative value for N 2 (as expected) at the rim shear rate (see Eq. 5a). According to Eq. 5a, N 2 at each shear rate can be calculated by extrapolating the pressure profile to its value at the rim of the plate. This was done by linear extrapolation to r = R on a plot of Π22 vs. ln(r) as shown in Fig. 3a. In order to also obtain N 1, according to Eq. 5b, the pressure value needs to be extrapolated to the center of the plate and then numerically differentiated with respect to the logarithm of the rim shear rate. As far as we know, such an extrapolation has not been attempted before, and the correct functional form to use for the parallel-plate pressure profile is not known. Therefore, for simplicity, we extrapolate Π22 to r = 0 in a linear plot as shown in Fig. 3b. These extrapolated values of Π22 at r = 0 were plotted against logarithm of rim shear rate. The derivative appearing in Eq. 5b was evaluated by fitting the data in this plot to a piecewise linear curve obtained by connecting successive data points with straight segments. Figure 4a contains N 1 values thus calculated using Eqs. 5a and 5b for the NIST fluid in the parallel-plate geometry as a function of shear rate, with error bars equated to the difference in values obtained from experiments with clockwise and counterclockwise rotation. The N 2 values obtained from the same calculations are shown in Fig. 4b. Figure 5 contains a comparison of the N 1 values measured by the PDCP method and the PDPP method. The agreement is excellent, despite the crude procedure used to extrapolate the parallel-plate pressure profile. Also shown for comparison are N 1 values obtained from the total vertical force measured using the semiconductor strain gage transducer in the cone–plate geometry (TFCP method) and the certified N 1 values given by NIST. The N 1 values calculated from the pressure profile, whether cone and plate or parallel plate, are 10–15% below the NIST-certified values for shear rates 40 s\(^{-^1}\) and above. This probably cannot be attributed to a sample flow instability such as shear banding because the TFCP values measured simultaneously are in good agreement with the NIST-certified values. Another possible explanation is that the calibration curve used for the pressure sensors was systematically low by about 10–15% at high pressure values.

Fig. 3
figure 3

a The local pressure ( − Π22) of the NIST fluid as a function of the logarithm of radial position at various shear rates as measured in parallel-plate shearing flow experiments. Also shown in the figure is the extrapolation of the pressure to the rheometer rim used to determine the second normal stress difference N 2 using Eq. 5a in the text. b The pressure profiles of Fig. 3a are replotted here in a linear scale. Also shown in the figure is the extrapolation of the pressure to the center of the plate used to determine the first normal stress difference N 1 using Eq. 5b in the text

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Fig. 4
figure 4

a The first normal stress differences N 1 for the NIST fluid as a function of shear rate as determined from the measured pressure profile in parallel-plate shearing flow (PDPP method). The error bars shown are equal to the difference in values obtained from experiments with CW and CCW rotation. b The second normal stress differences N 2 for the NIST fluid as a function of shear rate as determined from the measured pressure profile in parallel-plate shearing flow (PDPP method). The error bars shown are equal to the difference in values obtained from experiments with CW and CCW rotation

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Fig. 5
figure 5

The first normal stress differences N 1 for the NIST fluid as a function of shear rate as determined from the measured pressure profile in cone-and-plate (PDCP method) and in parallel-plate (PDPP method) shearing flow with the NSS. Also shown in the figure are the N 1 values obtained from the total vertical force measured using the strain gage instead of the NSS in cone–plate flow (TFCP method) and the certified values of N 1 from NIST. The error bars shown for the TFCP method results are equal to the difference in values obtained from experiments with CW and CCW rotation

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  1. 3.

    TFCPPP method

In these experiments, the torque and normal force transducer with semiconductor strain gage transducing elements (Hutcheson et al. 2004) was used to measure the torque and the total vertical thrust exerted by the NIST fluid in both cone–plate and parallel-plate steady shearing flows. Figure 6a shows the normalized total vertical force (2F CP/πR 2) measured by the strain gage as a function of \(\dot {\gamma }\), where \(\dot {\gamma }\) is the uniform shear rate for cone-and-plate flow. According to Eq. 1, this quantity is equal to N 1, and these N 1 results have already been shown to be in good agreement with the N 1 values supplied by NIST at shear rates up to 70 s − 1 in Fig. 5. Also plotted against \(\dot {\gamma }\) in Fig. 6a is the quantity \(F_{\mathrm{PP}} {\left[ {2+d\ln {\left( {{F_{\mathrm{PP}} } \mathord{\left/ {\vphantom {{F_{\mathrm{PP}} } {\pi R^2}}} \right. \kern-\nulldelimiterspace} {\pi R^2}} \right)} \mathord{\left/ {\vphantom {{\left( {{F_{\mathrm{PP}} } \mathord{\left/ {\vphantom {{F_{\mathrm{PP}} } {\pi R^2}}} \right. \kern-\nulldelimiterspace} {\pi R^2}} \right)} {d\ln \dot {\gamma }}}} \right. \kern-\nulldelimiterspace} {d\ln \dot {\gamma }}} \right]} \mathord{\left/ {\vphantom {{\left[ {2+d\ln {\left( {{F_{\mathrm{PP}} } \mathord{\left/ {\vphantom {{F_{\mathrm{PP}} } {\pi R^2}}} \right. \kern-\nulldelimiterspace} {\pi R^2}} \right)} \mathord{\left/ {\vphantom {{\left( {{F_{\mathrm{PP}} } \mathord{\left/ {\vphantom {{F_{\mathrm{PP}} } {\pi R^2}}} \right. \kern-\nulldelimiterspace} {\pi R^2}} \right)} {d\ln \dot {\gamma }}}} \right. \kern-\nulldelimiterspace} {d\ln \dot {\gamma }}} \right]} {\left( {\pi R^2} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\pi R^2} \right)}\), as measured by the strain gage in parallel-plate shear flow, where \(\dot {\gamma }\) in this case is the rim shear rate value. According to Eq. 2, this quantity is equal to N 1 − N 2. The second term within the brackets in Eq. 2 is a shear rate correction that must be determined by numerically differentiating total vertical force data with respect to rim shear rate. This term was evaluated by fitting the total force data to a nonlinear sigmoidal function of rim shear rate (Soskey and Winter 1984; Brown et al. 1995), as shown in Fig. 6b. This shear rate correction is significant, increasing the apparent value of N 1 − N 2 by over 30%. According to Eqs. 1 and 2, the vertical distance between the two curves in Fig. 6a, \({2F_{\mathrm{CP}} } \mathord{\left/ {\vphantom {{2F_{\mbox{CP}} } {\left( {\pi R^2} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\pi R^2} \right)}-F_{\mathrm{PP}} {\left[ {2+d\ln {\left( {{F_{\mathrm{PP}} } \mathord{\left/ {\vphantom {{F_{\mathrm{PP}} } {\pi R^2}}} \right. \kern-\nulldelimiterspace} {\pi R^2}} \right)} \mathord{\left/ {\vphantom {{\left( {{F_{\mathrm{PP}} } \mathord{\left/ {\vphantom {{F_{\mathrm{PP}} } {\pi R^2}}} \right. \kern-\nulldelimiterspace} {\pi R^2}} \right)} {d\ln \dot {\gamma }}}} \right. \kern-\nulldelimiterspace} {d\ln \dot {\gamma }}} \right]} \mathord{\left/ {\vphantom {{\left[ {2+d\ln {\left( {{F_{\mathrm{PP}} } \mathord{\left/ {\vphantom {{F_{\mathrm{PP}} } {\pi R^2}}} \right. \kern-\nulldelimiterspace} {\pi R^2}} \right)} \mathord{\left/ {\vphantom {{\left( {{F_{\mathrm{PP}} } \mathord{\left/ {\vphantom {{F_{\mathrm{PP}} } {\pi R^2}}} \right. \kern-\nulldelimiterspace} {\pi R^2}} \right)} {d\ln \dot {\gamma }}}} \right. \kern-\nulldelimiterspace} {d\ln \dot {\gamma }}} \right]} {\left( {\pi R^2} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\pi R^2} \right)}\), is equal to N 2. However, the curve from the cone–plate experiments consistently lies above the curve from the parallel-plate measurements in Fig. 6a, implying N 2 is positive, in disagreement with previous experiments on this fluid (Baek and Magda 2003) and with molecular theory (Bird et al. 1987).

Fig. 6
figure 6

a The total normalized axial force (2F CP/πR 2) vs. shear rate as measured for the NIST fluid in cone–plate flow (TFCP method) and the total normalized axial force \(\left( {F_{\mathrm{PP}} {\left[ {2+d\ln {\left( {{F_{\mathrm{PP}} } \mathord{\left/ {\vphantom {{F_{\mathrm{PP}} } {\pi R^2}}} \right. \kern-\nulldelimiterspace} {\pi R^2}} \right)} \mathord{\left/ {\vphantom {{\left( {{F_{\mathrm{PP}} } \mathord{\left/ {\vphantom {{F_{\mathrm{PP}} } {\pi R^2}}} \right. \kern-\nulldelimiterspace} {\pi R^2}} \right)} {d\ln }}} \right. \kern-\nulldelimiterspace} {d\ln }\dot {\gamma }} \right]} \mathord{\left/ {\vphantom {{\left[ {2+d\ln {\left( {{F_{\mathrm{PP}} } \mathord{\left/ {\vphantom {{F_{\mathrm{PP}} } {\pi R^2}}} \right. \kern-\nulldelimiterspace} {\pi R^2}} \right)} \mathord{\left/ {\vphantom {{\left( {{F_{\mathrm{PP}} } \mathord{\left/ {\vphantom {{F_{\mathrm{PP}} } {\pi R^2}}} \right. \kern-\nulldelimiterspace} {\pi R^2}} \right)} {d\ln }}} \right. \kern-\nulldelimiterspace} {d\ln }\dot {\gamma }} \right]} {\left( {\pi R^2} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {\pi R^2} \right)}} \right.)\) vs. shear rate measured for the NIST fluid in parallel-plate flow (TFPP method), both quantities obtained with the strain gage transducer. As discussed in the text, the first quantity should equal N 1, and the second quantity should equal N 1 − N 2. The error bars shown are equal to the difference in values obtained from experiments with CW and CCW rotation. b The total axial force (F PP/πR 2) measured by the strain gauge transducer for the NIST fluid as a function of rim shear rate in the parallel-plate shearing flow experiment (TFPP method). The solid line shows the best fit to the sigmoidal function used by Soskey and Winter (1984) and Brown et al. (1995) to obtain a Rabbinowitch-type correction factor which is the second term within the brackets of Eq. 2 in the text

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Discussion

Comparison between steady shear normal stress differences obtained from the three methods

We show in Fig. 7 a comparison between β = − N 2/N 1 vs. shear rate for the NIST fluid as measured using the three methods discussed in this paper. The results shown in Fig. 7 confirm the existence of negative values of N 2 obtained from the measured radial pressure profile in both cone-and-plate and parallel-plate shearing flows, where the range of − N 2/N 1 values is between 0.1 and 0.2. As observed in Fig. 7, our PDCP and PDPP results are consistent with those reported for another batch of the same reference poly(isobutylene) solution in previous work using the PDCP method (Baek and Magda 2003). The values of N 1 obtained using the PDCP method by Baek and Magda (2003) were in excellent agreement with the certified values from NIST, unlike the N 1 values reported here (Fig. 5), which are 10–15% low at higher shear rates, possibly due to a systematic error in pressure sensor calibration. If one assumes that the measured values of the local pressure were 15% low, then one can show using Eq. 3 that this would have had no effect on the apparent value of β. The values of N 1 reported here for the TFCPPP method are in good agreement with the NIST-certified values up to the maximum shear rate at which the certified values are available, 70 s − 1. Nonetheless, the values of N 2 measured by the TFCPPP method are positive, in strong disagreement with theory, with results from the PDPP and PDCP methods, and with previous experimental work on this material. Hence, the TFCPPP method does not give reliable results for the N 2 value of the NIST fluid in our experiments. For example, for a shear rate of 100 s − 1, using the TFCPPP method, we obtained N 2 = + 80 Pa for the clockwise rotation direction and N 2 = + 265 Pa for the counterclockwise rotation direction. One possible source of error for the TFCPPP experiments was the use of a strain gage of insufficient sensitivity to measure the total normal force of the moderately elastic NIST fluid. Using the load cell stability value quoted by the manufacturer (5 g), one calculates the resolution of our total force cone–plate N 1 measurements to be 200 Pa. This is probably a conservative estimate because the agreement between our strain gage N 1 measurements and the NIST-certified values in Fig. 5 is much better than this, with a standard deviation of only 30 Pa for shear rates below 15 s − 1. Since this standard deviation is smaller than the observed error in the N 2 measurement, which is approximately 300 Pa at 100 s − 1 based on comparison with the results of Baek and Magda (2003), the failure of the TFCPPP method cannot primarily be due to resolution limitations of the strain gage transducer. In our opinion, a more significant source of error with the TFCPPP method is the necessary use of both cone–plate and parallel-plate measuring geometries, with axial compliance errors associated with the cone–plate measurements even at steady state (Dutcher and Venerus 2008). The effect of axial compliance may be greater on the cone–plate measurements than on the parallel-plate measurements because the gap is less. This may explain why the error bars are much larger for the cone–plate results than for the parallel-plate results in Fig. 6a, indicating a larger discrepancy between the CW and CCW total thrust measurements. This discrepancy is responsible for the large error bars given for TFCPPP β values in Fig. 7. This being the case, the strain potential total force method (Sui and McKenna 2007a) is probably a better alternative than the TFCPPP method for measurements of β since the former method does not require two sample geometries. In the strain potential total force method, isochronal derivatives of the strain potential are calculated from the torque and total normal force response observed in step-strain tests and used to calculate β via the scaling laws for elastic materials proposed by Penn and Kearsley (1976).

Fig. 7
figure 7

The ratio − N 2/N 1 for the NIST fluid at different shear rates as measured using the three different methods discussed in this paper: pressure distribution cone–plate (PDCP) method, pressure distribution parallel-plate (PDPP) method, and total force cone–plate parallel-plate (TFCPPP) method. The dashed line summarizes the previous results of Baek and Magda (2003)

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Conclusions

In this paper, we report what we believe is the first successful application of the PDPP method (Adams and Lodge 1964) to measurement of both normal stress differences of a viscoelastic fluid. A novel pressure sensor plate was attached to an ARES rheometer and used to measure the radial dependence of the pressure (− Π22) for a standard viscoelastic fluid (SRM-1490) in parallel-plate shearing flow experiments. The value of Π22(0) was obtained by extrapolating Π22 to r = 0 assuming linear dependence on radial coordinate r; the value of Π22(R) was obtained by extrapolating Π22 to r = R assuming linear dependence on ln(r), and Π22(0) was differentiated with respect to the rim shear rate. Despite the potential numerical errors, the values of N 1 and the values of β = − N 2/N 2 ≈ 0.1 so obtained were in good agreement with the results of the PDCP method applied to the same sample, and the β values at the least were in good agreement with previously published results (Baek and Magda 2003). The TFCPPP method for measurement of both normal stress differences (Ginn and Metzner 1969) was also evaluated by measuring the total vertical force during shearing flow of the same fluid with a custom-made semiconductor strain gage transducer. Comparison was made between the values of N 1 and N 2 determined by the three different methods (PDPP method, PDCP method, and TFCPPP method) in order to provide insight into relative accuracy. The TFCPPP method gave results for N 1 vs. shear rate that showed the best agreement with NIST-certified values. However, the TFCPPP method yielded large positive values for N 2 at many shear rates, in strong disagreement with the other two methods and with most molecular theories. The anomalous N 2 results obtained with the TFCPPP method probably arose from errors associated with sample–sample or test–test variability that comes from trying to compare cone-and-plate total thrust measurements with parallel-plate total thrust measurements and from axial compliance-induced errors associated with the cone–plate measurements.